IdentityTickBioremediationPilotCert
plain-language theorem explainer
The IdentityTickBioremediationPilotCert structure bundles the positivity, upper bound below one, and numerical band 0.87 to 0.89 for the per-cycle reduction factor together with the base case, positivity, strict decrease, and multiplicative recurrence for the residual contaminant fraction after n cycles. Bioremediation modelers would cite it to certify that the exponential decay law matches the J-cost prediction of approximately 0.882. The definition is a direct aggregation of prior lemmas on the power function with no additional steps.
Claim. Let $r = 5/2 - φ$. The certificate asserts $0 < r < 1$, $0.87 < r < 0.89$, and that the residual fraction $f(n) = r^n$ satisfies $f(0) = 1$, $f(n) > 0$ for all natural $n$, $f(n) < 1$ for $n ≥ 1$, $f$ strictly decreasing, and the recurrence $f(n+1) = f(n) · r$.
background
In the Identity-Tick Bioremediation Pilot the activation barrier is reduced by the factor $1 - J(φ)$, yielding the per-cycle reduction factor $r = 5/2 - φ$ (approximately 0.882). The residual contaminant fraction after n cycles is defined as the power $r^n$. The structure collects the elementary properties of this geometric sequence: positivity and upper bound on $r$, the explicit band, the base case $f(0) = 1$, positivity for every n, strict monotonicity, and the multiplicative recurrence.
proof idea
The structure is introduced by enumerating its eight fields. Each field is supplied by a direct reference to an existing lemma: reductionFactor_pos supplies the positivity of the factor, reductionFactor_lt_one the upper bound, reductionFactor_band the numerical interval, residualFraction_zero the base case, and the remaining four fields are filled by the corresponding power-function lemmas.
why it matters
The certificate supplies the formal engineering claim for Track J8, confirming that the residual fraction follows the exponential law derived from the J-cost reduction. It is instantiated by the downstream definition identityTickBioremediationPilotCert. The construction ties the per-cycle factor to the Recognition Science prediction $1 - J(φ)$ with φ the self-similar fixed point and supplies the falsifiable scaling law for pilot-scale bioremediation of PFAS and microplastics.
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