Cybergraph

Formal specification of the cybergraph data structure, its axioms, derived operators, and theorems.

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. $\mathcal{T}$ and the karma function $\kappa$ are derived from $L$.

Each element $\ell \in L$ is a cyberlink -- a 5-tuple $(p, q, \tau, a, v)$ carrying two particles, a conviction stake, and an epistemic valence. The signing neuron $\nu$ and block height $t$ are attributes of the containing signal, not of the cyberlink itself. The cyberlink is the only primitive from which the entire graph is built. See cyberlink for the full field specification, box 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

Every signal $s$ carries a valid signature from its creating neuron: $\operatorname{Verify}(\operatorname{pk}(\nu(s)),\; H(s),\; \sigma(s)) = \top$. Authentication is at the signal level — cyberlinks within an invalid signal 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

$\phi^* \in \Delta^{|P|-1}$, i.e., $\sum_{p \in P} \phi^*_p = 1$ and $\phi^*_p > 0$ 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:

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

where $s_\ell$ is the signal containing $\ell$, $\kappa: N \to \mathbb{R}_+$ 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.

Tri-Kernel Composite

The weighted graph $A^{\text{eff}}$ is the input to the tru. Cybergraph produces the weighted graph; tru runs the three local operators (diffusion, springs, heat) over it to compute the focus fixed point $\phi^*$. The boundary is sharp: cybergraph defines the graph and its adjacency, tru defines what the graph computes.

Dynamics live in tru

Everything about the $\phi^*$ fixed point is computed and specified by the tru, not here: existence and uniqueness, conservation, geometric convergence, the locality radius, syntropy, the free-energy variational form, and effective semantic rank. Cybergraph defines the graph; tru defines the dynamics on it.

• tri-kernel — the three operators, the completeness argument, the locality radius
• collective focus theorem — existence, uniqueness, conservation, geometric convergence
• focus-flow — the inference process that computes $\phi^*$
• syntropy — the information-geometric measures (syntropy, free energy, effective rank)
• rewards — how a proven $\Delta\phi^*$ (impulse) becomes a self-minted reward

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 $\phi^*$ (molecular regime -- each neuron's contribution is individually trackable). Above $|P^*|$, individual contributions become statistically negligible -- only the full $\phi^*$ distribution remains informative (thermodynamic regime -- planetary superintelligence). This is the graph analog of the thermodynamic limit.

Properties at a Glance

The Graph Is the Protocol

The cybergraph 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 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.