Plain English statement: The theorem born_rule_normalized asserts that if two complex amplitudes α₁ and α₂ are assigned squared moduli equal to the normalized recognition-cost weights exp(−C₁) / (exp(−C₁) + exp(−C₂)) and exp(−C₂) / (exp(−C₁) + exp(−C₂)) respectively, then those squared moduli necessarily sum to 1.
Relevance to Recognition Science: It supplies the normalization step that turns raw J-cost exponentials into a valid probability distribution, linking the cost functional (T5) to the Born rule of quantum measurement without introducing extra parameters.
Reading the formal statement: The declaration takes four inputs—real costs C₁, C₂ and complex α₁, α₂—plus two hypotheses h₁ and h₂ that equate the normalized Boltzmann factors to ‖αᵢ‖². The conclusion is the single equality ‖α₁‖² + ‖α₂‖² = 1. The proof proceeds by substituting the hypotheses, applying the algebraic identity (a + b) / (a + b) = 1, and using the fact that the common denominator is nonzero because both exponentials are positive.
Visible dependencies in the supplied source: The proof relies only on Mathlib lemmas Real.exp_pos, add_pos, div_add_div_same, and div_self. No other declarations from the supplied modules are invoked; the theorem stands alone in IndisputableMonolith.Measurement.BornRuleLight.
What it does not prove: It assumes the identification of cost weights with |α|² and therefore does not derive the Born rule itself, does not address multi-outcome cases, and does not connect to the full forcing chain or any other module in the supplied source.