cert
The definition cert assembles a SentencingCert witness that encodes proportionality of punishment to harm under the J-cost model, with the ratio exceeding 1, adjacent severity ratios exceeding 2, and zero cost on matched scales. Recognition-science applications to jurisprudence would cite it to instantiate the prediction that the optimal ratio equals the recognition quantum φ. The construction is a direct structure literal invoking three upstream ratio and cost theorems.
claimLet proportionalityRatio be the punishment-to-harm ratio and adjacentSeverityRatio the ratio between adjacent severity categories. Then cert is the SentencingCert satisfying $1 <$ proportionalityRatio, $(2 : {R}) <$ adjacentSeverityRatio, and $∀ p : {R}, p ≠ 0 →$ sentencingCost $p$ $p = 0$.
background
The module Sentencing Proportionality from J-Cost treats proportionality (harm × culpability = punishment) as a foundational criminal-justice principle and derives it from the J-cost function, which quantifies departure from null cost. The recognition-science prediction is that the optimal punishment-to-harm ratio equals the recognition quantum φ, with adjacent severity ratios near φ² ≈ 2.618. Upstream, proportionalityRatio_gt_one follows from one_lt_phi, adjacentSeverityRatio_gt_two is proved by linarith on phi_squared_bounds, and sentencingCost_proportional reduces sentencingCost p p to Jcost_unit0 after unfolding.
proof idea
The definition cert is a one-line structure constructor that directly populates the three fields of SentencingCert by reference to the sibling theorems proportionalityRatio_gt_one, adjacentSeverityRatio_gt_two, and sentencingCost_proportional.
why it matters in Recognition Science
cert supplies the concrete witness required by the structural theorem of the module, confirming that sentencing proportionality follows from J-cost and realizes the RS claim that the ratio equals φ. It aligns with the phi-ladder and eight-tick octave landmarks while closing the certification step for jurisprudence. No open questions are left in the supplied results.
scope and limits
- Does not derive proportionalityRatio from empirical sentencing corpora.
- Does not obtain the J-cost function from the unified forcing chain T0-T8.
- Does not treat the p = 0 case in the cost-proportionality condition.
- Does not tighten the ratio bounds beyond the stated inequalities.
formal statement (Lean)
60noncomputable def cert : SentencingCert where
61 ratio_gt_one := proportionalityRatio_gt_one
proof body
Definition body.
62 adj_gt_two := adjacentSeverityRatio_gt_two
63 cost_proportional := sentencingCost_proportional
64