bioremediation_one_statement
plain-language theorem explainer
The declaration establishes that the per-cycle reduction factor 1 - J(φ) lies in (0.87, 0.89), the residual fraction equals one at zero cycles, and decreases strictly for larger cycle counts. Bioremediation engineers applying the identity-tick model to PFAS or microplastic degradation would reference this statement as the quantitative anchor. The proof assembles the interval bound, the base case, and the monotonicity lemma into a single conjunction via a term constructor.
Claim. Let $r = 1 - J(φ)$. Then $0.87 < r < 0.89$, $r^0 = 1$, and for all natural numbers $n < m$ one has $r^m < r^n$.
background
In the Identity-Tick Bioremediation Pilot module the reduction factor is defined as $5/2 - φ$, which equals $1 - J(φ)$ where J is the J-cost function. The residual fraction after n cycles is then defined as $r^n$ with this same r. This setup models the exponential decay of contaminants under repeated phantom-cavity bioremediation cycles that lower the activation energy barrier by the factor (1 - J(φ)). The upstream reductionFactor_band theorem supplies the numerical interval using bounds on φ, while residualFraction_zero and residualFraction_strict_anti establish the base case and strict decrease.
proof idea
The proof is a one-line term that packages the left and right parts of the reductionFactor_band conjunction, the residualFraction_zero identity, and the residualFraction_strict_anti lemma (with implicit arguments) into the required four-part conjunction.
why it matters
This theorem serves as the single-statement summary of the bioremediation pilot in Track J8 of Plan v5. It directly encodes the per-cycle degradation factor derived from the J-cost and the exponential residual scaling. The module falsifier is a pilot deployment whose degradation rate deviates from (1 - J(φ))^n scaling by more than 5σ. It connects the Recognition Science constants (φ, J) to an engineering application without introducing new hypotheses.
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