hadamardInv
plain-language theorem explainer
Componentwise inversion on n-dimensional real vectors supplies the operation needed to evaluate cost functionals on reciprocal coordinates. Researchers extending the scalar J-cost to vector models cite this definition when establishing reciprocity. The construction is a direct pointwise map that applies the real reciprocal to each coordinate.
Claim. Let $x$ be a vector in $R^n$. Its componentwise inverse is the vector $y$ satisfying $y_i = x_i^{-1}$ for each coordinate index $i$.
background
The module defines the N-dimensional reciprocal cost by lifting the scalar kernel through a weighted log aggregate. Vectors are formalized as maps from finite indices to reals, allowing coordinatewise operations. The Vec abbreviation supplies the type of n-component real vectors as coordinate functions, which serves as the sole upstream dependency.
proof idea
The definition is realized directly as a lambda expression that inverts each component independently using the real reciprocal operation.
why it matters
It is required by the log-aggregate theorem for inverses and the reciprocity theorem for the N-dimensional cost. These results support the framework's treatment of self-similar fixed points and dimensional extensions, aligning with the eight-tick octave structure in the forcing chain.
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