WEBVTT

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A smooth parabola has curvature one. But a tiny MLX example

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reports zero. The problem appears when differentiating through

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a zero incoming gradient.

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For one element, log cumulative sum exponential simply returns

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that element. Square it and divide by two. At zero, the first

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derivative is zero, but the second derivative is one.

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With two elements, the exact Hessian has nonzero entries. MLX

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returns all zeros. Finite differences of the actual gradient

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recover the expected matrix.

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A backward product must remain linear in its incoming

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gradient. Splitting that gradient into positive and negative

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branches loses the correct derivative at zero.

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A research C plus plus prototype supplies explicit reverse

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rules. All four hundred seventy seven checks pass. The

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original fails one hundred eighty nine. These are

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manifestations of one defect.

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This is a local CPU result, using a partial rebuild. GPU and

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large tensor performance remain untested. The report includes

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exact formulas, executable tests and logs.