A_toward_identity
plain-language theorem explainer
The theorem establishes that the actualization operator A strictly decreases J-cost for any configuration whose value differs from 1. Modal physicists modeling recognition state transitions would cite this to confirm that actualization selects lower-cost successors. The proof is a direct one-line application of the actualize_decreases_cost lemma.
Claim. Let $c$ be a configuration with value not equal to 1. Then the J-cost of the actualized configuration satisfies $J((A(c)).value) < J(c.value)$, where $A$ maps $c$ to the J-minimizer over its possibility set.
background
A configuration is a structure holding a positive real value (generalizing a bond multiplier), a positivity proof, and a time coordinate in ticks. The actualization operator A sends each configuration to the J-minimizer chosen from the set of possibilities P(c). The J function is the recognition cost, satisfying J(1) = 0 and J(x) > 0 for x ≠ 1, with the decrease property supplied by the upstream actualize_decreases_cost result.
proof idea
This is a one-line wrapper that applies the actualize_decreases_cost theorem from the Possibility module directly to the input configuration and the hypothesis that its value is not 1.
why it matters
The result feeds the actualization_status definition in the same module and confirms that A advances toward identity. It instantiates the J-minimization step within the modal layer, consistent with the Recognition Composition Law and the phi-ladder structure. The declaration closes a local gap in the actualization chain without addressing convergence speed.
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