Possibility
plain-language theorem explainer
The Possibility operator defines the set of configurations reachable from a given state c in Recognition Science modal logic. It consists of those y where time advances by exactly eight ticks and the total J-cost satisfies a non-increase condition under transition versus stasis. Modal logic researchers in the RS framework cite this definition when establishing the grammar of possibility and its adjoint relation to actualization. The definition is supplied directly by set comprehension enforcing the time and cost constraints.
Claim. $P(c) = { y : Config | y.time = c.time + 8 ∧ J(c.value) + J_transition(c.value, y.value, c.pos, y.pos) + J(y.value) ≤ J(c.value) + J_stasis(c.value) }$, where Config is a recognition state point with positive real value, time coordinate in ticks, and log-bound, J measures recognition cost, J_transition is the transition-state cost, and J_stasis is the unchanged-state cost.
background
A Config is a point in recognition state space: it carries a positive real value (generalizing bond multiplier), a positivity witness, a time coordinate in natural-number ticks, and a log-bound |log(value)| ≤ 16 that keeps values inside the physical range from roughly 10^{-7} to 10^6. The J-cost function quantifies recognition cost at a value; J_transition extracts the cost at the transition point while J_stasis records the cost of remaining unchanged. This definition lives in the Modal.Possibility module, which develops the modal operators on top of LawOfExistence and draws the eight-tick step from the octave definition (ratio 2, linked to φ^5 ≈ 11.09) supplied by upstream aesthetic and constant modules.
proof idea
The declaration is a direct set-comprehension definition. It intersects the time-advancement predicate y.time = c.time + 8 (one octave step) with the cost inequality that compares the sum of initial J, transition J, and final J against the stasis cost. No lemmas are invoked; the body simply assembles the two conjuncts.
why it matters
This supplies the possibility set P(c) that ActualizationPrinciple uses to select the cost-minimizing successor and that adjoint_from_cost_monotonic and collapse_automatic rely on for their cost-monotonic arguments. It realizes the modal grammar of RS, directly implementing the eight-tick octave (T7) and cost-nonincrease rule that underpins contingent versus determined properties downstream. The definition closes a scaffolding gap between the forcing chain and the modal layer; no open question remains inside the module.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.