Module continuation_path 

Reactive three-leg continuation for a non-finite K≥2 SAE outer seed.

Three endpoint legs coupled by one scalar parameter

One objective evaluation installs and solves all three homotopy legs at the same s:

1. log-ρ — the objective’s legal upper-box endpoint down to its literal target. At large penalty strength the penalized Hessian dominates the likelihood Hessian.
2. Assignment temperature τ — diffuse softmax / IBP relaxation (high τ) sharpened toward the objective’s literal target. High τ makes the assignment map smooth and far from the combinatorial argmax cliff.
3. Isometry weights — zero entry weights ramped to the objective’s literal per-penalty vector. A loose gauge leaves the decoder free to find a good fit before the gauge pins it.

ContinuationPath advances all three in lockstep along a single scalar path parameter s ∈ [1 → 0]. s = 1 is the entry regime: legal upper-box ρ, high τ, and loose isometry. s = 0 is the real objective: target ρ*, sharp τ, tight isometry. The path walks s monotonically down, advancing the three literal waypoint values together.

Entry is always the heavy-smoothing regime

There is no “solve cold at the real objective” entry. The only entry is s = 1, where every leg is at its smoothing extreme. A K≥2 SAE joint fit either reaches the literal target through accepted coupled waypoints or returns a typed refusal; it never fabricates arrival.

The accepted path is monotone; failed attempts refine their distance

If a downward step’s inner solve struggles, the last accepted waypoint is retained and only the next attempted distance is halved. No independent waypoint count, wall-clock deadline, or evaluation ceiling can fabricate an arrival. A successful solve at the literal s = 0 target is the only [ContinuationStep::Arrived] value. If repeated refinement can no longer produce a strictly smaller representable waypoint, ContinuationPath::step returns a typed non-convergence error.

Full objective checkpoints, rather than coefficient-only fallback state, make each accepted waypoint independent of mutations from refused trials.

Structs

ContinuationPath

Coupled continuation path. Owns the three endpoint legs and the scalar path parameter s, and drives the K≥2 SAE joint fit down the coupled homotopy. Entry is always the legal heavy endpoint at s = 1, and only a solved literal target can produce arrival.

ContinuationScalarContract

Objective-owned scalar endpoints for reactive domain entry. The target is the literal state of the real objective; the entry is a smoother state derived by that objective from its own routing and penalty geometry.

ContinuationScalarState

Literal scalar state of an objective at one coupled-domain waypoint.