the conviction-weighted adjacency matrix that the tri-kernel operates on. raw cyberlinks define the structural graph — which particles connect to which. effective adjacency transforms this binary structure into a weighted matrix where each edge carries the economic signal of staked tokens
$$A_{\text{eff}}(p, q) = \sum_{\ell \in L(p,q)} f(\tau_\ell, a_\ell, v_\ell, \kappa_\ell)$$
the weight function $f$ combines conviction amount $a$, valence $v$, token denomination $\tau$, and the linking neuron's karma $\kappa$. high-karma neurons with large stakes on positively-valenced links produce the strongest edge weights. negative valence ($v = -1$) reduces or inverts the weight, allowing the graph to encode disagreement
the tri-kernel — diffusion, springs, heat — computes focus over this weighted matrix. the effective adjacency is the input; cyberank is the output. changing the weights (by creating, withdrawing, or transferring cyberlinks) shifts the fixed-point distribution $\pi^*$ that determines every particle's rank
this design means the cybergraph is simultaneously a topological object (which nodes connect) and an economic object (how much conviction flows along each edge). the topology is append-only and permanent. the economics are dynamic — conviction can be added, withdrawn, or transferred at every step
the spectral properties of the effective adjacency matrix (eigenvalues, spectral gap) determine convergence speed of the tri-kernel and the conditions under which equilibrium is reached. see perron-frobenius theorem for the mathematical foundation
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