IndisputableMonolith.Foundation.Superposition_Cost_RS
Foundation module defining the Recognition Science cost of holding a superposition: a domain cost functional, its nonnegativity, and a canonical positive threshold, bundled as a SuperpositionCostCert. Measurement and collapse energetics work would cite the certificate. Structure is definitional plus elementary positivity lemmas from the Cost import.
claimThe module defines a domain cost $C(\Omega)$ for superposition support, proves $C\ge 0$ and an evaluation identity, introduces a canonical threshold $\tau_*>0$, and packages these facts in an inhabited certificate structure for RS-native superposition cost.
background
Recognition Science measures recognition effort by the J-cost $J(x)=(x+x^{-1})/2-1$, the unique solution forced at T5 of the unified forcing chain and obeying the Recognition Composition Law. Holding several recognizers in superposition incurs a cost over the amplitude support domain.
This Foundation module imports Constants (RS time quantum $\tau_0=1$ tick) and the Cost library. Sibling definitions introduce domainCost (cost of a superposition domain), elementary identities and nonnegativity, a canonicalThreshold with positivity, and a SuperpositionCostCert bundle that packages the static cost facts for downstream use.
proof idea
Definition-and-certificate module, not a deep theorem development. It declares the domain cost, records an evaluation identity and nonnegativity via the imported Cost API, defines the canonical threshold and proves it is positive, then assembles these into SuperpositionCostCert with an inhabited instance. No multi-step forcing or analytic argument; proofs are short wrappers and elementary sign checks.
why it matters in Recognition Science
Supplies the cost-side substrate for RS accounts of superposition, measurement, and collapse energetics. Downstream developments can import the certificate instead of re-proving nonnegativity or threshold positivity. No used_by edges are recorded yet, so the module is presently a Foundation leaf awaiting consumers in the quantum or measurement layer. It sits downstream of J-cost uniqueness (T5) and the Constants/Cost stack, and is the natural place to attach later links to the Berry creation threshold $\phi^{-1}$ or eight-tick scheduling costs.
scope and limits
- Does not derive unitary dynamics or the Schrödinger equation.
- Does not fix numerical superposition costs beyond threshold positivity.
- Does not treat entanglement or multipartite domain costs.
- Does not prove dynamical collapse, only static cost bounds.
- Does not invoke eight-tick octave or D=3 forcing results.