modifiedPoissonStub
plain-language theorem explainer
modifiedPoissonStub supplies a placeholder instance of ModifiedPoissonPDEFacts for use in relativity extensions of the Recognition Science framework. Developers would cite this when temporarily discharging sorries in Poisson equation analyses involving the lag coupling. The implementation consists of a trivial exact tactic for solution uniqueness and a simp reduction for the fundamental relation.
Claim. The definition instantiates the modified Poisson PDE facts by requiring that solutions for density $ρ$, weight $w$, and potentials $Φ_1$, $Φ_2$ differ by zero (witnessed by the pair $⟨0, rfl⟩$) and that the fundamental relation holds identically after simplification for parameters $ψ_0$, $ng$, $ρ$, $α$, $C_{lag}$, $x$, where $C_{lag} = φ^{-5}$.
background
Fixtures in the Relativity.NewFixtures module provide stub instances for hypothesis classes that replace sorries while keeping the trust boundary clear, per the module documentation. The upstream definition C_lag from EightTickResonance sets the lag coupling to $C_{lag} := φ^{-5} ≈ 0.09$, which enters the fundamental_modified_poisson field of this stub. This sits among sibling stubs such as gaugeFactsStub and curvatureFactsStub for building relativity and gravity results in the Recognition Science monolith.
proof idea
The definition constructs the instance directly. The poisson_solution_unique field is filled by introducing the parameters ρ, w, Φ₁, Φ₂, h₁, h₂, r, hr and then applying exact to the pair ⟨0, rfl⟩. The fundamental_modified_poisson field introduces ψ₀, ng, ρ, α, C_lag, x and reduces via simp.
why it matters
This definition serves as a temporary fixture to allow progress on modified Poisson analyses in the relativity domain of Recognition Science, replacing what would otherwise be sorries. It incorporates the lag constant C_lag from the eight-tick resonance structure. With no current downstream uses, it addresses the scaffolding need for PDE facts pending full integration with the forcing chain or Recognition Composition Law.
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