J_log
plain-language theorem explainer
The definition supplies the cost function in logarithmic coordinates as the hyperbolic cosine of the argument minus one. Researchers examining stability of configurations under the Recognition Science cost landscape cite this form when shifting analysis to additive variables. It is introduced by direct definition from the change of base in the original multiplicative cost.
Claim. The function mapping a real number $t$ to the value of the cost at $e^t$ is given by $J(t) = 2^{-1}(e^t + e^{-t}) - 1$, which equals $cosh(t) - 1$.
background
The Discreteness Forcing module develops the argument that the cost functional forces discrete ontology. Its documentation states that J(x) equals one-half times (x plus x inverse) minus one, with a unique minimum at x equals one, and that in log coordinates this becomes a convex bowl centered at t equals zero. The upstream definition from the ContinuumBridge expresses the identical quantity as the cost applied to the exponential of the argument.
proof idea
The declaration is a direct definition that equates the logarithmic cost to the hyperbolic cosine minus one. It supplies the base expression used by all later results in the module on non-negativity, zero location, and curvature.
why it matters
This supplies the explicit logarithmic expression of the cost that underpins the discreteness forcing principle. The principle theorem invokes the curvature of this function at zero to conclude that continuous spaces admit no isolated minima. In the Recognition Science framework it realizes the J-uniqueness step by writing the cost as cosh of the logarithm minus one, and it supports the minimum step cost of one that appears in the eight-tick octave analysis.
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