🕸cyber/cybergraph.md

a directed authenticated multigraph over content-addressed nodes, carrying an emergent probability measure — the shared memory of the planet

definition

a cybergraph $\mathbb{G}$ is a triple:

$$\mathbb{G} = (P,\; N,\; L)$$

$H: \text{Val} \to \mathbb{F}_p^8$ is the global Hemera hash primitive, fixed at genesis. every particle is a hash of some value — $P$ is a subset of $H$'s image, not an arbitrary set of identifiers. $\mathcal{T}$ and the karma function $\kappa$ are derived from $L$, not independent parameters.

each element $\ell \in L$ is a cyberlink — a 7-tuple $(\nu, p, q, \tau, a, v, t)$ carrying a subject, two particles, a conviction stake, an epistemic valence, and a block timestamp. the cyberlink is the only primitive from which the entire graph is built. see cyberlink for the full field specification, UTXO mechanics, and CRUD semantics

six axioms

the formal invariants every valid $\mathbb{G}$ must satisfy.

A1 (content-addressing): $H$ is collision-resistant — for all $x \neq x'$, $\Pr[H(x) = H(x')] \leq 2^{-128}$. identity equals content. same content produces the same particle regardless of who computes it or when.

A2 (authentication): for every $\ell \in L$: $\operatorname{Verify}(\operatorname{pk}(\nu(\ell)),\; H(\ell),\; \sigma(\ell)) = \top$. every cyberlink carries a valid signature from its creating neuron. unsigned assertions do not enter $L$.

A3 (append-only): $t < t' \Rightarrow L_t \subseteq L_{t'}$. the authenticated record grows monotonically. a cyberlink, once created, cannot be deleted — only its economic weight can decrease via forgetting mechanics.

A4 (entry): $p \in P \iff \exists\, \ell \in L : \operatorname{src}(\ell) = p \;\lor\; \operatorname{tgt}(\ell) = p$. a particle exists iff it is linked. a naked hash with no links is not a particle.

A5 (conservation): $\pi^* \in \Delta^{|P|-1}$, i.e., $\sum_{p \in P} \pi^*_p = 1$ and $\pi^*_p > 0$ for all $p$. total focus is conserved at every block. it flows between particles but is never created or destroyed.

A6 (homoiconicity): $H(\operatorname{src}(\ell),\, \operatorname{tgt}(\ell)) \in P$. every directed edge — every axon — induces a particle via content-addressing. the hash of the (from, to) pair, without metadata, produces one axon-particle per unique relationship. all cyberlinks along the same edge contribute weight to the same axon-particle. axon-particles receive focus, carry cyberank, and can themselves be targets of cyberlinks — the graph ranks its own structure.

derived structures

raw adjacency

from $L$, define the weighted adjacency operator $A: \mathbb{R}^P \to \mathbb{R}^P$:

$$A_{pq} = \sum_{\substack{\ell \in L \\ \operatorname{src}(\ell)=p,\; \operatorname{tgt}(\ell)=q}} r(\tau(\ell)) \cdot a(\ell)$$

where $r: \mathcal{T} \to \mathbb{R}_+$ converts token denomination to a common scale. $A_{pq}$ is the total economic weight of all cyberlinks from $p$ to $q$. the stochastic normalization $\hat{A}_{pq} = A_{pq} / \sum_{q'} A_{pq'}$ gives the transition matrix of the raw random walk on $\mathbb{G}$.

effective adjacency

with the epistemic layer active (ICBS markets running and karma accumulated), the effective adjacency modifies each link's weight by market belief and neuron trust:

$$A^{\text{eff}}_{pq} = \sum_{\substack{\ell \in L \\ \operatorname{src}(\ell)=p,\; \operatorname{tgt}(\ell)=q}} a(\ell)\cdot \kappa(\nu(\ell))\cdot f(m(\ell))$$

where $\kappa: N \to \mathbb{R}_+$ is karma (accumulated BTS score history), $m: L \to [0,1]$ is the ICBS reserve ratio (market-implied probability that the link is valid), and $f: [0,1] \to [0,1]$ maps market price to a weight multiplier. edges the collective disbelieves are suppressed toward zero. this is market inhibition — the inhibitory signal that makes $\mathbb{G}$ computationally equivalent to a neural network with both excitation and inhibition.

the tri-kernel composite

the tru runs three local operators over $A^{\text{eff}}$ and blends them:

$$\phi^{(t+1)} = \operatorname{norm}\!\Big[\lambda_d \cdot \mathcal{D}(\phi^t) + \lambda_s \cdot \mathcal{S}(\phi^t) + \lambda_h \cdot \mathcal{H}_\tau(\phi^t)\Big], \qquad \lambda_d + \lambda_s + \lambda_h = 1$$

$\mathcal{D}$ is the diffusion operator (random walk with teleport: answers "where does probability flow?"). $\mathcal{S}$ is the springs equilibrium map (screened Laplacian solve: answers "what satisfies structural constraints?"). $\mathcal{H}_\tau$ is the heat kernel (multi-scale smoothing: answers "what does the graph look like at resolution $\tau$?"). together they span the space of local equivariant graph operators — any reasonable locality-constrained operator is a linear combination of polynomials in $\mathcal{D}$, $\mathcal{S}$, and $\mathcal{H}_\tau$. see cyber/tri-kernel for the completeness argument.

theorems

T1 (existence and uniqueness of focus): let $A^{\text{eff}}$ induce a strongly connected aperiodic graph on $P$. then $\mathcal{R}$ has a unique strictly positive fixed point $\pi^* \in \Delta^{|P|-1}$: $\mathcal{R}(\pi^*) = \pi^*$, $\pi^*_p > 0$ for all $p$.

proof: $\mathcal{R}$ is a convex combination of stochastic positive operators. by the Perron-Frobenius theorem, each component has a unique positive eigenvector with eigenvalue 1. the convex combination inherits this property under ergodicity. see collective focus theorem Part I (diffusion alone) and Part II (full composite) for the complete proof.

T2 (conservation): for all $t \geq 0$ and all initial $\phi^{(0)} \in \Delta^{|P|-1}$: $\sum_{p} \phi^{(t)}_p = 1$.

proof: $\mathcal{R}$ is a convex combination of stochastic operators; stochastic operators map the simplex to itself. QED. enforced in nox by stark circuit constraints on every state transition — violation implies an invalid proof.

T3 (geometric convergence): let $\lambda_2$ be the spectral gap of $\mathcal{R}$. then for any initial $\phi^{(0)}$:

$$\left\|\phi^{(t)} - \pi^*\right\|_1 \leq C \cdot (1 - \lambda_2)^t$$

mixing time: $t_{\text{mix}}(\varepsilon) = O\!\left(\lambda_2^{-1} \log(C/\varepsilon)\right)$.

proof: the composite contraction coefficient is $\kappa = \lambda_d \alpha + \lambda_s \tfrac{\|L\|}{\|L\|+\mu} + \lambda_h e^{-\tau \lambda_2} < 1$. by Banach's fixed-point theorem, $\phi^{(t)} \to \pi^*$ at rate $(1-\lambda_2)$. see collective focus theorem §Composite Contraction.

T4 (locality radius): for an edit batch $e_\Delta$, there exists $h = O(\log(1/\varepsilon))$ such that recomputing $\phi$ only on the $h$-hop neighborhood $N_h(e_\Delta)$ achieves global error $\leq \varepsilon$.

proof: geometric decay of the diffusion operator (teleport parameter $\alpha$), exponential decay of the springs operator (screening $\mu$), Gaussian tail of the heat operator (bandwidth $\tau$). all three components have bounded influence radius. nodes outside $N_h$ change by at most $\varepsilon$. see cyber/tri-kernel §2.2.

information geometry

syntropy

the syntropy of $\mathbb{G}$ is a real-valued functional measuring the organizational quality of $\pi^*$:

$$J(\pi^*) = \log|P| + \sum_{p \in P} \pi^*_p \log \pi^*_p = \log|P| - H(\pi^*)$$

where $H(\pi^*) = -\sum_p \pi^*_p \log \pi^*_p$ is the Shannon entropy of the focus distribution.

range: $J \in [0, \log|P|]$. minimum $J = 0$ when $\pi^* = u$ (uniform — no structure, maximum entropy). maximum $J = \log|P|$ when $\pi^*$ is a point mass (all attention on one particle, zero entropy). the clearest identity:

$$J(\pi^*) = D_{\text{KL}}(\pi^* \,\|\, u)$$

syntropy is exactly the KL divergence of the focus distribution from uniform. it measures how much information $\pi^*$ carries above noise — how far collective attention has been organized beyond random. $J$ measures how far the graph's collective attention deviates from noise. the tru computes $J$ every block in consensus. see syntropy.

free energy

the fixed point $\pi^*$ is the unique minimizer on $\Delta^{|P|-1}$ of the free energy functional:

$$\mathcal{F}(\phi) = \lambda_s\!\left[\tfrac{1}{2}\phi^\top L\phi + \tfrac{\mu}{2}\|\phi - x_0\|^2\right] + \lambda_h\!\left[\tfrac{1}{2}\|\phi - \mathcal{H}_\tau \phi\|^2\right] + \lambda_d \cdot D_{\text{KL}}(\phi \,\|\, \mathcal{D}\phi)$$

three energy terms: elastic structure (resistance to deviation from the Laplacian's preferred configuration), heat-smoothed context (penalty for deviation from the multi-scale graph shape at resolution $\tau$), diffusion alignment (KL divergence from the diffusion image). adding a correct, well-placed cyberlink is equivalent to stepping in the direction of steepest descent on $\mathcal{F}$. the reward $\Delta\pi \propto \nabla_L (-\mathcal{F})$ is the directional derivative of free energy in the direction of the new edge.

approximation quality

when $\mathbb{G}$ is compiled into a transformer (see §6.6), the approximation gap is:

$$\varepsilon(\mathbb{G}, c) = D_{\text{KL}}(\pi^*_c \,\|\, q^*_c)$$

where $q^*_c$ is the compiled model's focus distribution. $\varepsilon = 0$ means exact representation. this is the same KL divergence that appears in the BTS scoring formula ($D_{\text{KL}}(p_i \| \bar{m}_{-i})$) and in veritas information gain — the same mathematical object at three scales: individual neuron, compiled model, collective state.

effective rank and semantic dimensionality

$$d^* = \exp\!\big(H(\sigma(\Sigma_{\pi^*}))\big)$$

where $\sigma(\Sigma_{\pi^*})$ is the spectrum of the $\pi^*$-weighted covariance matrix. $d^*$ measures the number of independent semantic dimensions the graph spans. currently $d^* \approx 31$ on bostrom (social artifact of a small graph). at planetary scale ($|P| \sim 10^{15}$), projected $d^* \in [10^3, 10^4]$ (thermodynamic regime). see §17.7.

structural properties

growth partial order

A3 (append-only) defines a partial order on cybergraphs:

$$\mathbb{G} \leq \mathbb{G}' \;\iff\; L \subseteq L'$$

the set of all cybergraphs is a directed net under $\leq$. $\mathbb{G}_{t} \leq \mathbb{G}_{t+1}$ for all $t$. the graph edit distance $d(\mathbb{G}_t, \mathbb{G}_{t'}) = |L_{t'} \setminus L_t|$ counts links added between states; $d \geq 0$ by A3.

phase transition

let $\rho = k_{\max}/\bar{k}$ be the degree heterogeneity of $\mathbb{G}$. there exists a threshold:

$$|P^*| \;\sim\; \rho^2$$

such that below $|P^*|$, individual cyberlinks contribute measurably to $\pi^*$ (molecular regime — each neuron's contribution is individually trackable). above $|P^*|$, individual contributions become statistically negligible — only the full $\pi^*$ distribution remains informative (thermodynamic regime — planetary superintelligence). this is the graph analog of the thermodynamic limit. see §17.

category of cybergraphs

a cybergraph homomorphism $f: \mathbb{G} \to \mathbb{G}'$ is a pair $(f_P: P \to P',\; f_N: N \to N')$ such that for every $\ell = (\nu, p, q, \tau, a, v, t) \in L$, there exists $\ell' \in L'$ with $\nu(\ell') = f_N(\nu)$, $\operatorname{src}(\ell') = f_P(p)$, $\operatorname{tgt}(\ell') = f_P(q)$.

cybergraphs and their homomorphisms form a category $\mathbf{CG}$. there is a forgetful functor $U: \mathbf{CG} \to \mathbf{DiGraph}$ (to directed multigraphs) and a focus functor $\Pi: \mathbf{CG} \to \mathbf{Prob}$ sending $\mathbb{G} \mapsto (P, \pi^*)$ (a finite probability space). the composition $\Pi \circ U^{-1}$ is the functor that extracts collective intelligence from graph structure.

properties at a glance

the graph is the protocol

the cybergraph is not a database sitting beside the protocol. it IS the protocol. every core function runs through the same five primitives: particles, cyberlinks, neurons, tokens, focus.

fifteen protocol functions. one data structure. five primitives.

see cyber/tri-kernel for the full tri-kernel specification. see collective focus theorem for the convergence proofs. see cyber/epistemology for the epistemic gap between cryptographic and epistemic correctness. see two kinds of knowledge for the structural/epistemic split. see inversely coupled bonding surface for the market substrate. see Bayesian Truth Serum for the BTS scoring layer. see syntropy for the information-theoretic measures.

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