Root values and sensitivities

Research question. When does a computed root value support a parameter sensitivity claim?

Prerequisites. What is a derivative? and From mathematical relations to differentiable programs.

Predict¶

For $x^2-\theta=0$ on the positive branch, predict the root, derivative sign, units, conditioning, bracket behavior, and failure if uniqueness or slope evidence is rejected. Distinguish the value-first branch history from the ideal implicit sensitivity before running code.

Compute¶

Run the repository-owned example:

uv run --no-sync python -m examples.investigations.root_values_and_sensitivities

The command prints a complete report: prediction, the required units-explicit metric table, audit status and evidence, and the calibrated warranted claim. A failed audit is visible and gives the command a nonzero exit status. The source retains typed solver status, final bracket, evaluation count, derivative certificate, and the analytic fixture.

Audit¶

Check the analytic square-root identity, signed residual, final bracket width, certificate, and analytic implicit derivative independently. Then inspect the registered rootfinding.performance and rootfinding.implicit-gradients evidence.

Misconception check¶

A converged value-first root does not inherit the derivative of an ideal root. The certified API makes that separate claim only after its gates pass.

State the warranted claim¶

The analytic positive branch independently supplies uniqueness and smoothness. The runtime certificate records those caller assertions and checks convergence, finite state, residual, width, and slope conditioning. Together they support the fixture’s derivative; they do not prove the assumptions for arbitrary residuals or downstream physical models.