# Toroidal Field–Vector Equilibrium Coupling Reference V1: measured results

**Decision: retain as an optional geometric and visualization primitive. Production integration is not recommended.**

The bounded operators replay exactly in this JavaScript runtime. The FCC shell provides deterministic addresses and an exact local cuboctahedron. These are useful implementation properties, but the comparisons do not demonstrate a reusable advantage attributable to binding toroidal modes to the Vector Equilibrium. No production component or authoritative contract is used.

Reproduce with `npm test` and `npm run measure` from this standalone directory. [The methodology](METHODOLOGY.md) defines units, equations, limitations and control fairness. [Raw results](reference-results.json) contain all finite differences, quadrature levels, candidate metrics, ledgers and scale histories. Timing values are machine observations and change between runs.

## Seven-scenario comparison

| Scenario | Anchors | Field RMS | Divergence RMS | Curl RMS | Held-out relative RMS | Anchor visits / FCC pass | Numeric payload bytes |
| --- | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| One toroidal anchor | 1 | 0.2027 | 1.355e-6 | 1.0064 | 0.5144 | 171 | 21784 |
| Sparse interacting anchors | 5 | 0.3915 | 2.483e-6 | 1.7195 | 0.5543 | 855 | 22848 |
| Every-vertex torus stress control | 171 | 2.4295 | 5.642e-6 | 3.6905 | 0.3658 | 29241 | 60816 |
| Spherical radial flow | 1 | 0.2247 | 1.0960 | 7.989e-7 | 0.3886 | 171 | 22536 |
| Regularized dipole flow | 1 | 0.0781 | 0.4999 | 0.0284 | 0.8667 | 171 | 22544 |
| Ordinary vortex flow | 1 | 0.2114 | 7.801e-7 | 0.8368 | 0.3854 | 171 | 22520 |
| Sparse randomized axes | 5 | 0.4187 | 3.035e-6 | 1.8939 | 0.5733 | 855 | 23272 |

All seven cases passed same-state, reconstruction and evolved-anchor replay: **true**. Each FCC pass samples 171 bounded cells. Sparse modes cost 5 times the one-anchor evaluator visits; every-vertex activation costs 171 times. The experiment operator never allocates torus meshes. Rendering a naive 32 × 16 torus at each stress anchor would add 87552 diagnostic vertices (2101248 position payload bytes, before indices or object overhead).

The seven cases share authored amplitude and support-radius settings, not equal integrated energy or occupied support volume. A smaller vector magnitude alone is not better sampling. Relative interpolation error divides held-out vector RMS error by that case’s reference field RMS. The dipole is softened and compactly cut off; its nonzero divergence and curl are measured features of that chosen control.

## Flux, circulation and accounted loss

| Scenario | Outward boundary flux | Inward boundary flux | Net flux | Integrated divergence | Gauss residual | Net-flux change (6→12) |
| --- | ---: | ---: | ---: | ---: | ---: | ---: |
| One toroidal anchor | 1.5143 | 1.5144 | -1.341e-4 | -2.807e-7 | -1.338e-4 | 0.0016 |
| Sparse interacting anchors | 2.8675 | 2.8677 | -1.615e-4 | -1.789e-7 | -1.613e-4 | 0.0030 |
| Every-vertex torus stress control | 5.7044 | 5.7059 | -0.0016 | -6.028e-7 | -0.0016 | 0.0103 |
| Spherical radial flow | 10.5645 | 0.0000 | 10.5645 | 10.5836 | -0.0191 | 0.0806 |
| Regularized dipole flow | 0.1040 | 0.1139 | -0.0099 | -0.0110 | 0.0011 | 8.814e-5 |
| Ordinary vortex flow | 1.5671 | 1.5671 | 6.571e-7 | 2.084e-10 | 6.569e-7 | 2.053e-6 |
| Sparse randomized axes | 4.4735 | 4.4756 | -0.0021 | -8.134e-7 | -0.0021 | 0.0036 |

For the one-anchor torus, 192-segment toroidal circulation is 6.3343 and poloidal circulation is 0.3592. Their 96→192 changes are 0.0101 and 4.907e-5. This is line-quadrature convergence; there is no fluid equation establishing material-loop conservation. The largest absolute activation-proxy accounting residual in the seven short evolution runs is 0.0000. Damping loss and coupling change are recorded separately.

The actual fixed-loop toroidal circulation also changes from 6.3343 to 6.2437 after 12 evolution steps. Its damping contribution is -0.0906. Phase/axis/time change, coupling-activation change, damping change and arithmetic residual are separately recorded for both loops in all seven cases. This attribution uses a phase-only state and an undamped evolution replay; it accounts observed changes without imposing a conservation law.

Cube midpoint quadrature can miss structure at coarse resolution. The raw report exposes residuals and resolution changes rather than interpreting numerical residuals as physical sources. The analytic neutral toroidal and poloidal terms have zero divergence; their finite-difference tests use interior points and a step-refinement check. Radial source/sink controls have deliberately accounted local divergence. A compact source-shaped field has compensating divergence near its support edge, so it is not a global matter source.

## Composition, perturbations and scale

Over 80 steps of 0.04, disordered sparse phase order changes from 0.0600 to 0.1824; without coupling it stays 0.0600. The same graph with an ordinary vortex reaches 0.1824, and translating anchors off FCC reaches 0.1824. The synchronization law is shared, so its result cannot support a toroidal or FCC-specific advantage. Randomized axes are a separate orientation ablation: they change sampled geometry without creating a different activation law.

| Geometric scale | Seeded active anchors | Active anchors at final time | First observed neighbor activation | Final field RMS | Proxy ledger residual |
| --- | ---: | ---: | ---: | ---: | ---: |
| 0.5 | 1 | 5 | 0.4000 | 0.0989 | 0.0000 |
| 1 | 1 | 5 | 0.4000 | 0.0989 | 0.0000 |
| 2 | 1 | 5 | 0.4000 | 0.0989 | 0.0000 |
| 4 | 1 | 5 | 0.4000 | 0.0989 | 0.0000 |

These histories actually step the activation/phase rule over time, beginning from one active mode and dormant neighbors. Their scale invariance follows from scaling graph distances with geometry. This supports deterministic authored graph diffusion over the measured horizon; it does not establish stable object-, organism- or world-scale field propagation. An all-dormant neutral test remains exactly zero with null axes.

A separate geometric scale study at scales 0.5, 1, 2, 4 preserves the same FCC cell count and vector values at corresponding positions; maximum covariance error is 0.0000. It must not be confused with temporal propagation. Halving the FCC spacing at fixed world geometry increases cells from 171 to 1099 and changes held-out absolute error from 0.2013 to 0.0864.

## Form and optical candidates

| Scenario | Final displacement RMS | Final roughness RMS | Uniform intensity RMSE | Adaptive intensity RMSE | Uniform evals | Adaptive pilot + retained evals |
| --- | ---: | ---: | ---: | ---: | ---: | ---: |
| One toroidal anchor | 0.0575 | 0.1072 | 0.0181 | 0.0035 | 128 | 1328 |
| Sparse interacting anchors | 0.1162 | 0.2545 | 0.0798 | 0.0658 | 128 | 1328 |
| Every-vertex torus stress control | 0.3863 | 0.2683 | 0.1135 | 0.0567 | 128 | 1328 |
| Spherical radial flow | 0.2757 | 1.710e-16 | 6.245e-17 | 3.469e-17 | 128 | 1328 |
| Regularized dipole flow | 0.0056 | 0.0046 | 1.221e-4 | 3.927e-5 | 128 | 1328 |
| Ordinary vortex flow | 0.0537 | 8.868e-4 | 0.0105 | 0.0014 | 128 | 1328 |
| Sparse randomized axes | 0.2293 | 0.2992 | 0.0980 | 0.0816 | 128 | 1328 |

The observer receiver explicitly addresses the full 4π solid angle. Growth is a bounded authored field-to-displacement transform; “roughness” is a diagnostic, not an optimization target or biological plausibility score. Optical samples contain synthetic intensities, RGB values, directions, solid angles and IDs. Adaptive projection keeps a bounded sample count but spends extra pilot evaluations; any error reduction in this table is not an equal-operation performance win. Fixed held-out directions assess reconstruction, not visual similarity alone.

## Integration gate and nonclaims

Integration would require an independently useful task objective, equal-budget controls, a demonstrated toroidal/FCC-specific gain, longer and larger interaction studies, and a reviewed bounded shared contract. This V1 supplies no such result. Keep it isolated and optional. It does not claim complete electromagnetism, cosmological proof, consciousness physics, physical zero-point extraction, or recovery of Robert Edward Grant’s exact construction. A raster does not reveal its exact authored generator; the reconstruction in the laboratory is explicitly a candidate.

Measurement runtime: 3956.3657 ms on darwin/arm64, v24.10.0.