Explain the Lean theorem `recidivismCost_reciprocal` in module `IndisputableMonolith.CriminalJustice.RecidivismFromJCost`. Write for an educated reader who knows science and programming but may not know Lean. Cover: (1) what the declaration says in plain English, (2) why it matters in Recognition Science, (3) how to read the formal statement, (4) visible dependencies or certificates in the supplied source, and (5) what this declaration does not prove. Cite only declarations present in the supplied Recognition source.

1. Plain English Meaning

The theorem states that the informational "cost" of a recidivism ratio is perfectly symmetric. If you evaluate the cost of a specific reoffense rate against a baseline rate, you obtain the exact same mathematical value as if you flipped them and evaluated the baseline against the reoffense rate, provided both rates are strictly greater than zero.

2. Significance in Recognition Science

In RS, every structure is downstream of a single invariant logic formalized via the $J$-cost function. A core theorem of this foundation is reciprocal symmetry: $J(r) = J(1/r)$. The theorem recidivismCost_reciprocal maps this universal invariant onto criminal justice. It establishes that an intervention pushing the recidivism ratio $r < 1$ (effective rehabilitation) is structurally governed by the same symmetric cost landscape as deterioration ($r > 1$). This casts recidivism tracking not as arbitrary statistics, but as an admissible setting of the fundamental recognition ledger.

3. Reading the Formal Statement

theorem recidivismCost_reciprocal (reoffense baseline : ℝ)
    (hr : 0 < reoffense) (hb : 0 < baseline) :
    recidivismCost reoffense baseline = recidivismCost baseline reoffense

• theorem recidivismCost_reciprocal: Declares a formally proved mathematical fact.
• (reoffense baseline : ℝ): Declares two variables as real numbers representing the measured rate and the baseline rate.
• (hr : 0 < reoffense) and (hb : 0 < baseline): The formal hypotheses. The proof requires both rates to be strictly positive.
• : : Separates the premise from the conclusion.
• recidivismCost ... = recidivismCost ...: The claim. The function recidivismCost yields identical results regardless of argument order.

4. Visible Dependencies and Certificates

In the supplied source, the proof unfolds the definition of recidivismCost and algebraically applies the imported foundational theorem Jcost_reciprocal.

This theorem is then required by RecidivismCert and satisfied by the cert instance. This certificate acts as a bundled formal guarantee that the recidivism metric behaves as a valid RS cost model, fulfilling four structural requirements: equilibrium at zero, non-negativity, reciprocal symmetry, and the $\phi$-step threshold.

5. What This Declaration Does Not Prove

This declaration is a structural THEOREM about the mathematical properties of the defined model. It does not prove that any real-world intervention actually reduces recidivism. Furthermore, it does not mathematically prove the HYPOTHESIS outlined in the module documentation—that the minimum detectable difference in human behavioral change sits at a one-$\phi$-step departure ($J(\phi) \approx 0.118$). That is an empirical prediction requiring large-N randomized controlled trials to falsify, not a logical tautology.