residualFraction_zero
The residual contaminant fraction after zero cycles equals one by the power definition. Bioremediation engineers cite this base case to anchor the exponential decay model in the identity-tick pilot protocol. The proof is a one-line wrapper that unfolds the definition and simplifies the zero-exponent identity.
claimLet $r(n) = ρ^n$ denote the residual contaminant fraction after $n$ cycles, where $ρ$ is the per-cycle reduction factor. Then $r(0) = 1$.
background
In the Identity-Tick Bioremediation Pilot the residual fraction after $n$ cycles is defined by unfolding to the $n$-th power of the reduction factor, which the module identifies with $(1 - J(φ)) ≈ 0.882$. This supplies the exponential decay law for PFAS and microplastic degradation under repeated phantom-cavity treatments that lower the activation barrier from $E_a$ to $E_a · (1 - J(φ))$. The upstream definition is exactly residualFraction n := reductionFactor ^ n.
proof idea
The proof is a one-line wrapper that unfolds residualFraction and applies simp to reduce the zero-exponent case to the identity 1 = 1.
why it matters in Recognition Science
This base case is invoked by the bioremediation_one_statement theorem, which packages the reduction band, the zero-residual identity, and strict anti-monotonicity into a single engineering claim. It also supplies the residual_zero field of the IdentityTickBioremediationPilotCert structure. The declaration closes the n = 0 anchor for track J8 of the Recognition Science engineering derivation, where the reduction factor itself traces to J-uniqueness and the phi fixed point; the module falsifier remains empirical inconsistency with the $(1 - J(φ))^n$ scaling beyond 5σ.
scope and limits
- Does not establish the numerical interval for the reduction factor.
- Does not address decay rates or monotonicity for positive cycle counts.
- Does not invoke J-cost, defect distance, or the phi-ladder from the core framework.
- Does not prove any convergence bound or empirical validation.
Lean usage
have h0 : residualFraction 0 = 1 := residualFraction_zeroformal statement (Lean)
55theorem residualFraction_zero : residualFraction 0 = 1 := by
proof body
One-line wrapper that applies unfold.
56 unfold residualFraction; simp
57