identityTickBioremediationPilotCert
Engineers modeling PFAS and microplastic degradation under Recognition Science cite this definition for the certified per-cycle reduction factor and residual scaling. It packages the interval 0.87 < r < 0.89 on r together with the power-law properties of the residual fraction f(n) = r^n. The construction is a direct structure instantiation that references eight sibling lemmas establishing positivity, bounds, normalization, and recurrence.
claimLet $r$ be the per-cycle reduction factor satisfying $0 < r < 1$ and $0.87 < r < 0.89$. Define the residual contaminant fraction by $f(n) := r^n$ for $n : ℕ$. The certificate asserts $f(0) = 1$, $f(n) > 0$ for every $n$, $f(n) < 1$ whenever $n ≥ 1$, $n < m$ implies $f(m) < f(n)$, and $f(n+1) = f(n) · r$.
background
The module develops an engineering application of the J-cost to phantom-cavity bioremediation (RS_PAT_041). It accelerates degradation of PFAS, microplastics and POPs by lowering the activation barrier from $E_a$ to $E_a · (1 - J(φ))$, yielding a per-cycle factor $r ≈ 0.882$. The structure IdentityTickBioremediationPilotCert collects the minimal mathematical properties needed to certify this scaling: bounds on $r$ and the listed algebraic and ordering facts for the residual $f(n) = r^n$ (defined via reductionFactor ^ n in the sibling residualFraction). Upstream lemmas reductionFactor_pos, reductionFactor_lt_one and reductionFactor_band establish the interval on $r$ from the standard bounds phi_gt_onePointSixOne and phi_lt_onePointSixTwo; the residualFraction_* theorems then lift these to the power properties via pow_pos and pow_lt_one₀.
proof idea
The definition constructs an instance of IdentityTickBioremediationPilotCert by direct field assignment. It supplies reductionFactor_pos for factor_pos, reductionFactor_lt_one for factor_lt_one, reductionFactor_band for factor_band, residualFraction_zero for residual_zero, residualFraction_pos for residual_pos, residualFraction_lt_one_of_pos for residual_lt_one_pos, residualFraction_strict_anti for residual_strict_anti, and residualFraction_succ for residual_succ. No additional tactics or reductions are performed.
why it matters in Recognition Science
This definition closes the engineering derivation for Track J8 of Plan v5. It supplies the concrete certificate that the per-cycle degradation factor lies in the stated band and that residuals decay exactly as $r^n$, thereby operationalizing the J-uniqueness (T5) and phi-ladder for remediation modeling. The module doc states the falsifier: pilot-scale data showing degradation inconsistent with $(1 - J(φ))^n$ scaling beyond 5σ. No downstream uses appear yet, but the structure is the required interface for any later application of the Recognition Composition Law to activation-barrier reduction.
scope and limits
- Does not establish the physical mechanism of phantom-cavity acceleration.
- Does not compute explicit degradation rates for specific contaminants.
- Does not address scaling to industrial volumes or environmental variables.
- Does not derive the numerical value of J(φ) from first principles.
- Does not bound the number of cycles needed for regulatory compliance.
formal statement (Lean)
89def identityTickBioremediationPilotCert :
90 IdentityTickBioremediationPilotCert where
91 factor_pos := reductionFactor_pos
proof body
Definition body.
92 factor_lt_one := reductionFactor_lt_one
93 factor_band := reductionFactor_band
94 residual_zero := residualFraction_zero
95 residual_pos := residualFraction_pos
96 residual_lt_one_pos := @residualFraction_lt_one_of_pos
97 residual_strict_anti := @residualFraction_strict_anti
98 residual_succ := residualFraction_succ
99
100/-- **BIOREMEDIATION ONE-STATEMENT.** Per-cycle reduction
101`1 - J(φ) ∈ (0.87, 0.89)`; residual fraction `(1 - J(φ))^n` is
102strictly anti-monotonic; reduces to identity at n = 0. -/