Coincident and near-coincident roots: decide multiplicity structurally, not with fixed thresholds

[siddhss5]

Umbrella for one root cause diagnosed independently in four places. Two genuine IK branches coincide or nearly coincide, and the pipeline resolves that with a fixed constant or an arbitrary basis instead of from the problem's structure. The result is silently missing branches (never wrong ones: FK certification holds everywhere).

Where	Stage	Current handling	Symptom
#595 general-6R RR	eigensolve	reads an arbitrary basis vector of a 2-D eigenspace	1–2 branches lost at a shared linearity-joint value, finite or at π; near-repeats lost for δ ≲ 5e-9
#597, #566 (#559) spherical SP3/SP4 near a fold	final dedup	merges anything within subproblem_dedup = 1e-3 rad, and the survivor is chosen by 1e-16 residual noise	8 → 4 solutions within ~5e-4 rad of a fold; seed recovery is a coin flip
#598 (part 2) SRS SP4 near full extension	subproblem gate	rejects |ratio| > 1 + 1e-9	a near-tangent double root, pushed past 1 by approximation drift, gives no candidates

The principle

Multiplicity is a property of the problem at the root. Decide it from the local conditioning, and resolve it structurally:

1. Root level, detect. Use a scale-relative gap test: singular-value gap of M(x) for RR, discriminant relative to its conditioning for SP3/SP4. No absolute constants.

2. Root level, resolve.

   • k-fold eigenspace → split it with the k×k shift eigenproblem (Manocha–Canny / Möller–Stetter). A scratch prototype recovered 12 of 12 branches; see General 6R: repeated eigenvalues (shared linearity-joint value) lose branches on both backends #595.
   • Near-tangent subproblem → keep both roots when they are separated; clamp to the double root when drift pushes the ratio past tangency within a conditioning-scaled band (see seven_r.srs_polished: no IK for a reachable j2s7 pose (pre-existing, fails on main) #598). The prototype gave 0 failures in 3000 seeds, against 2 on main.

3. Solution level: dedup merges only the same root. Merge two candidates only if they are the same solution, for example:

   • the midpoint also closes FK (twins in spherical_two_intersecting: cached Hypothesis counterexample not recovered (pre-existing, fails on main) #597: endpoints 1e-16, midpoint 2e-8), or
   • the separation is below a radius scaled by σ_min(J).

   Never merge on a fixed 1e-3 alone. When a merge does happen, pick the representative deterministically, not by residual noise.

This belongs in the shared subproblem, eigensolve and dedup layers of both backends. The C++ side carries the same 1e-3 default (subproblems.hpp:26, rescue.hpp:62).

Public-contract decision needed first

Today subproblem_dedup = 1e-3 effectively means "solutions closer than 1 mrad are one solution". Under this principle, poses near folds return 8 distinct solutions instead of 4. Either:

• (A) adopt the "same root" semantics (more solutions near folds, and seed recovery becomes reliable), or
• (B) keep 1e-3 as a documented "physically indistinguishable" radius, choose the representative deterministically, and set seed-recovery tolerances ≥ 1e-3.

Related

• Harden empirical branch-golden generation and fixture trust invariants #582: the branch oracle dedups at the same 1e-3, so goldens cannot catch this.
• General 6R: multi-chart reconstruction and end-to-end representation selection #572 (multi-chart reconstruction).
• Native general_6r_tensor_solve segfaults on JACO 2 with linearity joint 0 #599 (native crash when re-solving under another linearity choice, which blocks the fallback route).
• Test strategies should screen on σ_min(J) rather than per-joint sin filters, which cannot see offset-elbow or shoulder folds.

[siddhss5]

Decision: (A). A solution is a distinct exact root. Dedup merges two candidates only when they are the same root, never merely because they are closer than a fixed radius. Near folds, callers get every distinct branch (for example 8 rather than 4). subproblem_dedup stops meaning 'physically indistinguishable'. Its remaining role, if any, must be documented with the change.