module module high

`IndisputableMonolith.CrossDomain.JPositivityUniversality`

The module delivers the universal non-equilibrium cost claim as a definitional specialization of Jcost positivity. It applies this to cross-domain cost functions including turbulence, disease, and market arbitrage. Researchers in non-equilibrium systems cite it to enforce consistent J-based costing across fields. The module structure consists of direct definitional instantiations with no new proofs.

claimFor any non-equilibrium cost ratio $x$ with $x ≠ 1$, the specialized J-cost satisfies $J(x) > 0$, obtained by definitional specialization of the base positivity result.

background

The Recognition Science framework measures non-equilibrium deviation through the J function, defined as $J(x) = (x + x^{-1})/2 - 1$. The upstream Cost module supplies the general result that J-cost is positive whenever the ratio differs from unity. This module introduces domain-specific cost objects such as turbulent_cost, disease_cost, off_target_cost, and market_arbitrage_gap, each obtained by specializing the base positivity claim.

proof idea

This is a definition module, no proofs. It consists of direct definitional specializations that instantiate the general Jcost positivity result for each listed cross-domain cost function.

why it matters in Recognition Science

The module supplies the universal non-equilibrium cost claim that supports all sibling definitions in the CrossDomain section. It directly specializes the parent result Jcost_pos_of_ne_one from the Cost module, enabling uniform application of the positivity property to turbulent, biological, and economic systems.

scope and limits

Does not derive the J function or its positivity from first principles.
Does not compute numerical values or bounds for specific cost parameters.
Does not treat equilibrium cases where the cost ratio equals one.
Does not contain independent proofs beyond the imported base result.

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Cost module_import

declarations in this module (19)

def TurbulentCost L34
def DiseaseCost L35
def OffTargetCost L36
def OffEquilibriumGameCost L37
def MarketArbitrageGap L38
def BiasedReasoningCost L39
def RecognitionDeficit L40
theorem turbulent_cost L42
theorem disease_cost L43
theorem off_target_cost L44
theorem off_equilibrium_game_cost L45
theorem market_arbitrage_gap L47
theorem biased_reasoning_cost L49
theorem recognition_deficit L51
theorem all_seven_are_one L56
theorem symmetry_at_equilibrium L66
theorem minimum_at_one L70
structure JPositivityUniversalityCert L77
def jPositivityUniversalityCert L94