SilverSight/docs/research/COLLECTIVE_INTELLIGENCE_OPTIMIZATION.md

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

---

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.

---

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

---

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

---

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.

---

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

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