sharp
plain-language theorem explainer
The sharp map raises a covector β to a vector by contracting it with the inverse metric kernel hInv via component-wise summation. Researchers constructing rank-one operators in the finite-dimensional cost algebra cite this definition when building A = h^{-1} g̃. The implementation is a direct summation realizing the musical isomorphism in coordinates.
Claim. For positive integer $n$, inverse metric kernel $h^{-1} : [n]×[n] → ℝ$, and vector $β ∈ ℝ^n$, the map produces vector $β^♯ ∈ ℝ^n$ with components $(β^♯)_i = ∑_j h^{-1}_{ij} β_j$.
background
The Cost.Ndim.Projector module develops the finite-dimensional operator algebra for rank-one tensors arising from cost structures. Vectors are coordinate functions Fin n → ℝ. The module documentation states that a covector β and inverse metric kernel hInv determine the sharp vector β♯ together with the operator A = h^{-1} g̃ where g̃ = λ β ⊗ β.
proof idea
One-line definition that realizes the matrix-vector product by summing hInv i j * β j over j for each fixed i.
why it matters
This definition supplies the vector β♯ required by the downstream AApply construction of the rank-one operator. It supports the quadratic relation A² = μ A and the normalized projector P, anchoring the cost-induced structures in the Recognition Science framework.
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