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IndisputableMonolith.Foundation.DiscretenessForcing

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DiscretenessForcing module defines the J-cost function in logarithmic coordinates as J(e^t) = cosh(t) - 1, establishing it as a convex bowl centered at t=0. This setup is used by downstream modules on inevitability structures and phi forcing to derive discrete ledger properties from cost minimization. It builds directly on the Law of Existence equating existence with zero defect. The module supplies supporting definitions and lemmas rather than a single central proof.

claimThe J-cost in log coordinates satisfies $J(e^{t}) = cosh(t) - 1$, which is a convex function with minimum at $t = 0$.

background

This module operates in the Foundation domain of Recognition Science, importing the J-cost definition from the Cost module and the Law of Existence from its sibling module. The Law of Existence states that an entity x exists if and only if its defect is zero. The key innovation is re-expressing the J function, originally J(x) = (x + x^{-1})/2 - 1, in exponential coordinates to reveal its hyperbolic cosine form, facilitating analysis of stability and discreteness.

proof idea

This is a definition module with no proofs. It introduces J_log as the composition of J with the exponential map, then derives basic properties such as J_log_zero, J_log_nonneg, symmetry, and second derivative at zero. These follow directly from the definition of cosh and standard calculus identities.

why it matters in Recognition Science

The module supplies the discreteness foundation for the InevitabilityStructure and PhiForcing modules, which derive the golden ratio from self-similarity under J-cost. It also supports NumberTheory results on zero composition laws induced by the Recognition Composition Law. In the unified forcing chain, this corresponds to the transition toward T6 phi forcing and T7 eight-tick octave by enabling discrete ledger analysis.

scope and limits

used by (11)

From the project-wide theorem graph. These declarations reference this one in their body.

depends on (2)

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declarations in this module (20)