Bath Layer Redesign (UserExponents-first)

[yjmaxpayne]

Overview

Reorients the bath abstraction so that an arbitrary user-supplied exponential
set (Bath::Kind::UserExponents, i.e. arbitrary (c_k, γ_k) pairs) is the
core primitive that the hierarchy and temperature-dependence machinery
consume, with the existing Pade-based spectral decomposition becoming one
provider on top of that primitive rather than the only path. This unblocks
arbitrary bath spectral densities (1/f, underdamped Brownian, user-fit
exponentials) without hard-coding a specific decomposition method into the
numerical core.

Background

The field has moved from purely analytic bath decomposition
(Matsubara/Pade expansions) toward numerical fitting methods (AAA/Prony) for
constructing exponential bath representations. Keeping the core primitive as
"a set of exponentials" rather than "a Pade-generated table" means new
decomposition methods can be added without touching hierarchy or
temperature-coefficient code. The current Pade coefficient generator is a
compact runtime eigenvalue-method implementation; it is reframed as the
first bath provider rather than replaced from scratch.

Core Objectives

1. Design the UserExponents interface surface

Decouple bath spectral-density output from the temperature-dependence
coefficient pipeline so any (c_k, γ_k) sequence can flow through.

2. Land UserExponents as a first-class Bath::Kind

Hierarchy and temperature-coefficient machinery consume arbitrary exponent
sets.

3. Re-platform the existing Pade generator as a provider

Not a special case in the core — a provider implementation on top of
UserExponents.

4. Resolve the double-precision activation blocker

Determine the real scope of templating the device-side numeric pipeline for
double precision (host-side Pade coefficients are already double-precision;
the remaining gap is device-side pipeline templating).

5. Keep numerical fitting (AAA/Prony) out of the core

Stays a Python-layer concern, reusing existing fitting tooling — the core
only needs to consume exponential sets, not fit them.

Success Criteria

Metric	Target
UserExponents path produces correct results on a non-Pade exponential set	within stated tolerance vs. reference
Existing Pade-derived results reproduce prior generator output	within numerical tolerance + legacy baseline preserved
Precision::Double scope decision executed	decision recorded; double path activated or explicitly scoped out with reason

Dependencies

Depends on Backend Abstraction Architecture (bath/coefficient code needs
to live in the backend-agnostic layer). Feeds Physics Validation & Test
Infrastructure (new bath kinds need oracle coverage) and Public API
Evolution (Bath::Kind enum activation).

Risks

Precision assumptions embedded in the current single-precision-fitted
coefficient path could produce silently degraded double-precision results if
double-precision activation isn't gated on this redesign landing first.