Date: 2026-08-06 Purpose: Why a single-node gauge pin ("bearing") leaves the Jacobian of an h-φ system near-singular, why that paralyzes Newton while Picard is immune, and what the remedies are. Module: src/fem/kernel (cl_FEM_Bearing, cl_FEM_DofMgr_BearingData), with consequences for the nonlinear Controller (see nonlinear_controller_theory.md)

1. The gauge freedom and the bearing

In the h-φ formulation the magnetic scalar potential φ in the air region enters the physics only through ∇φ: the solution is determined up to an additive constant per connected air component. BELFEM pins that constant strongly at a single mesh vertex — the bearing (bearing { nodes : <vertex id> ; } in the input deck), implemented as a Dirichlet fix of the node's φ dof (Bearing::impose_dirichlet). An id that does not resolve is no longer silent: an empty bearing is legitimate on a worker rank, since linking is master-only, but on the master it raises BELFEM_ERROR (cl_FEM_Bearing.cpp:104-120). When the id resolved to nothing at all it names the deck's nodes list without echoing the id; when the point exists but carries no dof of the requested field it names the point and node. The old silent return is what let a dead bearing run for months (DR-126).

The pin is correct: in exact arithmetic one point condition removes the constant nullspace. The problem is its strength. A point Dirichlet is a measure-zero constraint for a 3-D Laplacian: the discrete mode

φ = c everywhere, forced to 0 at the bearing node

costs only O(h)·c² of energy, where h is the mesh size at the bearing (often the coarsest region — the far boundary). The stiffness/Jacobian therefore keeps one near-zero eigenvalue whose eigenvector is essentially the constant-φ mode, and the condition number grows without bound under mesh refinement or coefficient contrast.

2. Why Newton dies and Picard does not

Newton solves J·Δx = r for a correction. In exact arithmetic r is orthogonal to the nullspace; in floating point it carries a roundoff-level component along the mode, which the solve amplifies by 1/λ_min. The direct solvers do not fail — their tiny-pivot replacement (STRUMPACK) or pivot patching (MUMPS) returns SUCCESS — they return a Δx dominated by a huge, physically meaningless constant-φ content. Observed on the Garber CORC deck (2026-08-06, BELFEM_PROBE_NEWTON_DX):

‖Δx‖ ≈ 102 for ‖r‖ ≈ 5.7·10⁻⁶ (amplification ~2·10⁷);
Δx components ≈ 0.43, near-identical across tens of thousands of φ nodes (the constant mode), largest entries in the finest-meshed region;
the residual frozen to within ±0.01 dB while ω swept 0.5 → 0.004 — the mode is invisible to the lagged operator, so the line search can neither use nor reject the step, and the ill-conditioning destroys the useful part of Δx as well (the whole factorization loses its digits, not just the null component).

The result is the "Newton does nothing" signature: bit-flat accepted iterates at any relaxation, until the stagnation guard falls back to Picard. Note the accepted iterates do drift along the mode — harmless if the mode is pure gauge.

Picard solves A·x_G = b with the physical right-hand side. The solution's component along the near-null mode is set once by the factorization (an arbitrary but stable constant offset) and cancels from every physical quantity; the fixed-point iteration converges as if the mode did not exist.

This behavior is independent of thin-shell edge fusing and identical on all branches; it is a property of the gauge pin, not of any particular formulation detail.

3. What does NOT work: surface gauging with net transport current

The tempting fix — Dirichlet φ = 0 on the outer boundary sidesets — is only admissible for problems with zero net current (background-field/screening problems). With net transport current I, the far field is H_θ = I/(2πr) and any loop around the conductor carries ∮H·dl = I: φ is multivalued out there (jump I across the cut, φ ≈ −I·θ/2π along the boundary). A constant-φ surface condition clamps that circulation to zero and fights the current constraint. The point bearing exists precisely because it pins only the constant and makes no field statement.

4. Remedies

Remedy	Status	Notes
Run pure Picard (algorithm : Picard)	RECOMMENDED for net-current h-φ decks (2026-08-06)	Immune to the mode; the hybrid controller's Newton stage buys nothing when every Newton solve is blind
Mean-value gauge row (Σφ = 0 as one dense constraint row, replacing the point pin)	future work	Pins the same constant with O(1) stiffness; same infrastructure class as the free-cut λ rows; physically identical to the bearing
Deflation (project the known constant mode out of Δx after the Newton solve)	future work	Cheap, but the factorization has already lost precision at solve time — inferior to fixing the operator
Absolute magnetic tolerance (absolute tolerance; opt-in — consumed when set, disabled at 0.0 by default, cl_FEM_Controller.hpp:203-211)	partly landed	Would stop iterating when ‖Ax−b‖ reaches machine floor; addresses a second, benign Newton-flat regime (solver returns Δx = 0 on machine-epsilon residuals), not the near-null mode itself

Caveat (open): the observed mode has not been fully discriminated between (a) the global constant against the point pin, and (b) a bore mode — the air inside a CORC former couples to the pinned outer air only through the narrow tape gaps, so its own constant is nearly floating regardless of the bearing. Both produce the same Newton signature and both are gauge-like for Picard; they differ in remedy ((a) mean-value row; (b) a second bearing inside the bore, at an error of the order of the neglected gap coupling). A one-shot dump of the full Δx mode shape (unimodal vs bimodal in φ) decides; the probe used on 2026-08-06 has been removed, so it would have to be re-instrumented in the Newton branch of cl_FEM_DofMgr_SolverData.cpp.

5. Recognizing the mode

The signature, should it need re-confirming (the diagnostic probe used in the 2026-08-06 investigation has been removed again): print ‖Δx‖ of the Newton solve next to ‖r‖. A ‖Δx‖ orders of magnitude above ‖r‖ (observed: 2·10⁷×), carried by near-identical values on the φ dofs of most of the mesh, with the reported residual frozen under any relaxation sweep, is this mode. Newton iterates in that state are accepted-but-inert; Picard from the same state converges normally.

6. Literature

Messe et al. 2023 — h-φ formulation and cut constraints; the gauge constant and its pinning are implicit in the formulation.
Alves et al. 2022b — cohomology cuts; the multivalued φ around net-current conductors that rules out surface gauging (Section 3 above).
Bathe, §8.2.4 — conditioning of near-singular systems and the loss of accuracy of direct solves.