mp_from_cost_and_logic
plain-language theorem explainer
The meta-principle that nothing cannot recognize itself receives dual derivations from infinite cost of vanishing ratios and positive cost of contradictions. Researchers in foundational unification cite it to show existence as the cost-minimizing state where logic emerges structurally. The proof is a direct term construction that packages three upstream lemmas into the required conjunction.
Claim. Let $d$ be the defect function on positive reals. Then $ (∀ C ∈ ℝ, ∃ ε > 0, ∀ x > 0 (x < ε → C < d(x))) ∧ (∀ c, c$ a configuration asserting both a proposition and its negation at complementary ratios, the cost of $c$ is positive or both ratios equal 1) ∧ $(d(1) = 0)$.
background
This module establishes that logical consistency arises as a cost-minimizing configuration inside the Recognition Science cost model. A contradiction configuration requires a proposition P at ratio r and its negation at complementary ratio 1/r, both asserted as stable. IsLogicalContradiction holds precisely when both ratios equal 1, forcing P ∧ ¬P. The defect function d(x) measures departure from unit ratio, with d(1) = 0 by upstream result and d(x) → ∞ as x → 0⁺. The local setting uses classical metalanguage to prove that positive-cost object configurations cannot be simultaneously asserted and negated.
proof idea
Term-mode proof that directly constructs the conjunction. It applies the lemma establishing infinite cost for arbitrarily small positive ratios, the lemma showing contradiction configurations carry positive cost or satisfy the logical-impossibility predicate, and the lemma fixing zero defect at ratio 1.
why it matters
The declaration unifies the cost derivation and logic derivation of the meta-principle inside the Foundation module. It supports the summary table that contradictions are unstable while unit-ratio consistency is free, confirming that the cost landscape coincides with the logical landscape. This step grounds the emergence of logic from J-minimization and feeds the broader claim that reality is logical because logic is cheap.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.