Strategy Self Knowledge Factorization

Abstract

Strategy self-knowledge factorization refines the current-state observation kernel.

Theorem 1.1 (Factorization refines the strategy kernel).

$$\forall x3\in \left(\forall x3\in \cdot ,\cdot \right),\forall x4\in \left(\forall x4\in \cdot ,\cdot \right),\forall x5\in \left(\forall x5\in \cdot ,\cdot \right),\left(\forall x6\in \cdot ,x4\left(x6\right)=x5\left(x3\left(x6\right)\right)\right)\Rightarrow \left(\forall x7\in \cdot ,\forall x8\in \cdot ,x3\left(x7\right)=x3\left(x8\right)\Rightarrow x4\left(x7\right)=x4\left(x8\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/StrategySelfKnowledgeFactorization.factorization_refines_strategy_kernel (✓ std3). ∎

Source. Repository-derived.

Commentary.

Assume the strategy profile factors pointwise through the current-memory readout, and fix histories with equal current values.

Substituting the factorization on both histories transports current equality to equality of their strategy profiles.

Theorem 1.2 (A visible profile adds no pairwise separation).

$$\begin{gathered} \forall current:H\to M,profile:H\to P,factor:M\to P, \\ x,y:H,((\forall h:H,\mathrm{profile}\left(h\right)=\mathrm{factor}\left(\mathrm{current}\left(h\right)\right))\wedge \mathrm{current}\left(x\right)=\mathrm{current}\left(y\right))\Rightarrow \mathrm{pair}\left(\mathrm{current}\left(x\right),\mathrm{profile}\left(x\right)\right)=\mathrm{pair}\left(\mathrm{current}\left(y\right),\mathrm{profile}\left(y\right)\right). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Observer/AgencySelf/StrategySelfKnowledgeFactorization.visible_profile_pair_equality (✓ std3). ∎

Source. Repository-derived.

Commentary.

Under the same factorization and current-equality hypotheses, both the current and profile components agree.

The paired agency-completion values are therefore equal for the displayed histories; no global kernel equality is asserted.

References

• Truth anchor: D5/S3/Observer/AgencySelf/StrategySelfKnowledgeFactorization.factorization_refines_strategy_kernel
• Truth anchor: D5/S3/Observer/AgencySelf/StrategySelfKnowledgeFactorization.visible_profile_pair_equality