identity_prefers_stasis
The configuration with value 1 is the unique point at which stasis cost vanishes while every transition to a different positive value incurs nonnegative cost. Modal extensions of Recognition Science cite this result to anchor the possibility operator at the fixed point of the dynamics. The proof is a direct nonnegativity argument that unfolds the cost definitions and applies the normalization J(1)=0 together with nonnegativity of J and its derived quantities.
claimThe configuration $x=1$ satisfies prefers_stasis: $J_ {stasis}(1)=0$ and $J_{change}(1,y)≥0$ for every $y>0$.
background
In the Modal.Possibility module the possibility operator collects configurations reachable at finite cost with non-increasing J-cost. The auxiliary definitions are J_stasis(x) := 8 J(x), the recognition cost of holding x unchanged over one octave, and J_change(x,y) := J_transition(x,y) + J_stasis(y), the total cost of moving from x to y and then maintaining y. Upstream, J_at_one supplies the normalization J(1)=0, the algebraic statement that the multiplicative identity carries zero recognition cost.
proof idea
The tactic proof introduces an arbitrary y>0, replaces J_stasis(1) by 0 via J_stasis_at_one, unfolds J_change, and splits the sum with add_nonneg. Each summand is shown nonnegative by J_at_one.ge, J_nonneg, J_stasis_nonneg, and the elementary facts that absolute value and division by a positive constant preserve nonnegativity.
why it matters in Recognition Science
This supplies the base case for the possibility_status report in the same module and confirms that only the identity is stable under J-cost. It instantiates the doc-comment claim that existence (x=1) is the unique stable state, aligning with the self-similar fixed point at T6 of the UnifiedForcingChain. The result closes one link in the modal extension by exhibiting the sole configuration that prefers to remain unchanged.
scope and limits
- Does not prove the full uniqueness statement for any x≠1.
- Does not quantify the speed of evolution toward 1.
- Does not incorporate continuous-time or differential dynamics.
- Does not treat multi-particle or higher-dimensional configurations.
formal statement (Lean)
206theorem identity_prefers_stasis : prefers_stasis 1 one_pos := by
proof body
Term-mode proof.
207 intro y hy _
208 -- J_stasis(1) = 0, and J_change(1,y) ≥ 0 for all y > 0
209 have hJ1 : J_stasis 1 = 0 := J_stasis_at_one
210 rw [hJ1]
211 unfold J_change
212 apply add_nonneg
213 · unfold J_transition
214 apply mul_nonneg (abs_nonneg _)
215 apply div_nonneg
216 · apply add_nonneg J_at_one.ge (J_nonneg hy)
217 · norm_num
218 · exact J_stasis_nonneg hy
219
220/-! ## Possibility: What Could Follow -/
221
222/-- **POSSIBILITY OPERATOR**: P(c) is the set of configurations reachable from c.
223
224 A configuration y is possible from x iff:
225 1. y has positive value (exists in RS)
226 2. The transition cost is finite
227 3. The change would be cost-advantageous or neutral
228
229 P : Config → Set Config -/