convergence
the process by which iteration approaches a destination that iteration itself defines. the tri-kernel iterates until focus stabilizes; neurons approach knowledge; the protocol approaches intelligence. convergence is the execution model of tru — what nox is to derivation, tru is to convergence.
a system applies the same operation over and over and arrives somewhere specific — not because anything told it where to go, but because the structure of the operation leaves no alternative. the destination is the attractor, not a logical consequence. $\phi^*$ is not derived from the graph; it emerges from it.
convergence vs derivation
derivation proceeds from axioms to conclusions in bounded depth — every formal system, every program execution, every transformer forward pass reaches only what its starting axioms can produce. convergence proceeds by iteration toward equilibrium. this is why tru sits outside the gödel confinement that binds derivation engines: the fixed point is a limit, not a proof.
| vm | execution model |
|---|---|
| nox | derivation |
| zheng | verification |
| glia | inference |
| tru | convergence |
the guarantee
the tri-kernel composite operator $\mathcal{R}$ is a contraction with coefficient $\kappa < 1$:
$$\|\mathcal{R}\phi - \mathcal{R}\psi\| \le \kappa\,\|\phi - \psi\|, \qquad \kappa = \lambda_d\,\alpha + \lambda_s\,\tfrac{\|L\|}{\|L\|+\mu} + \lambda_h\,e^{-\tau\lambda_2} < 1$$
by the Banach fixed-point theorem, iteration from any start reaches a unique fixed point $\phi^*$ at linear rate. uniqueness is what makes the cybergraph a shared memory rather than a collection of disagreeing views: every validator that iterates arrives at the same $\phi^*$. see collective focus theorem for the full proof.
rate — the spectral gap
how fast convergence happens is set by the spectral gap $\lambda = 1 - |\lambda_2|$, the gap between the largest and second-largest eigenvalues of the transition operator:
$$\|\phi^{(t)} - \phi^*\| \le C\,(1-\lambda)^t, \qquad t_{\mathrm{mix}}(\varepsilon) = O\!\left(\frac{\log(n/\varepsilon)}{\lambda}\right)$$
a larger gap means faster mixing, faster finality in foculus, and a tighter locality radius. the gap also governs the architecture parameters a model compiled from $\phi^*$ inherits (see ct0).
locality
convergence is local: an edit batch affects only an $h = O(\log(1/\varepsilon))$-hop neighborhood. all three operators decay — diffusion geometrically via teleport, springs exponentially via screening, heat by Gaussian tail. a new cyberlink in one corner of a planetary graph does not require recomputing $\phi^*$ everywhere; only its neighborhood updates. this is what makes the impulse $\Delta\phi^*$ a sparse, provable object.
see tri-kernel for the operators and the five-way reading of $\phi^*$ · collective focus theorem for the proofs · focus for the destination · spectral gap for the rate · foculus for consensus timing.
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