J_transition_symm
Symmetry of transition costs between positive real configurations holds in the modal possibility space. Researchers working on modal geometry cite this when establishing that distances in configuration space are symmetric. The proof proceeds by unfolding the definition of J_transition and applying properties of the logarithm to equate the absolute values of the log ratios.
claimFor all positive real numbers $x, y$, $J_{transition}(x,y) = J_{transition}(y,x)$, where $J_{transition}(x,y) := |ln(y/x)| · (J(x) + J(y))/2$.
background
In the Modal.Possibility module, configurations are positive reals equipped with the J-cost function from Recognition Science. The transition cost is defined as the absolute log-ratio of the configurations multiplied by the average J-cost of the endpoints. This setup models the action for direct transitions in possibility space. Upstream results include the definition of J from the foundation and constants like tick and octave that set the scale for recognition cycles. The J function satisfies the Recognition Composition Law.
proof idea
The proof unfolds the definition of J_transition. It then applies congr to reduce to showing equality of the log absolute value and the averaged J terms. For the log part, it uses Real.log_div twice to establish that log(y/x) = -log(x/y), followed by abs_neg. The averaged term is handled by ring.
why it matters in Recognition Science
This lemma supports the modal_distance_symm result in ModalGeometry, which establishes symmetry of distances in configuration space. It fills the symmetry requirement for the possibility topology in the Recognition framework, aligning with the eight-tick octave structure where costs are paid symmetrically. No open questions are directly touched here.
scope and limits
- Does not extend to zero or negative configurations.
- Does not prove triangle inequality or other metric properties.
- Does not incorporate multi-step transitions.
- Does not address quantum or continuous limits.
Lean usage
lemma modal_distance_symm (c1 c2 : Config) : modal_distance c1 c2 = modal_distance c2 c1 := by unfold modal_distance; exact J_transition_symm c1.pos c2.posformal statement (Lean)
135lemma J_transition_symm {x y : ℝ} (hx : 0 < x) (hy : 0 < y) :
136 J_transition x y hx hy = J_transition y x hy hx := by
proof body
Tactic-mode proof.
137 unfold J_transition
138 congr 1
139 · -- Show |log(y/x)| = |log(x/y)|
140 have h1 : Real.log (y / x) = -Real.log (x / y) := by
141 rw [Real.log_div (ne_of_gt hy) (ne_of_gt hx)]
142 rw [Real.log_div (ne_of_gt hx) (ne_of_gt hy)]
143 ring
144 rw [h1, abs_neg]
145 · ring
146
147/-! ## Cost of Stasis vs Cost of Change -/
148
149/-- **COST OF STASIS**: The cost for a configuration to remain unchanged.
150
151 In RS, even "staying the same" has a recognition cost because:
152 1. The universe is constantly recognizing itself (8-tick cycle)
153 2. Maintaining identity requires continuous cost payment
154 3. J(x) is paid per tick, not just for change
155
156 J_stasis(x) = 8 · J(x) (cost over one octave to maintain x) -/