identity_unique_attractor
plain-language theorem explainer
A configuration whose value equals the multiplicative identity admits no reachable possibility with strictly lower cost. Modal logicians in Recognition Science cite the result to establish the identity as a stable attractor under the possibility operator. The term proof rewrites the value hypothesis to invoke cost normalization at unity followed by non-negativity of J.
Claim. Let $c$ be a configuration with value $1$. For every configuration $y$ belonging to the possibility set of $c$, the cost function satisfies $J(y.value) ≥ J(1) = 0$.
background
Config is a structure representing a point in recognition state space: it carries a positive real value (generalizing bond multiplier), a positivity witness, a time coordinate in ticks, and a logarithmic bound ensuring physical scale. Possibility(c) is the set of configurations y reachable from c in one octave step (time increased by 8) whose total J-cost does not increase. J(x) = ½(x + x⁻¹) − 1 is the fundamental cost function normalized so that J(1) = 0 and J(x) ≥ 0 for all x > 0. The lemma lives inside the Modal.Possibility module, which develops the possibility operator as a precursor to actualization by J-minimization.
proof idea
The proof is a one-line term-mode wrapper. It introduces c and the hypothesis that c.value = 1, rewrites the goal via that hypothesis together with the normalization theorem J_at_one, and closes by applying the non-negativity lemma J_nonneg to the positivity witness of y.
why it matters
The lemma shows that the identity configuration is a unique attractor: no possibility offers a cost reduction. It directly supports the actualization comment in the same module, where selection proceeds by J-minimization. In the Recognition framework it reinforces T5 (J-uniqueness) and the fixed-point property of the identity under the Recognition Composition Law. No downstream theorems are recorded yet.
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