the constraint that every operator must compute from neighbors only
at planetary scale (10¹⁵ nodes), any algorithm requiring global state is physically impossible. locality is the filter that selects which operators can exist in the tri-kernel
why only three operators survive
the tri-kernel architecture begins with all known graph operators and applies one test: can this operator produce a correct answer by reading only the h-hop neighborhood, where h = O(log(1/ε))?
three families pass:
diffusion — geometric decay via teleport parameter α. a random walker forgets its origin exponentially fast. influence beyond O(log(1/ε)) hops falls below ε
springs — exponential decay via screening parameter μ. the Green's function of the screened Laplacian $(L + μI)^{-1}$ decays as $e^{-\sqrt{μ} \cdot d}$ with graph distance d
heat — Gaussian tail decay via temperature τ. the heat kernel $H_τ = \exp(-τL)$ concentrates mass within O(√τ) hops
every other operator family (global spectral methods, all-pairs shortest paths, full matrix inversions) fails the locality test. they require reading the entire graph and cannot scale
consequence
locality means edits are cheap: when a neuron creates a cyberlink, only the local neighborhood needs recomputation. the rest of the cybergraph is unaffected up to error ε. this is what makes collective focus computable in real time on a planetary network
see tri-kernel architecture for the full derivation. see collective focus theorem for the locality radius proof
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