modal_T_weak
plain-language theorem explainer
The weakened modal T axiom states that if a property holds in all possible futures of a configuration then it holds after actualization via cost minimization. Modal logicians adapting necessity operators to time-evolving systems with J-cost selection would cite this when the standard reflexivity axiom fails due to mandatory advancement. The proof is a one-line wrapper that instantiates the universal quantification in □p at the identity configuration, which is always possible.
Claim. Let $p$ be any property of configurations. If $p$ holds for every configuration in the possibility set of $c$, then $p$ holds at the actualized configuration obtained from $c$.
background
Config is a structure with positive real value, natural-number time coordinate, and log-bound constraint, serving as a point in recognition state space. ConfigProp is the type of predicates on configurations. Possibility(c) collects successor configurations whose time is exactly c.time + 8 and whose total J-cost does not exceed the stasis cost; the identity configuration always belongs to this set. Actualize(c) returns the identity configuration at the advanced time, implementing the J-minimizing selection. □p at c means p holds throughout Possibility(c), i.e., necessity is cost-forced. The local setting is the modal layer of Recognition Science, where full T reflexivity fails because the current configuration lies outside its own possibility set once time advances by one octave.
proof idea
The proof is a one-line wrapper. After introducing the assumption that p holds on every member of Possibility(c), the tactic applies that assumption directly to the actualized state by supplying the membership fact that the identity configuration (selected by Actualize) always lies in Possibility(c).
why it matters
This supplies the weakened reflexivity axiom required once time emergence forces an eight-tick advance, enabling consistent modal reasoning inside the Recognition Science framework. It sits inside the Modal.Possibility module and precedes the counterfactual section; it aligns with the T5 J-uniqueness and T7 octave structure that determine which successor is actualized. No downstream theorems are recorded yet, leaving open how the axiom composes with the Recognition Composition Law in full modal derivations.
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