Selection Theory

Competitive Dynamics, Cascades, Uncertainty, Equilibria, and Selection Speed

Selection Inequality·Five Derived Families

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THE INSIGHT

The selection inequality Sel(A) = CL(A) - 〈Λ, B(A)〉 ≥ 0, treated as a parametric family indexed by Λ, generates a complete theory of competitive dynamics. Game theory, market dynamics, and technology selection are DERIVED, not assumed.

Five bodies: competitive reversal from Λ variation, cascade dynamics with critical fragility, Λ estimation error as formal B_leak, multi-pattern SEP as Nash equilibrium, and selection speed decomposition deriving the innovator's dilemma.

1. Competitive Reversal

Theorem S2 — Dominance depends on Lambda

A's dominance over B depends on Λ. There exists a reversal surface in R³₊ such that changing the environment swaps which pattern dominates.

Dominance reverses when the Lambda vector crosses the reversal hyperplane

A pattern can dominate through EITHER higher coherence OR lower costs

Environmental change alone can flip the winner

SO WHAT A recession is a λ_th increase. It does not make companies worse — it makes the THRESHOLD higher. Some companies that were dominant at old Λ become dominated at new Λ.

2. Cascade Dynamics

Critical fragility phase transition

When a pattern fails (Sel drops below 0), its neighbors lose a coherence source. Their Sel decreases. If any neighbor crosses Sel = 0, it fails too, propagating the cascade. There is a critical fragility threshold: below it, failures are local; above it, a single failure can propagate through the entire niche.

3. Lambda Estimation Error

Bounded rationality as corollary

No pattern has perfect knowledge of Λ. The estimation error ΔΛ is itself a form of B_leak — information about the true environment that leaks away at the observation boundary. Bounded rationality (Herbert Simon) follows as a corollary: organisms act on estimated Λ, not true Λ, and the estimation cost is a formal budget term.

4. Multi-Pattern SEP as Nash Equilibrium

Price of anarchy and competitive transitions

When multiple patterns coexist in a niche, each optimizing its own Sel, the multi-pattern SEP is a Nash equilibrium: no pattern can unilaterally improve its Sel by changing its budget allocation. The price of anarchy (gap between cooperative and Nash optima) is bounded by B6 (quadratic tangent law).

5. Selection Speed Decomposition

Deriving the innovator's dilemma

Selection speed decomposes into three components: challenge rate (how fast Λ changes), propagation speed (how fast information about the change reaches patterns), and reorganization speed (how fast patterns adapt their budget allocation).

THE INNOVATOR'S DILEMMA (DERIVED)

Large organisms have high reorganization B_cx (many internal dependencies to coordinate). Small organisms have low B_cx but low CL (less coherence buffer). The scale-speed tradeoff creates a regime where small patterns can outpace large ones in rapidly changing environments (Λ shifting fast), while large patterns dominate in stable environments (Λ stationary).

Falsifiable Predictions

Economics

P1: Reversal Surface

When λ_th increases sharply (recession, supply shock), patterns with lower B_th will gain dominance regardless of their CL advantage.

CONFIRMS IF

Market leadership changes correlate with Lambda shifts

FALSIFIES IF

Market leadership is independent of environmental price changes

Ecology / Markets

P2: Cascade Threshold

Systems near the critical fragility threshold will show power-law distributed cascade sizes. Systems far below will show exponentially truncated cascades.

CONFIRMS IF

Cascade size distribution shows phase transition at critical fragility

FALSIFIES IF

Failure cascades are uniformly distributed regardless of system fragility

Source: CT_RESEARCH_SELECTION_THEORY.md · Five families of competitive theory derived from the selection inequality with no game-theoretic assumptions.