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IndisputableMonolith.Cost

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The Cost module defines the J-cost function as J(x) = (x-1)^2/(2x). This algebraic form is the basis for all downstream applications in acoustics and mechanics. It is cited by researchers deriving just-noticeable differences in pitch or reverberation times from recognition cost. The module provides the core definitions and basic properties without any theorem proofs.

claimThe J-cost function is given by \( J(x) = \frac{(x-1)^2}{2x} \) for \( x > 0 \).

background

The Cost module introduces the J-cost as the fundamental recognition cost on positive reals. The supplied expression is algebraically equivalent to the hyperbolic form from J-uniqueness. It imports only Mathlib and lists sibling definitions including Jcost, CostRequirements, Jcost_symm, Jcost_nonneg, and AveragingDerivation.

Downstream modules apply this cost to concrete ratios: frequency ratios for pitch JND, absorption ratios for Sabine reverberation time, and signal-to-noise ratios for speech intelligibility. The same cost enters the definition of the J-action functional in the Action modules.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

The module feeds the structural theorems in IndisputableMonolith.Acoustics.MusicPitchJNDFromJCost, RoomAcousticsSabineFromJCost, SpeechIntelligibilityFromJCost, and the Action modules EulerLagrange, Hamiltonian, PathSpace. It supplies the explicit cost expression required to derive those results from the Recognition Composition Law and the cost manifold.

scope and limits

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