# eccentric_coax

Family 12: the capacitance of a coaxial line whose inner conductor sits off centre.

Exact anchor everywhere in the space: C' = 2 pi eps / arccosh((a^2 + b^2 - e^2) / (2 a b)) for radii a, b and offset e.
The peak field on the inner conductor has an exact form too (bipolar coordinates), the number a designer actually needs:
how much closer to breakdown an off-centre conductor is. Half the section is solved as a slice; the mirror plane and the
ends carry no flux.

## Parameters (nondimensional)

| parameter | range |
|---|---|
| `a_over_b` | 0.1 - 0.6 |
| `e_over_gap` | 0.0 - 0.7 |

## Answers and their error bands

| answer | error band (median / 95 % / max) | similarity x2 / x4 | CalculiX (same mesh) |
|---|---|---|---|
| `c_star` | 0.0019 % / 0.021 % / 0.021 % | max 0.00069 % | max - |
| `emax_star` | 0.077 % / 0.18 % / 0.18 % | max 0.033 % | max - |

## Checks

- points: 200 solved, 0 failed
- three-mesh studies: 40 (every 5th point); the band of each point is 1.25 x the largest standard-mesh error among its 3 nearest studies
- symmetry check (max): 0 %
- neighbour check: flagged 0, re-solved on a finer mesh 0, unresolved 0
- direction checks: c_star vs a_over_b ok (rho 0.981), c_star vs e_over_gap ok (rho 0.17), emax_star vs e_over_gap ok (rho 0.678)

## Against hand theory

| ratio | median | 95 % | max |
|---|---|---|---|
| `c_over_exact` | 1 | 1 | 1 |
| `emax_over_exact` | 1 | 1 | 1.001 |

## Columns of points.csv

parameters, the answers, `<answer>_band` (relative), the hand-theory ratios, `mirror_err`, `equilibrium_err` (where a single reaction exists), `<answer>_loo_err` / `_neighbour_flag` (leave-one-out prediction from all other points), `<answer>_recheck_diff` / `_flag_resolved` (finer re-solve of a flagged point), `has_gci` / `has_similarity` / `has_calculix`, `nodes`, `solve_s`, `studio` (version).

Solved with NovaSolver Studio through the same commands an AI client uses over MCP; the first section says which solver and mesh. The claim is verification (the equations are solved within the band), not validation against tests.

## License

This data is licensed under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/
Credit: NovaSolver, https://novasolver.jp/studio/verified/
Provided as is, without warranty. Design decisions remain the user's responsibility.