15 · The fourth cell: an instrument for blame

Chapter 4 gave you NEC’s bet — piecewise-sinusoidal currents, matched at segment centres. Chapter 5 gave you the B-spline alternative — a different basis, tested by integration (Galerkin) instead of point-sampling. When the two solvers disagreed on an antenna, which ingredient did you blame?

You couldn’t. The two solvers differed in two things at once — basis and testing — so every discrepancy was a confounded experiment. For years this tree’s issues filed such gaps under “basis effect” for lack of a way to know better. momwire#182 built the missing instrument: a sinusoidal-Galerkin solver — the same three-term basis and junction tables as chapter 4, but with test rows formed by integrating the field against each basis function, the way chapter 5 does. The basis × testing matrix got its fourth cell, and any gap could suddenly be attributed: persists under both testings → basis; vanishes → testing.

	point-matched	Galerkin
sinusoidal	SinusoidalSolver (ch. 4)	SinusoidalGalerkinSolver — the new cell
B-spline	—	BSplineSolver (ch. 5)

The free oracle

A correct Galerkin fill has a property collocation cannot fake: the matrix is symmetric by construction — reciprocity, to machine precision. That made ‖Z − Zᵀ‖/‖Z‖ < 1e-10 the gate for the whole build: sloppy quadrature breaks it, and no amount of plausible-looking impedance can restore it. The hard 20% was where chapter 6 said it would be — the near-singular integrals. The kernel’s endpoint terms put a spike of width ~a (the wire radius) into every node-sharing test integral; a graded rule that shrinks panels dyadically toward the endpoints, with the panel count derived from the geometry’s own a/h, met the gate at default settings at a fraction of brute-force cost.

Two instrument-grade side-findings from that gate work, both worth keeping:

• Symmetry is a uniform-radius property. The thin-wire kernel regularizes with the observer’s radius, so mixed per-wire radii make the kernel itself asymmetric — no testing scheme can restore a symmetry the kernel doesn’t have. Where radii mix, the oracle is brute-force quadrature refinement, not reciprocity.
• A drifting feed fakes non-convergence. The delta-gap snaps to the nearest segment centre, so a feed that isn’t a segment centre at every mesh density moves by up to h/2 as you refine — an O(h) perturbation of the problem that can make a perfectly good solver appear to walk away from its own converged answer. Every convergence sweep in this act pins the feed to a point that is a segment centre at every N.

And one epistemic one: “the converged impedance” does not exist. The delta-gap’s width is the segment length, so refining the mesh shrinks the source, and the gap reactance drifts logarithmically without limit. Every error figure below is stated against a named reference — fine solves plus Richardson extrapolations — and gated on the verdict that survives the whole family.

What the instrument read

The full sweep — the antennaknobs#521 residue cluster, the #478 near-open designs, eleven designs by four solver columns — lives in the instrument report, tables generated verbatim by a checked-in harness. The taxonomy it produced:

Testing effects — the big class, and the surprise. The entire T/X-junction cluster (hentenna, hourglass, their arrays, the discone) is testing-limited: point-matched↔B-spline gaps of 1.2–23% collapse to 0.01–0.33% when only the testing changes. On the hentenna, Galerkin at 119 segments holds an answer collocation hasn’t reached at 1835. This also closed an arbitration open since momwire PR #45: the hentenna’s anti-convergent impedance columns — the sinusoidal solver and PyNEC in lockstep, fitted exponent ≈ −0.5 — were the point matching, not the basis.

Feed-model effects — the axis nobody knew was there. The near-open, high-Q designs (#478’s lazy-H, the V-beam) barely respond to the testing swap; what collapses their residual — 55–74×, down to ~0.02% — is matching the source model (a point delta-gap versus a segment-wide one). On a clean dipole the matched-feed gap between the sinusoidal-Galerkin and B-spline solvers is 4×10⁻⁸: with basis and testing and feed matched, there is essentially nothing left.

Basis effects — real, small, and rate-only. The high-Q star and L-shape geometries are genuinely basis-limited: swapping the testing buys ~1.01×, swapping the basis buys ~1.4×. But refined, all schemes agree to ≤4×10⁻⁴ — so a “basis effect” in this tree is a statement about coarse-mesh cost, never about what the solvers converge to.

Not attributable — and saying so is the result. The helix’s builder emits one segment per winding chord and ignores the mesh knob entirely, so its convergence “ladder” re-solved one identical mesh at every rung. Until it has a real mesh parameter, no attribution is honest, and the report says exactly that.

Instrument artifacts — the things you must never re-attribute to a solver. The bridged-gap port oracle carries a ~1.4% residue that is its own linear-in-δ extrapolation tail (halving the gap ladder halves it, for both port implementations); the fill’s 8×10⁻¹² reciprocity floor amplifies to ~10⁻⁸ visibility in port networks; and the two experiment-design traps above (drifting feeds, phantom “converged” references) will cheerfully manufacture solver defects out of nothing if ignored.

Inherited defects — one, pinned: a wire ending in a finite (Fresnel or Sommerfeld) ground is unsound on both sinusoidal solvers, because the ground-connected basis completes the end current with an exact mirror image that only a PEC plane actually provides. Recorded with a test, not patched.

The port detour: refuted, then lifted

The plan’s junction-port milestone produced the best kind of failure first. The obvious construction — a port basis column carrying unit current into a node — was built, measured, and refused: a current that terminates at a point deposits a charge there, and this family’s field kernel prices that charge’s self-energy at the wire radius, Z_pp ≈ 1/(jω·4πε·a) — matched to 7% with no fitted constant, immovable under mesh refinement. The mechanism was then dissolved by a measurement on the B-spline side: a Lagrange-style port’s current leaves through an ideal unmodelled lead, so the lumped node charge doesn’t belong in the port physics at all. Holding it outside the reaction integral — plus a second, charge-free through-port formulation for two-terminal cases — lifted the refusal. The decisive check isn’t the oracle: it’s entrywise agreement between two port implementations sharing no basis, no testing, and no port algebra — 3×10⁻⁵, self-terms included. The payoff downstream: PortAtEnd designs in antennaknobs — the Sterba curtain fed through a modelled transmission line, chapter for chapter the reason ports had to exist — now solve on a second basis, agreeing with the reference to 0.28% in impedance and 0.001 dB in gain.

What changed, practically

• SinusoidalGalerkinSolver sits in the tree beside chapter 4’s solver — same constructor, all three grounds of Act III by evaluator reuse, junction ports in free space — and is wired into the simulator as the sinusoidal-galerkin backend.
• The point-matched solver is deliberately untouched: its 0.1–0.3 Ω tracking of NEC is collocation heritage — shared basis, shared point matching — and it keeps the NEC-parity-probe role. The new cell will never track NEC that tightly, by design.
• The milestone’s Python-first rule has since paid out and retired (momwire#194): a profile indicted the far fill for ~85 % of wall-clock and the O(N²·n_qp) scratch that OOM-killed census rungs past ~2000 segments, so the fill is now blocked on the numpy path and served by a fused C++ kernel — field evaluation and test reduction in one pass, the same math twice — for 7–12× end-to-end. A 4000-segment Galerkin solve runs in under 4 GiB; the numpy fill remains the reference and the oracle the kernel is gated against.
• Every claim above is pinned by a test or a checked-in reproduction script; the follow-ups the work generated are momwire#191 (ports over a PEC ground) and momwire#192 (the point-gap feed model as an option — it re-baselines pinned numbers, so it needs its own gate).

The moral is the same one chapter 7 ended on, one level up: a disagreement between two solvers is not information until you can vary one ingredient at a time. Building the fourth cell cost a solver; it bought a taxonomy.