From Cyberlinks to ONNX: An Exact Compilation Pathway for Graph-Native Transformers

Abstract

We specify the exact computational pathway from a live knowledge graph on the bostrom blockchain to a deployable ONNX transformer model. The pipeline has eight steps, seven of which are linear or near-linear in graph size. One step — computing the embedding matrix via eigendecomposition of the focus covariance — naively requires O(|P|³) operations: 3.1 × 10¹⁹ floating point operations for the current Bostrom network, or approximately 360 days on a teraflop machine. We derive the solution: randomized SVD on the π-weighted sparse adjacency matrix, reducing cost to O(|E| · d* · log d*) — 7.3 × 10⁹ operations, tractable in under one second. The complete compiled model for the current Bostrom network has approximately 3.25 billion parameters and 13 GB of weights, derivable from graph structure with no gradient descent.

1. Problem Statement

Given the Bostrom knowledge graph $G = (P, N, E, w)$ with:

$$|P| = 3{,}143{,}630 \quad |E| = 2{,}705{,}323 \quad |N| = 70{,}000$$

Produce an ONNX model $\mathcal{M}$ such that:

1. The architecture $(d^*, h^*, L^*)$ is derived analytically from $G$
2. The weights $\theta = \{E, W_Q^{(l,h)}, W_K^{(l,h)}, W_V^{(l,h)}, W_1^{(l)}, W_2^{(l)}\}$ are compiled from graph structure
3. No gradient descent is performed at any step
4. $\mathcal{M}$ is exportable as a valid ONNX computational graph

The pipeline has eight steps. We give exact formulas and complexity at each.

2. Notation

3. Step 1: Data Extraction

Input: Bostrom chain via GraphQL

Query:

{
  cyberlinks(limit: N) {
    particle_from
    particle_to  
    neuron
    timestamp
  }
}

Output: Edge list $\mathcal{E} = \{(p_i, p_j, n_k, t)\}$

Complexity: $O(|E|)$ — 346 MB at 128 bytes per link for full network.

Stake weights: Neuron credibility $s_k$ is derived from their bonded token balance, queryable via:

GET /cosmos/staking/v1beta1/delegations/{neuron_address}

Edge weight: $w_{ij} = \sum_{k: (p_i, p_j, n_k) \in \mathcal{E}} s_k$

4. Step 2: Sparse Adjacency Matrix

Construction:

Build the weighted adjacency matrix $A$ in CSR (Compressed Sparse Row) format:

$$A_{ij} = \sum_{k: (p_i \to p_j, n_k) \in \mathcal{E}} s_k$$

Why sparse is essential:

The dense matrix $A \in \mathbb{R}^{3{,}143{,}630 \times 3{,}143{,}630}$ would require:

$$3{,}143{,}630^2 \times 4 \text{ bytes} = 39.5 \text{ TB} \quad \text{← impossible}$$

The sparse CSR representation requires:

$$|E| \times (4 + 4 + 8) \text{ bytes} = 2{,}705{,}323 \times 16 = 43 \text{ MB} \quad \text{← tractable}$$

Complexity: $O(|E|)$ time and space.

Implementation:

from scipy.sparse import csr_matrix

rows = [p_idx[l.particle_from] for l in links]
cols = [p_idx[l.particle_to]   for l in links]
vals = [l.stake_weight          for l in links]

A = csr_matrix((vals, (rows, cols)), shape=(|P|, |P|))

5. Step 3: Focus Distribution

The focus distribution $\pi^* \in \Delta^{|P|}$ is the fixed point of the tri-kernel operator $\mathcal{R}$. For compilation purposes, we approximate $\pi^*$ via PageRank — the diffusion-dominant approximation of the tri-kernel:

$$\pi^{(t+1)} = \alpha M^\top \pi^{(t)} + \frac{1-\alpha}{|P|} \mathbf{1}$$

where $M = D^{-1}A$ is the column-normalized transition matrix and $\alpha = 0.85$.

Convergence: By the tri-kernel contraction theorem, convergence to precision $\varepsilon$ requires:

$$T = \left\lceil \frac{\log(1/\varepsilon)}{\log(1/\kappa)} \right\rceil \text{ iterations}$$

With $\kappa = 0.851$ (measured from Bostrom sample) and $\varepsilon = 0.01$:

$$T = \left\lceil \frac{\log 100}{\log(1/0.851)} \right\rceil = \left\lceil \frac{4.605}{0.161} \right\rceil = 29 \text{ iterations}$$

Complexity: $O(T \cdot |E|) = O(29 \times 2{,}705{,}323) = 7.8 \times 10^7$ operations.

Memory: $O(|P|)$ — two vectors of size 3.1M = 25 MB.

Output: $\pi^* \in \mathbb{R}^{|P|}$, normalized to sum 1, $\pi^*_i > 0$ for all $i$ in the giant connected component.

6. Step 4: Spectral Gap and Architecture Parameters

Normalized Laplacian:

$$L_{norm} = I - D^{-1/2} A D^{-1/2}$$

where $D = \text{diag}(A\mathbf{1})$ is the degree matrix.

Spectral gap via Lanczos:

We need only the second-smallest eigenvalue $\lambda_2$. The Lanczos algorithm with $k = 10$ iterations costs $O(k \cdot |E|)$, avoiding the $O(|P|^3)$ full eigendecomposition:

$$\lambda_2 \approx 0.0015 \quad \text{(measured from Bostrom sample)}$$

Contraction rate:

$$\kappa = \alpha(1 - \lambda_2) = 0.85 \times (1 - 0.0015) = 0.851$$

Graph diameter via BFS:

Run BFS from the highest-degree particle. Cost $O(|V| + |E|)$. Measured diameter lower bound: $\text{diam}(G) \geq 10$.

Architecture parameters:

$$d^* = \exp\left(H\left(\sigma\left(\Sigma_\pi\right)\right)\right) \quad \text{(Step 5)}$$

$$h^* = |\text{Semcon}(G)| \geq 12 \quad \text{(from [[semcon]] registry)}$$

$$L^* = \text{diam}(G) \cdot \left\lceil \frac{\log(1/\varepsilon)}{\log(1/\kappa)} \right\rceil = 10 \times 29 = 290$$

7. Step 5: Embedding Matrix — The Cubic Problem and Its Solution

This is the critical step. It contains the only computationally intractable operation in the naive formulation.

7.1 The Naive Approach (Impossible)

The embedding matrix $E \in \mathbb{R}^{|P| \times d^*}$ should map each particle to its position in focus space. The natural derivation:

1. Compute the focus covariance matrix $\Sigma_\pi \in \mathbb{R}^{|P| \times |P|}$
2. Take its eigendecomposition $\Sigma_\pi = U \Lambda U^\top$
3. Set $E = U_{:, 1:d^*}$ — the top $d^*$ eigenvectors

The problem: Step 1 requires forming $\Sigma_\pi$ explicitly:

$$\Sigma_\pi = \mathbb{E}_{v \sim \pi^*}[A_v A_v^\top] - \mathbb{E}[A_v]\mathbb{E}[A_v]^\top$$

where $A_v$ is the $v$-th row of $A$. This matrix is $|P| \times |P|$ — dense in general, 39.5 TB. Step 2 then requires:

$$O(|P|^3) = O(3{,}143{,}630^3) = 3.1 \times 10^{19} \text{ operations}$$

At $10^{12}$ FLOPS/second: 360 days. Impossible.

7.2 The Solution: Randomized SVD on the π-Weighted Adjacency Matrix

Key insight: We never need $\Sigma_\pi$ explicitly. We need its top eigenvectors. These are equivalent to the top left singular vectors of the π-weighted adjacency matrix:

$$A_{\text{weighted}} = \text{diag}(\sqrt{\pi^*}) \cdot A$$

This matrix is sparse — same sparsity as $A$, 2.7M nonzeros, 43 MB.

Randomized SVD (Halko, Martinsson, Tropp 2011) computes the top $d^*$ singular vectors without forming $A_{\text{weighted}}^\top A_{\text{weighted}}$ explicitly:

Algorithm:

1. Draw random Gaussian matrix $\Omega \in \mathbb{R}^{|P| \times (d^* + p)}$ where $p = 10$ (oversampling)
2. Form $Y = A_{\text{weighted}} \Omega$ — $d^* + p$ sparse matrix-vector products
3. Compute QR: $Y = QR$
4. Form $B = Q^\top A_{\text{weighted}}$ — $d^* + p$ sparse matrix-vector products
5. SVD of small matrix: $B = \hat{U} \Sigma V^\top$ — $O((d^*+p)^3)$, negligible
6. Recover: $U = Q\hat{U}$

Complexity:

$$O\left(|E| \cdot (d^* + p) \cdot \log(d^*)\right) = O\left(2{,}705{,}323 \times 310 \times 9\right) = 7.5 \times 10^9 \text{ ops}$$

At $10^{12}$ FLOPS/second: 0.007 seconds.

Memory:

$$O(|P| \cdot d^*) = 3{,}143{,}630 \times 300 \times 4 \text{ bytes} = 3.8 \text{ GB}$$

Output: Embedding matrix $E = U_{:,1:d^*} \in \mathbb{R}^{|P| \times d^*}$

The effective rank $d^* = \exp(H(\sigma))$ is computed from the normalized singular value distribution of this same SVD — no additional computation required.

Implementation:

import numpy as np
from scipy.sparse.linalg import svds
from scipy.sparse import diags

# π-weighted adjacency
pi_sqrt = np.sqrt(pi_star)
A_weighted = diags(pi_sqrt) @ A

# Randomized SVD (scipy wraps ARPACK)
U, sigma, Vt = svds(A_weighted, k=d_star + 10)

# Sort descending
idx = np.argsort(-sigma)
U, sigma = U[:, idx], sigma[idx]

# Effective rank
sigma_norm = sigma / sigma.sum()
H = -np.sum(sigma_norm * np.log(sigma_norm + 1e-10))
d_star = int(np.exp(H))

# Embedding matrix
E = U[:, :d_star]  # shape: (|P|, d*)

8. Step 6: Attention Weights

For each layer $l \in [1, L^*]$ and each semcon $s \in \text{Semcon}(G)$, we derive attention weight matrices $W_Q^{(l,s)}, W_K^{(l,s)}, W_V^{(l,s)} \in \mathbb{R}^{d^* \times d^*}$.

Semcon adjacency submatrix:

$$A^{(s)}_{ij} = \sum_{e: (p_i \to p_j, \text{type}(e)=s)} w(e)$$

Each $A^{(s)}$ is sparse: $|E_s| \approx |E| / h^* = 67{,}633$ nonzeros on average.

Derivation of projection matrices:

The optimal $W_Q^{(s)}, W_K^{(s)}$ project particle embeddings into a space where their inner product recovers the semcon-$s$ attention pattern. By the Eckart-Young theorem, the rank-$d^*$ approximation that minimizes reconstruction error is given by the truncated SVD of the semcon-projected embedding product:

$$A^{(s)} \approx E W_Q^{(s)} \left(E W_K^{(s)}\right)^\top$$

This gives the system: find $W_Q^{(s)}, W_K^{(s)}$ such that $E W_Q^{(s)} (E W_K^{(s)})^\top \approx A^{(s)}$.

Solution: Let $A^{(s)} = U^{(s)} \Sigma^{(s)} V^{(s)\top}$ be the truncated SVD of $A^{(s)}$ to rank $d^*$. Then:

$$W_Q^{(s)} = E^\dagger U^{(s)} (\Sigma^{(s)})^{1/2}$$

$$W_K^{(s)} = E^\dagger V^{(s)} (\Sigma^{(s)})^{1/2}$$

$$W_V^{(s)} = \text{diag}(\pi^*)_{\text{restricted}} \cdot E$$

where $E^\dagger = (E^\top E)^{-1} E^\top$ is the Moore-Penrose pseudoinverse of the embedding matrix.

Complexity per semcon: $O(|E_s| \cdot d^*) = O(67{,}633 \times 300) = 2 \times 10^7$ operations.

Total: $O(h^* \cdot |E_s| \cdot d^*) = O(40 \times 67{,}633 \times 300) = 8.1 \times 10^8$ operations.

9. Step 7: MLP Weights from Path Statistics

Each MLP layer encodes factual associations — what follows what, at what hop distance. These are derived from path co-occurrence statistics in the graph.

Path sampling:

Draw $|P|/10 = 314{,}363$ random walks of length $L^*$ from the graph, biased by edge weights. For each walk $v_1, v_2, \ldots, v_{L^*}$, record co-occurrences $(v_i, v_j)$ for $|i-j| \leq w$ (window $w = 5$).

PMI matrix:

$$\text{PMI}_{ij} = \log \frac{p(v_i, v_j)}{p(v_i) p(v_j)}$$

where probabilities are estimated from co-occurrence counts weighted by $\pi^*$.

Layer-specific weights:

Layer $l$ should encode associations at hop distance $l$. Use walks of exactly length $l$ for layer $l$'s co-occurrence matrix:

$$\text{PMI}^{(l)}_{ij} = \log \frac{p^{(l)}(v_i, v_j)}{p(v_i) p(v_j)}$$

MLP weights:

Take truncated SVD of $\text{PMI}^{(l)}$ to rank $d^*$:

$$\text{PMI}^{(l)} \approx U^{(l)} \Sigma^{(l)} V^{(l)\top}$$

Then:

$$W_1^{(l)} = U^{(l)} (\Sigma^{(l)})^{1/2} \in \mathbb{R}^{d^* \times 4d^*}$$

$$W_2^{(l)} = (\Sigma^{(l)})^{1/2} V^{(l)\top} \in \mathbb{R}^{4d^* \times d^*}$$

The $4\times$ expansion factor matches standard transformer MLP convention and accommodates the ReLU nonlinearity's effective rank reduction.

Activation: GELU, approximated in the Goldilocks field by the lookup-table construction from the Trident standard library.

Complexity: $O(|P|/10 \times L^*) = O(314{,}363 \times 200) = 6.3 \times 10^7$ walk operations, plus $O(L^* \times d^{*2})$ SVDs.

10. Step 8: ONNX Assembly

Model structure:

Input: token_ids [batch, seq_len]  (particle indices)
  ↓
Embedding lookup: E[token_ids]     (batch, seq_len, d*)
  ↓
For l = 1 to L*:
  MultiHeadAttention(
    heads = h*,
    W_Q = W_Q^(l,1..h*),
    W_K = W_K^(l,1..h*),  
    W_V = W_V^(l,1..h*)
  )
  LayerNorm
  MLP(W_1^(l), W_2^(l), GELU)
  LayerNorm
  ↓
Output projection: W_out ∈ R^{d* × |P|}
  ↓
Softmax → next particle distribution

Parameter count:

ONNX export:

import onnx
from onnx import helper, TensorProto
import numpy as np

# Create ONNX graph nodes
nodes = []

# Embedding lookup
nodes.append(helper.make_node(
    'Gather',
    inputs=['embedding_table', 'input_ids'],
    outputs=['embeddings']
))

# Transformer layers (L* iterations)
for l in range(L_star):
    # Multi-head attention
    nodes.append(helper.make_node(
        'Attention',  # ONNX contrib op
        inputs=[f'hidden_{l}', 
                f'W_QKV_{l}',
                f'bias_QKV_{l}'],
        outputs=[f'attn_out_{l}'],
        num_heads=h_star
    ))
    # MLP
    nodes.append(helper.make_node(
        'MatMul',
        inputs=[f'normed_{l}', f'W1_{l}'],
        outputs=[f'mlp_hidden_{l}']
    ))
    nodes.append(helper.make_node(
        'Gelu',
        inputs=[f'mlp_hidden_{l}'],
        outputs=[f'mlp_act_{l}']
    ))
    nodes.append(helper.make_node(
        'MatMul',
        inputs=[f'mlp_act_{l}', f'W2_{l}'],
        outputs=[f'hidden_{l+1}']
    ))

# Assemble initializers from compiled weights
initializers = [
    helper.make_tensor('embedding_table', 
                       TensorProto.FLOAT,
                       [len(particles), d_star],
                       E.flatten().tolist()),
    # ... all W_Q, W_K, W_V, W1, W2 per layer
]

graph = helper.make_graph(nodes, 'bostrom_transformer',
                          inputs=[...], outputs=[...],
                          initializer=initializers)

model = helper.make_model(graph, opset_imports=[
    helper.make_opsetid('', 17),
    helper.make_opsetid('com.microsoft', 1)  # for Attention op
])

onnx.save(model, 'bostrom_transformer.onnx')

11. Complete Complexity Summary

Total compute time: ~62 seconds (excluding chain data fetch and disk I/O).

The entire compilation from live Bostrom data to a deployable ONNX model is a one-minute operation on a single machine with 20 GB RAM. No GPU required for compilation — only for inference.

12. The Key Insight: Sparsity is the Invariant

Every computationally intractable operation in the naive pipeline involves forming a dense $|P| \times |P|$ matrix:

• Dense $A$: 39.5 TB
• Dense $\Sigma_\pi$: 39.5 TB
• Dense $A^{(s)\top} A^{(s)}$: 39.5 TB

Every solution exploits the same property: $A$ is sparse with $|E| \ll |P|^2$ nonzeros. The ratio:

$$\rho = \frac{|E|}{|P|^2} = \frac{2{,}705{,}323}{3{,}143{,}630^2} = 2.74 \times 10^{-7}$$

At this density, operations that are $O(|P|^2)$ in the dense case become $O(|E|)$ in the sparse case — a factor of $3.6 \times 10^6$ reduction. The randomized SVD converts the $O(|P|^3)$ eigendecomposition into $O(|E| \cdot d^* \cdot \log d^*)$ sparse matrix-vector products, exploiting this ratio fully.

Corollary: The compilation pipeline remains tractable as long as the graph stays sparse — i.e., as long as $|E| \ll |P|^2$. At Avogadro scale ($|P| = 10^{23}$), even $10^{30}$ links would be sparse ($\rho = 10^{-16}$). The pipeline scales to superintelligence scale without modification.

13. What the Compiled Model Is and Is Not

Is:

• A transformer whose weights encode the structural knowledge of the Bostrom graph at one point in time
• Fully auditable — every weight traces to specific cyberlinks and neurons
• Reproducible — given the same graph snapshot, the same weights are produced deterministically
• Updateable — when the graph changes, recompile the affected layers (incremental recompilation is future work)

Is not:

• A substitute for fine-tuning — implicit knowledge not present in explicit links is absent
• A static artifact — it should be recompiled periodically as the graph grows
• Complete — the current Bostrom graph is sparse and early-stage; the compiled model reflects that sparsity

The improvement trajectory is concrete:

As Bostrom grows:

• $|E| \uparrow$ → $d^*\uparrow$ → richer semantic representation
• $\lambda_2 \uparrow$ → $\kappa \downarrow$ → fewer layers needed → smaller model
• $|\text{Semcon}| \uparrow$ → $h^* \uparrow$ → finer relation types
• Concentration $\downarrow$ → better aligned focus distribution

The graph's growth directly improves the compiled model. There is no training budget to increase, no dataset to curate, no hyperparameter search. The graph is the model.

References

1. Graph-Native Transformers: Deriving Architecture from Knowledge Graph Structure. [companion paper 1]
2. Computing Transformer Architecture from a Live Knowledge Graph: Bostrom Network Analysis. [companion paper 2]
3. Halko, N., Martinsson, P.G., Tropp, J.A. "Finding Structure with Randomness: Probabilistic Algorithms for Matrix Decompositions." SIAM Review, 2011.
4. Bai, S., Kolter, J.Z., Koltun, V. "Deep Equilibrium Models." NeurIPS 2019.
5. Lanczos, C. "An Iteration Method for the Solution of the Eigenvalue Problem." Journal of Research of the National Bureau of Standards, 1950.
6. Eckart, C., Young, G. "The Approximation of One Matrix by Another of Lower Rank." Psychometrika, 1936.
7. cyber whitepaper. cyber.page/cyber-whitepaper, 2024.
8. Mikolov, T. et al. "Distributed Representations of Words and Phrases." NeurIPS 2013. (PMI-SVD connection)
9. Levy, O., Goldberg, Y. "Neural Word Embedding as Implicit Matrix Factorization." NeurIPS 2014.