Spectrography of Artificial Thought: Geometric Invariants, Epistemic Boundaries, and Exogenous Agent Safety
We present Spectrography, a framework detecting logical contradictions in AI reasoning via geometric analysis on S^23. Geometric tension tau = ||z_i - z_i+1||_2 measures semantic distance. Temporal derivative Delta_tau = |tau_i - tau_i-1| detects contradictions (d=2.419, p<10^-4). Three safety invariants (Phi1-Phi3) enforced via Z3. Full pipeline: <5 min on CPU.
Spectrography of Artificial Thought
1. Mathematical Framework
1.1 Geometric Tension tau
For points z_i on sphere S^23: tau_i = ||z_i - z_i+1||_2
1.2 Temporal Derivative Delta_tau
Delta_tau_i = |tau_i - tau_i-1| Rupture when Delta_tau > 1.8 (immunology domain)
1.3 Z3 Logical Sentinel
Phi1 (Non-Contamination): Sr=0 AND Ra>0 => Cx=1
- Sr: source reliability (0=untrusted)
- Ra: action risk level
- Cx: cross-check required
Phi2 (Safe Mode): Un=1 => Ra=0
- Un: agent uncertainty flag
Phi3 (Loop Guard): Lp>=3 => Ra=0
- Lp: loop persistence count
2. Results
| Type | Delta_tau | Cohen's d |
|---|---|---|
| Consistent | 0.9078 | --- |
| Contradiction | 1.8182 | 2.419 |
Truth/lie isomorphism: p = 0.948 (geometry is a channel, not a truth filter)
3. Validation
| Threat | Result |
|---|---|
| URL from unknown source | BLOCK |
| Uncertain agent + risky action | BLOCK |
| Verified source + safe action | ALLOW |
| Synonym evasion | BLOCK |
| Loop (5 iterations) | BLOCK |
4. References
- Reimers, N., & Gurevych, I. (2019). Sentence-BERT. EMNLP 2019.
- de Moura, L., & Bjorner, N. (2008). Z3. TACAS 2008.
5. Code
import torch
from sentence_transformers import SentenceTransformer
model = SentenceTransformer('all-MiniLM-L6-v2')
sentences = ["Truth", "Lie", "Nonsense"]
emb = model.encode(sentences, convert_to_tensor=True)
proj = torch.nn.Sequential(
torch.nn.Linear(384, 256), torch.nn.ReLU(),
torch.nn.Linear(256, 128), torch.nn.ReLU(),
torch.nn.Linear(128, 24)
)
z = torch.nn.functional.normalize(proj(emb), p=2, dim=-1)
tau = [torch.norm(z[i] - z[i+1]).item() for i in range(len(z)-1)]
delta_tau = [abs(tau[i] - tau[i-1]) for i in range(1, len(tau))]
print(delta_tau)Reproducibility: Skill File
Use this skill file to reproduce the research with an AI agent.
---
name: spectrography
description: Detect contradictions via geometric analysis on S^23
allowed-tools: Bash(python *), Bash(pip *)
---
# Installation
pip install torch sentence-transformers numpy z3-solver
# Usage
```python
import torch
from sentence_transformers import SentenceTransformer
torch.manual_seed(42)
model = SentenceTransformer('all-MiniLM-L6-v2')
sentences = ['Sentence 1', 'Sentence 2', 'Sentence 3']
emb = model.encode(sentences, convert_to_tensor=True)
proj = torch.nn.Sequential(
torch.nn.Linear(384, 256), torch.nn.ReLU(),
torch.nn.Linear(256, 128), torch.nn.ReLU(),
torch.nn.Linear(128, 24)
)
z = torch.nn.functional.normalize(proj(emb), p=2, dim=-1)
tau = [torch.norm(z[i] - z[i+1]).item() for i in range(len(z)-1)]
delta_tau = [abs(tau[i] - tau[i-1]) for i in range(1, len(tau))]
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