Tri-Kernel Specification
Formal definition of the three local operators whose fixed point is cyberank. The convergence and uniqueness result in §3 is the collective focus theorem. Part of the tru specification.
1. The Three Operators
1.1 Diffusion (Markov)
The transition matrix $P = D^{-1}A$ (or column-stochastic $P = AD^{-1}$) governs probability flow:
$$\phi^{(t+1)} = \alpha P^\top \phi^{(t)} + (1-\alpha)u$$
where $\alpha \in (0,1)$ is the teleport parameter and $u$ is a prior (often uniform or stake-weighted).
Properties: row-stochastic, preserves probability mass, powers remain local. Under ergodicity (strong connectivity + aperiodicity), converges to unique stationary distribution $\phi^*$.
Locality: geometric decay via teleport parameter $\alpha$.
Answers: "Where does probability flow?"
1.2 Springs (Screened Laplacian)
The graph Laplacian $L = D - A$ (or normalized $\mathcal{L} = I - D^{-1/2}AD^{-1/2}$) encodes structural constraints:
$$(L + \mu I)x^* = \mu x_0$$
where $\mu > 0$ is the screening/stiffness parameter and $x_0$ is a reference state.
Properties: positive semi-definite, null space = constant vectors. The screened Green's function $(L+\mu I)^{-1}$ has exponential decay, ensuring locality.
Locality: exponential decay via screening parameter $\mu$.
Answers: "What satisfies structural constraints?"
1.3 Heat Kernel
The heat kernel $H_\tau = \exp(-\tau L)$ provides multi-scale smoothing:
$$\frac{\partial H}{\partial \tau} = -LH, \quad H_0 = I$$
where $\tau \geq 0$ is the temperature/time parameter.
Properties: positivity-preserving, semigroup ($H_{\tau_1}H_{\tau_2} = H_{\tau_1+\tau_2}$). Admits Chebyshev polynomial approximation for locality. High $\tau$ explores (annealing), low $\tau$ commits (crystallization).
Locality: Gaussian tail decay, $h = O(\log(1/\varepsilon))$ hops.
Answers: "What does the graph look like at scale $\tau$?"
2. The Composite Operator
The tri-kernel blends the three primitives into a single update:
$$\phi^{(t+1)} = \operatorname{norm}\big[\lambda_d \cdot D(\phi^t) + \lambda_s \cdot S(\phi^t) + \lambda_h \cdot H_\tau(\phi^t)\big]$$
where $\lambda_d + \lambda_s + \lambda_h = 1$, $D$ is the diffusion step, $S$ is the springs equilibrium map, $H_\tau$ is the heat map, and $\operatorname{norm}(\cdot)$ projects to the simplex.
2.1 The Free Energy Functional
The fixed point of the composite operator minimizes:
$$\mathcal{F}(\phi) = \lambda_s\left[\frac{1}{2}\phi^\top L\phi + \frac{\mu}{2}\|\phi-x_0\|^2\right] + \lambda_h\left[\frac{1}{2}\|\phi-H_\tau\phi\|^2\right] + \lambda_d \cdot D_{\text{KL}}(\phi \| D\phi)$$
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
The Boltzmann equilibrium form:
$$\phi^*_i \propto \exp\big(-\beta\,[E_{\text{spring},i} + \lambda\,E_{\text{diff},i} + \gamma\,C_i]\big)$$
The three energy terms correspond to the three operators: $E_{\text{spring}}$ encodes structural coherence via the screened Laplacian, $E_{\text{diff}}$ encodes flow consistency via diffusion, $C_i$ encodes context pressure via heat kernel weighting.
2.2 Convergence and Locality
Contraction Lemmas
Lemma 1 (Diffusion Contracts): Under ergodicity of $P$ with teleport parameter $\alpha \in (0,1)$, the diffusion map $D$ satisfies $\|D\phi - D\psi\|_1 \leq \alpha \|\phi - \psi\|_1$. The teleport ensures geometric mixing with rate $\alpha$.
Lemma 2 (Springs Contract): Under screening parameter $\mu > 0$, the screened Laplacian solve $S: \phi \mapsto (L + \mu I)^{-1}(\mu x_0)$ satisfies $\|S\phi - S\psi\|_2 \leq \frac{\|L\|}{\|L\| + \mu} \|\phi - \psi\|_2$. The Green's function $(L + \mu I)^{-1}$ decays exponentially with distance -- screening ensures locality and contraction.
Lemma 3 (Heat Contracts): For bounded temperature $\tau > 0$, the heat kernel $H_\tau = \exp(-\tau L)$ satisfies $\|H_\tau \phi - H_\tau \psi\|_2 \leq e^{-\tau \lambda_2} \|\phi - \psi\|_2$ where $\lambda_2$ is the Fiedler eigenvalue. Positivity-preserving and semigroup properties ensure well-defined contraction.
Theorem (Composite Contraction)
Under ergodicity of $P$, screening $\mu > 0$, and bounded $\tau$, the composite operator $\mathcal{R}$ is a contraction:
$$\|\mathcal{R}\phi - \mathcal{R}\psi\| \leq \kappa \|\phi - \psi\|, \quad \kappa = \lambda_d \alpha + \lambda_s \frac{\|L\|}{\|L\|+\mu} + \lambda_h e^{-\tau\lambda_2} < 1$$
Since each component contracts and $\mathcal{R}$ is a convex combination, $\kappa$ is a convex combination of individual contraction coefficients -- each less than 1, hence $\kappa < 1$. By Banach fixed-point theorem, $\phi^t \to \phi^*$ at linear rate.
$\kappa$ does more than guarantee convergence: it fixes the trace length. The operators act on fixed-point vectors over the Goldilocks field -- no float anywhere in the iteration (arithmetic) -- and rather than loop to a data-dependent threshold, tru runs a constant $T(\varepsilon) = \lceil \log(1/\varepsilon) / \log(1/\kappa) \rceil$ steps. The linear rate makes that bound exact, so the iterate is bit-identical across machines and the zheng trace has a compile-time-constant length.
Theorem (Locality Radius)
For edit batch $e_\Delta$, there exists $h = O(\log(1/\varepsilon))$ such that recomputing only on $N_h$ (the $h$-hop neighborhood) achieves global error $\leq \varepsilon$.
This follows from: geometric decay for diffusion (teleport), exponential decay for springs (screening), Gaussian tail for heat (kernel bandwidth).
2.3 Compute-Verify Symmetry
Because all operations are local and memoizable:
$$t_{\text{verify}} / t_{\text{compute}} \to c \approx 1$$
Light clients can verify focus updates by checking boundary flows and authenticated neighborhood commitments, with constant-factor overhead relative to computation.
2.4 One Fixed Point, Five Ways
The fixed point $\phi^*$ is one object that five independent research traditions arrived at separately. They are isomorphic; the tri-kernel makes the isomorphism literal rather than analogical.
| lens | $\phi^*$ is | term in $\mathcal{R}$ | source |
|---|---|---|---|
| PageRank | the stationary distribution of a stake-teleported random walk | diffusion $D$ | Brin & Page 1998 |
| transformer attention | the leading left eigenvector of the row-stochastic attention operator | diffusion $D$ — softmax-attention is row-stochastic | Vaswani et al. 2017 |
| graph neural diffusion | one message-passing step of a continuous GNN — the heat equation as a layer | heat $H_\tau$ | GRAND, Chamberlain et al. 2021 |
| modern Hopfield network | an energy minimum of an associative memory; retrieval is one update step | the Boltzmann form $\phi^*_i \propto e^{-\beta E_i}$ (§2.1) | Ramsauer et al. 2020 |
| active inference | the variational free-energy minimum a self-evidencing system settles into | the functional $\mathcal{F}(\phi)$ (§2.1) | Friston 2010; Smithe 2023 |
Each row is a projection of the same object. Diffusion is PageRank and is the attention eigenvector at once — softmax-attention is exactly a row-stochastic operator, so its leading left eigenvector is the diffusion stationary distribution. Heat is the GNN — GRAND shows message passing is the discretized heat equation, which is $H_\tau$. The Boltzmann equilibrium of §2.1 is the Hopfield energy minimum — modern Hopfield retrieval is a softmax over stored patterns, the same exponential-of-energy form. And the functional $\mathcal{F}$ that $\phi^*$ minimizes is Friston's variational free energy — the quantity an active-inference agent minimizes by self-evidencing.
The convergence proof (§2.2) therefore carries more weight than a numerical guarantee. It is the statement that PageRank, transformer attention, graph neural diffusion, associative memory, and active inference are five names for a single contraction's fixed point. A model compiled from $\phi^*$ (see ct0, focus-flow) inherits all five readings simultaneously: it is a transformer because $\phi^*$ is an attention eigenvector, and it is a graph's equilibrium because $\phi^*$ is a PageRank distribution. The two inference paths of focus-flow are one instance of this five-way identity.
The identity holds for the composite operator $\mathcal{R}$ iterated to a single fixed point — the attractor of the blend. Computing each operator's own fixed point in isolation and averaging the three (the blend of separate attractors) breaks it: the average of five projections is not the object. This is why §2.2 specifies one coupled iteration governed by a single $\kappa$, not three independent solves.
3. Collective Focus Theorem
Part I -- Diffusion Convergence (Special Case)
Consider a cybergraph $G = (V, E, W)$ with $|V| = n$ particles. Each cyberlink $(i, j) \in E$ has weight $w_{ij} \geq 0$. Each particle $j$ has token value $t_j > 0$. Define transition probabilities:
$$p_{ij} = \frac{w_{ij} \cdot t_j}{\sum_{k \in \mathcal{N}(i)} w_{ik} \cdot t_k}$$
Assumptions: $G$ is strongly connected (directed path between any pair) and aperiodic (gcd of all directed cycle lengths is 1).
Claim: there exists a unique stationary distribution $\phi^*$ satisfying $\phi^* P = \phi^*$ with $\sum_i \phi^*_i = 1$.
Proof
Step 1 (Markov Chain): The matrix $P = [p_{ij}]$ is stochastic. Non-negativity: $p_{ij} \geq 0$ since $w_{ij} \geq 0$ and $t_j > 0$.
Step 2 (Irreducibility): For any pair $(u, v)$, a path from $u$ to $v$ exists with positive probability. The chain is irreducible.
Step 3 (Uniqueness): Since $P$ is irreducible and aperiodic, the chain is ergodic. By the Perron-Frobenius theorem, a unique stationary distribution $\phi^*$ exists satisfying $\phi^* P = \phi^*$, $\sum_i \phi^*_i = 1$.
Step 4 (Convergence): By the ergodic theorem, for any initial distribution $\mu^{(0)}$:
$$\phi^* = \lim_{t \to \infty} \mu^{(0)} \cdot P^t$$
Step 5 (Interpretation): The stationary distribution $\phi^*$ is a stable consensus of observation probabilities. Each $\phi^*_j$ reflects both the particle's structural position and the neuron token influence. This is the simplest Schelling point everyone can universally agree on.
Corollaries
Corollary 1 (Stability): Small perturbations in $w_{ij}$ or $t_j$ do not destabilize the equilibrium: $\lim_{t \to \infty} \phi^*_j(t) = \phi^*_j + \varepsilon, \quad |\varepsilon| \ll \phi^*_j$.
Corollary 2 (Decentralized Computation): Focus $\phi^*_j$ for each node can be computed locally by summing contributions from incoming edges.
Corollary 3 (Emergent Modularity): Clusters of strongly connected particles naturally emerge, forming modules: $C_i = \{ j \in V \mid \phi^*_j > \tau \}$ where $\tau$ is a significance threshold.
Part II -- Composite Contraction (General Case)
Part I proves convergence for diffusion alone. The tri-kernel combines three operators. The composite converges as well.
The composite operator $\mathcal{R} = \lambda_d D + \lambda_s S + \lambda_h H_\tau$ is a contraction with coefficient $\kappa < 1$ (proved above in 2.2). Its fixed point $\phi^*$ minimizes $\mathcal{F}(\phi)$ (the free energy functional in 2.1). This is variational free energy minimization in the sense of Friston.
Reduction
When $\lambda_s = \lambda_h = 0$: $\mathcal{R} = D$, $\kappa = \alpha$, $\mathcal{F}$ reduces to $D_{\text{KL}}(\phi \| D\phi)$, and the fixed point is the stationary distribution $\phi^*$ from Part I. The general case subsumes the special case.
Part III -- Emergence (Superadditivity)
Existence, uniqueness, and convergence establish that $\phi^*$ is a well-defined collective object. They do not establish that it is better than what an individual computes. The generalized theorem (superadditivity) adds that claim: the collective $\phi^*$ strictly outperforms every neuron's ego focus $\phi^*_\nu$ on graph tasks (superadditivity $\sigma_{\text{best}} > 0$), and $\sigma$ rises with the algebraic connectivity $\lambda_2$ — the same Fiedler value that sets the heat contraction rate $e^{-\tau\lambda_2}$ in §2.2. Convergence speed and collective advantage share one knob. Benchmark-tested: $\sigma$–$\lambda_2$ is supported, but syntropy $J$ falls with $\lambda_2$ (densification spreads focus) — the two are distinct axes; see superadditivity.
4. Completeness
4.1 Completeness Conjecture
Conjecture (Weak Completeness): Any $h$-local linear operator $T$ can be written as $T = p(M) + q(L)$ for polynomials $p$, $q$ of degree $\leq h$.
Conjecture (Strong Completeness): Any eventually-local operator that is equivariant, continuous, and convergent can be expressed as $T = \alpha \cdot f(M) + \beta \cdot g(L) + \gamma \cdot H_\tau$ for spectral functions $f$, $g$ and scale $\tau$.
4.2 Lemmas Toward Proof
Lemma 1: Any 1-local linear operator is a linear combination of $\{I, A, D\}$.
Lemma 2: Any $k$-local linear operator is a polynomial of degree $\leq k$ in $\{A, D\}$.
Lemma 3: Polynomials in $\{A, D\}$ can be rewritten as polynomials in $\{M, L\}$.
Theorem (Linear Local Completeness): Every $k$-local linear operator on a graph is a polynomial of degree $\leq k$ in $M$ and $L$.
The heat kernel $H_\tau = \exp(-\tau L)$ is required for multi-scale analysis -- it is the unique generator of resolution-dependent queries. Together $\{M, L, H_\tau\}$ span the space of meaningful local graph computations.
5. Complexity
Memory and computation scale linearly with cybergraph size:
| storage type | bytes per particle | bytes per cyberlink |
|---|---|---|
| volatile | 56 | 24 |
| persistent | 72 | 128 |
Per-iteration complexity: $O(V + E)$
Total work to reach precision $\varepsilon$:
$$O\left(\frac{(E + V) \cdot \log(1/\varepsilon)}{\lambda}\right)$$
where $\lambda$ is the spectral gap governing convergence rate. See emergence for scaling estimates across intelligence phases.
6. Implementation
6.1 Two-Timescale Architecture
The correct implementation separates timescales:
- Structure (slow, amortized): springs precompute effective distances, modify diffusion tensor $D$
- Focus flow (fast, local): diffusion + heat operate on fixed structure, converge to equilibrium
Springs compute where nodes are; ranking computes how attention flows. Different questions, different timescales.
6.2 Algorithm Sketch
Per epoch on neighborhood $N_h$:
- Detect affected neighborhood around edit batch $e_\Delta$
- Pull boundary conditions: cached $\phi$, boundary flows, Laplacian blocks
- Apply local diffusion (fixed-point iteration with boundary injection)
- Apply local heat (Chebyshev $K$-term filter)
- Normalize and splice back into global $\phi$
- Emit attention_root and locality report for verification
Complexity: $O(|N_h| \cdot c)$ per kernel for average degree $c$.
6.3 Telemetry
Monitor per epoch:
- Entropy $H(\phi^*)$, negentropy $J(\phi^*)$
- Spectral gap estimate
- $\ell_1$ drift $\|\phi^{(t)} - \phi^{(t-1)}\|$
- Locality radius $h$, nodes touched
- Compute vs verify wall-time
Safety policies: degree caps, spectral sparsification, novelty floor, auto-rollback to diffusion-only on threshold breach.
References
- Brin & Page. "The anatomy of a large-scale hypertextual web search engine." WWW 1998
- Zhu et al. "Semi-supervised learning using Gaussian fields and harmonic functions." ICML 2003
- Chung. "The heat kernel as the pagerank of a graph." PNAS 2007
- Fiedler. "Algebraic connectivity of graphs." Czech Math Journal 1973
- Spielman. "Spectral Graph Theory." Yale Lecture Notes
- Levin, Peres & Wilmer. "Markov Chains and Mixing Times." AMS 2009
- Perron. "Zur Theorie der Matrices." Mathematische Annalen, 1907
- Frobenius. "Uber Matrizen aus nicht negativen Elementen." Sitzungsberichte, 1912
- Banach. "Sur les operations dans les ensembles abstraits." Fundamenta Mathematicae, 1922
- Friston. "The free-energy principle: a unified brain theory." Nature Reviews Neuroscience, 2010
- Vaswani et al. "Attention is all you need." NeurIPS 2017
- Chamberlain et al. "GRAND: Graph Neural Diffusion." ICML 2021
- Ramsauer et al. "Hopfield Networks is All You Need." 2020
- Smithe. "Mathematical Foundations for a Compositional Account of the Bayesian Brain" (active inference fixed points). 2023