Parametric parameter forms for axis-aware parameters

[jc-macdonald]

Summary

Allow parameters to declare a functional form (e.g., polynomial, piecewise-linear) rather than requiring pre-computed per-coordinate scalar values.

Problem

For continuous axes, parameters like beta[age] currently require the user to supply a discrete value for every coordinate bin. As resolution increases (e.g., 20 or 50 age bins), maintaining per-bin parameter tables becomes unwieldy. A parametric form would let users declare beta[age]: {form: polynomial, degree: 2, coeffs: [a0, a1, a2]} and have op_system evaluate it at each coordinate.

Scope

• Define a parameter-form schema in the spec (e.g., form, degree, coeffs, breakpoints)
• Support at least: constant, polynomial, piecewise-linear
• Evaluate the form at axis coordinates during normalization or compilation
• Expose the resolved per-coordinate values in the same way as manually specified parameters
• Add tests for each supported form

Non-Goals (for now)

• Fitting parameter forms to data (that belongs in the inference layer)
• Time-varying parametric forms (handled by the engine/backend)

Related

• Listed as "Proposed" in InsightNet talk slide 4
• Complements continuous axis support and integrate_over() (more bins = more need for parametric forms)