1. Plain English Meaning
The theorem recidivismCost_nonneg establishes that the structural cost associated with a recidivism ratio is always greater than or equal to zero, provided both the reoffense rate and the baseline rate are strictly positive numbers.
2. Role in Recognition Science
In Recognition Science, physical and systemic structures are treated under a MODEL that minimizes a universal J-cost, where equilibrium corresponds to zero cost. A pre-intervention societal state has a reoffense ratio of r = 1 and thus J = 0. By providing a THEOREM that the cost of any positive ratio is non-negative, RS ensures that deviations from equilibrium accumulate as a positive, absolute structural metric. This absolute baseline is required to define a meaningful "recognition threshold". The framework makes an empirical HYPOTHESIS that effective rehabilitation interventions push the system by at least a one-φ-step departure (costing J(φ) ≈ 0.118).
3. Reading the Formal Statement
theorem recidivismCost_nonneg (reoffense baseline : ℝ)
(hr : 0 < reoffense) (hb : 0 < baseline) :
0 ≤ recidivismCost reoffense baseline
(reoffense baseline : ℝ): Declares the variables representing the two rates as real numbers.(hr : 0 < reoffense)and(hb : 0 < baseline): The hypotheses. The theorem requires both rates to be strictly greater than zero.0 ≤ recidivismCost reoffense baseline: The conclusion. It asserts that evaluating the recidivismCost function on these inputs yields a non-negative result.- The proof script (
by unfold recidivismCost; exact Jcost_nonneg (div_pos hr hb)) expands the definition into J(reoffense / baseline), uses Lean's Mathlib to note that the division of two positive reals is positive (div_pos), and applies the general J-cost non-negativity theorem.
4. Visible Dependencies and Certificates
The declaration directly depends on the definition recidivismCost, which bridges the ratio to the underlying Jcost.
Structurally, it acts as a required pillar for the module's certificate. It fulfills the cost_nonneg requirement of the RecidivismCert structure. This certificate is instantiated via cert, which is then used to prove cert_inhabited, certifying that recidivism ratios legally obey the required RS cost symmetries.
5. What This Does Not Prove
This THEOREM is purely structural. It does not prove:
- The exact value of the cost at any specific ratio (e.g., that r=1 yields zero is separately proved by recidivismCost_at_equilibrium).
- What happens if the baseline rate is exactly zero (division by zero is structurally excluded by the hypotheses).
- Any empirical claim. It does not prove that a specific cognitive-behavioral therapy intervention actually achieves a recidivism reduction. That remains an empirical HYPOTHESIS subject to falsification via randomized controlled trials.