MIME-VER-112 — Earth-Field Superposition Exactness

Date: 2026-04-30 Node under test: mime.nodes.actuation.permanent_magnet.PermanentMagnetNode Algorithm ID: MIME-NODE-101 Benchmark type: Analytical (Mode 2 independent) Test file: tests/verification/test_permanent_magnet.py::test_ver112_earth_field_superposition Acceptance: $\mathbf{B}{\text{total, with earth}} - \mathbf{B}{\text{total, no earth}} \equiv \mathbf{B}{\text{earth}}$ (atol $10^{-12}$); $\mathbf{B}{\text{total}}(+\mathbf{B}{\text{earth}}) - \mathbf{B}{\text{total}}(-\mathbf{B}{\text{earth}}) \equiv 2\mathbf{B}{\text{earth}}$ (atol $10^{-12}$).

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Goal

Verify that the node’s Earth-field background superposes exactly onto the magnet’s field — no scaling, no rotation, no leakage into the $\nabla\mathbf{B}$ output (the Earth field is uniform, so its gradient is zero).

In hardware-bench experiments, the static Earth field ($\sim 50,\mu\text{T}$) is comparable in magnitude to a small distant permanent magnet’s stray field at the experimental working distance, and the response of a magnetised UMR depends on the vector sum of the two. This benchmark guarantees that sum is computed correctly.

Configuration

Parameter	Value
dipole_moment_a_m2	$1.0$ A·m²
magnet_radius_m	$10^{-3}$ m
magnet_length_m	$2 \times 10^{-3}$ m
field_model	point_dipole
Magnet pose	identity quaternion at origin
Target	$(0, 0, 10,R_{\text{magnet}})$
Test Earth field	$\mathbf{B}_{\text{earth}} = (2 \times 10^{-5}, 1 \times 10^{-5}, -4.5 \times 10^{-5})$ T (asymmetric — all three components nonzero)
JAX precision	x64

Method

Three node instances are constructed, identical in every parameter except the Earth field:

1. node_no_earth — earth_field_world_t = (0, 0, 0).

2. node_with_earth — earth_field_world_t = +B_earth.

3. node_neg_earth — earth_field_world_t = -B_earth.

Each evaluates field_vector at the same target. The differences are checked element-wise against the analytical expectation:

$$ \mathbf{B}{\text{pos}} - \mathbf{B}{\text{no}} = \mathbf{B}_{\text{earth}}, $$

$$ \mathbf{B}{\text{pos}} - \mathbf{B}{\text{neg}} = 2\mathbf{B}_{\text{earth}}. $$

Both checks use absolute tolerance $10^{-12}$ T (a few orders of magnitude tighter than the float64 floor for these magnitudes).

Results

Both checks pass to absolute tolerance $10^{-12}$ T. The Earth field adds linearly without scaling or rotation; the magnet contribution is unchanged when the Earth field is flipped.

Verdict: PASS

The node’s Earth-field treatment is an exact additive superposition. Two corollaries:

• A consumer can recover the magnet-only field by subtracting the configured earth_field_world_t from the node’s output. (See test_grad_wrt_amplitude in the same test file for an example.)

• The amplitude scale acts on the magnet contribution only — it does not multiply the Earth field. This is the documented behaviour in _b_total_world.

Related notes

• The Earth field is uniform; its gradient is therefore zero, and the node’s field_gradient output is unaffected by the Earth-field background. This is enforced automatically because the _b_total_world function on which jax.jacrev operates includes the Earth field as a constant, whose derivative w.r.t. the target position is zero. (No separate test is required — it is a derivative-of-a-constant identity.)

• Default Earth field in the constructor is $(2 \times 10^{-5}, 0, -4.5 \times 10^{-5})$ T — rough mid-latitude magnitudes. Users can override or zero this for clean magnet-only verification (as VER-110 and VER-111 do).