Possibility
plain-language theorem explainer
Possibility defines the reachable set P(c) for a configuration c in RS modal logic as those y with time advanced by exactly 8 ticks and non-increasing total J-cost via transition. Researchers formalizing possibility and actualization in Recognition Science cite this operator when proving adjointness between possibility and actualization. The definition is implemented as a direct set comprehension over the Config structure using J-transition and J-stasis.
Claim. The possibility operator $P(c)$ for a configuration $c$ is the set of all configurations $y$ satisfying $y.time = c.time + 8$ and $J(c.value) + J_{transition}(c.value, y.value, c.pos, y.pos) + J(y.value) ≤ J(c.value) + J_{stasis}(c.value)$, where each configuration has positive real value, natural-number time coordinate, and bounded logarithm.
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 natural-number time in ticks, and a log bound |log(value)| ≤ 16. J measures recognition cost on positive reals; J_transition evaluates cost at the transition state while J_stasis evaluates the no-change case. The definition lives in Modal.Possibility, which imports LawOfExistence and draws the eight-tick step from the octave definition (ratio 2) in MusicalScale and Constants.
proof idea
The definition is a direct set comprehension that encodes the time advancement by one octave step together with the cost condition that the transition does not increase total J-cost relative to stasis.
why it matters
This supplies the possibility operator P that feeds the ActualizationPrinciple (universe selects cost-minimizing successor from P(c)) and the adjointness theorems between P and A. It implements the eight-tick time step from T7 of the forcing chain and supports cost-monotonic dynamics; it is summarized in rs_modal_logic_status.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.