C_equals_2A
plain-language theorem explainer
Recognition action equals twice the residual rate action at geodesic separation angle theta_s. Quantum measurement theorists bridging gravitational coherence to the Born rule cite this identity as the central bridge. The proof is a one-line reflexivity that follows immediately from the definitional relation between the two actions.
Claim. Let $theta_s in mathbb{R}$ be the geodesic separation angle. The recognition action $C(theta_s)$ equals twice the residual rate action $A(theta_s)$, so $C(theta_s) = 2 A(theta_s)$.
background
The CoherenceCollapse module formalizes the link between gravitational coherence and quantum measurement via recognition cost. Recognition action is the path integral $C[gamma] = int J(r(t)) dt$ along a path gamma. Residual rate action is defined as $A = -ln(sin theta_s)$ for the separation angle theta_s, with the module stating the central identity C = 2A as the bridge to Born rule emergence: $P(I) = exp(-C_I) / sum exp(-C_J) = |alpha_I|^2$ and the mesoscopic threshold $m_coh approx 0.2$ ng, $tau approx 1$ s for $A approx 1$ (from MODULE_DOC).
proof idea
The proof is a term-mode reflexivity. It equates the two sides directly from the definitions of recognition_action and rate_action without invoking external lemmas or tactics beyond rfl.
why it matters
This identity supplies the C_is_2A field inside coherence_collapse_cert, which assembles the full collapse certificate including positivity and normalization. It is also invoked by born_rule_from_C to derive that probabilities match amplitude squares. In the Recognition Science framework it realizes the connection from recognition cost to the Born rule, consistent with the J-uniqueness and phi-ladder structure.
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