# Collective Intelligence Optimization Theorem — Formal Framework

**Date:** 2026-06-29  
**Based on:** Fugu's collective intelligence insight, formalized in Lean 4  
**Connection:** Dual quaternion χ ratio as geometric alignment measure  

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## Theorem 1: Capability Space Partition

Let $\mathcal{M} = \{M_1, \dots, M_n\}$ be models with capability vectors $c_i \in \mathbb{R}^d$.  
Partition capability space into $k$ orthogonal sectors $\mathcal{S}_1, \dots, \mathcal{S}_k$.  
For task $t$ with requirement $r_t \in \mathbb{R}^d$:

$$\max_{M \in \mathcal{M}} \langle c_M, r_t \rangle \leq \sum_{j=1}^k \max_{M \in \mathcal{M} \cap \mathcal{S}_j} \langle c_M, r_t^{(j)} \rangle$$

**Prediction:** Optimal single model ≤ sum of optimal sector specialists.

## Theorem 2: Orchestration Advantage

$$\Delta_{\text{orch}} \geq \sum_{j=1}^k \left( \max_{M \in \mathcal{S}_j} \langle c_M, r_t^{(j)} \rangle - \langle c_{M^*}, r_t \rangle \right)$$

**Prediction:** Orchestration advantage ≥ sum of performance gaps between sector specialists and best generalist.

## Theorem 3: Cost-Performance Tradeoff

For budget $B$, optimal set $S^*(B) = \arg\max_{S \subseteq \mathcal{M}, C(S) \leq B} P(S)$.  
Greedy algorithm achieves $\geq 1 - 1/e \approx 63\%$ of optimal.  

## Theorem 4: Dynamic Adaptation Inequality

$$\mathbb{E}[P_{t+1}(S_{t+1})] \geq \mathbb{E}[P_t(S_t)] + \alpha \cdot \text{Var}(P_t(S_t))$$

**Prediction:** Adaptive orchestrators improve proportionally to performance variance.

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## Connection to Dual Quaternion χ

The χ ratio is geometric alignment in capability space:

| Lean Structure | Dual Quaternion | Purpose |
|---------------|-----------------|---------|
| `Capability` | Real part (compressive) | H, I, C, quality |
| `Task.requirements` | Dual part (anti-compressive) | R, L, noise, cost |
| `alignment` | χ = real² / (real² + dual²) | Geometric alignment |

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## Empirical Predictions (Falsifiable)

1. Coding tasks: optimal set always includes a model with χ > 0.8 for `code_generation`
2. 3-model ensemble improvement bounded by $1 + \sqrt{2}$ (information-theoretic)
3. For tasks requiring $\geq 3$ sectors, greedy achieves $\geq 75\%$ of optimal

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## Lean Formalization Path

| Module | Content | Status |
|--------|---------|--------|
| `Semantics.CollectiveIntelligence.lean` | Structures, theorems, bounds | 🔴 NOT STARTED |
| `model_panel.py` | Empirical validation harness | ✅ EXISTS |
| `dual_quat_model_selector.py` | Dual quaternion selection | ✅ EXISTS |

**Target:** `formal/SilverSight/CollectiveIntelligence/` in SilverSight.

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## Transparent Cost Framework — No Hidden Fees

### Theorem 0: Cost Transparency Axiom

For any model $M_i$ and task $T$, the cost $C(M_i, T)$ must be expressible as:

$$C(M_i, T) = c_{\text{in}} \cdot |I| + c_{\text{out}} \cdot |O| + c_{\text{fixed}}$$

- $c_{\text{in}}$ = known input token cost (published API rate)
- $c_{\text{out}}$ = known output token cost (published API rate)
- $|I|, |O|$ = measurable input/output token counts
- $c_{\text{fixed}}$ = known fixed overhead (zero if none)

No hidden terms. No per-call multipliers. No surprise fees.

### Theorem 1: Cost-Performance Pareto Frontier

For any budget $B$, the optimal set $S^*(B)$ lies on the Pareto frontier:
- $C(S^*(B)) \leq B$ (never exceeds budget)
- No model outside the set is both cheaper AND better than one inside

### Theorem 2: Economies of Scale Bound

$$\frac{P(S)}{C(S)} \leq \max_i \frac{p_i}{c_i}$$

Ensemble's cost-performance ratio never exceeds the best individual model's ratio. **No magical "ensemble synergy" that makes multi-model cheaper per unit performance.**

### Theorem 3: Cost Synergies Forbidden

Coordination cost is explicit and bounded:

$$C(\{M_1, M_2\}, T) = C(M_1, T) + C(M_2, T) + C_{\text{coord}}$$

where $0 \leq C_{\text{coord}} \leq 0.10$ (hard bound, no hidden routing fees).

### Explicit Cost Models

```python
COST_MODELS = {
    "deepseek-v4-pro":  {"in": 0.27, "out": 1.10, "fixed": 0.0},
    "claude-opus":     {"in": 15.0, "out": 75.0, "fixed": 0.0},
    "gpt-5.5":         {"in": 10.0, "out": 30.0, "fixed": 0.0},
}

COORDINATION_COST = {
    "single": 0.0, "pair": 0.01, "multi": 0.02, "max": 0.10
}
```

### Comparison: Fugu (Hidden) vs This Framework (Explicit)

| Aspect | Fugu | This Framework |
|--------|------|---------------|
| Model costs | Proprietary | Published API rates |
| Routing fees | Hidden | Fixed $0.01–0.02 |
| Coordination overhead | Hidden | ≤ 10% (theorem) |
| Token counting | Opaque | Explicit in/out |
| Budget enforcement | "Sakana manages this" | Mathematical proof |

### Falsifiable Cost Predictions

1. Optimal set cost within 5% of budget when admissible sets exist
2. Coordination cost ≤ 10% of total for up to 8 models
3. Single-model is cost-optimal when best model's P/C ratio > 2× second-best
