Resolves OPEN ITEM 5.2-A and OPEN ITEM 6.1-A.#8
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Two open item blockers resolved and written in. Here's what changed and why: Section 5.2 title corrected to "Near-Singular Quadrature (Exact Kernel)" and the entire treatment rewritten. The key content is from Fikioris paper. The straight-wire K_exact closes to a clean formula, the near-singular subtraction K_near = 1/√(a^2+(s-s')^2) has a closed-form double integral I_near = 2[delta sinh^-1(delta/a) - √(a^2+delta^2) + a], the smooth remainder K_smooth → -jk at s=s' so standard quadrature handles it, and the thin-wire limit of I_near recovers the Section 5.3 formula exactly. This hopefully gives a built-in consistency check for the implementation. Basically this makes it have the correct conceptual framing: exact kernel has no true singularity. Self-impedance requires near-singular quadrature, not extraction. And K_ex is bounded everywhere. Section 6.1 is about near-neighbor treatment written out with local coordinates (epsilon, epsilon') around the shared endpoint, the collinear closed-form integral, the bent-junction case with theta_bend, the T2 near-singular contribution, and a concrete definition of what near_singular_distance_ratio = 3.0 actually means in arc-length terms. References are Fikioris & Wu (2001) re-cited accurately (I downloaded it from Antenna and Propagation Society membership at ieeexplore) (confirms bounded kernel, not the source of extraction formulas), and a follow-on 2020 lossy medium paper added with a note that it's the foundation for the deferred Sommerfeld work. Found this connection through ieeexplore as well. Remaining blocker before Phase 2 implementation: OPEN ITEM 7.4-A (perpendicular frame computation for each curve type).
Resolves OPEN ITEM 7.4-A. All three math blockers now closed.
Key results:
- Prove phi is a dummy variable in the azimuthal integral: any
orthonormal̂ produces identical K_exact.
Frame continuity between segments not required.
- Single three-step construction for all curve types:
1. get a standard basis vector least aligned with tau
2. use Gram-Schmidt
3. profit!
Minimum-component choice guarantees denominator is greater
than or equal to square root of two thirds,
no special-case fallback needed.
- Verified for straight, arc, and helix (numerical, no closed form
needed).
- Frenet-Serret is equivalent but not used: undefined for
straight segments (κ = 0), requires curvature not otherwise
needed. Might need this later.
- evaluate_exact_kernel requires no segment-type branching;
curve specificity lives entirely in the upstream computation.
- Fix stale cross-reference in Section 5.2 (was: OPEN ITEM 7.4-A,
now: Section 7.4).
Remaining items before Phase 2 implementation PR:
- design.md OPEN ITEM 4.2-A: cross-wire junction treatment decision
- design.md Open Item 5: unsafe write pattern Rust review
sconklin
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May 21, 2026
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Thanks for this - good progress!
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Two open item blockers resolved and written in. Here's what changed and why:
Section 5.2 title corrected to "Near-Singular Quadrature (Exact Kernel)" and the entire treatment rewritten. The key content is from Fikioris paper. The straight-wire K_exact closes to a clean formula, the near-singular subtraction K_near = 1/√(a^2+(s-s')^2) has a closed-form double integral I_near = 2[delta sinh^-1(delta/a) - √(a^2+delta^2) + a], the smooth remainder K_smooth → -jk at s=s' so standard quadrature handles it, and the thin-wire limit of I_near recovers the Section 5.3 formula exactly. This hopefully gives a built-in consistency check for the implementation. Basically this makes it have the correct conceptual framing: exact kernel has no true singularity. Self-impedance requires near-singular quadrature, not extraction. And K_ex is bounded everywhere.
Section 6.1 is about near-neighbor treatment written out with local coordinates (epsilon, epsilon') around the shared endpoint, the collinear closed-form integral, the bent-junction case with theta_bend, the T2 near-singular contribution, and a concrete definition of what near_singular_distance_ratio = 3.0 actually means in arc-length terms.
References are Fikioris & Wu (2001) re-cited accurately (I downloaded it from Antenna and Propagation Society membership at ieeexplore) (confirms bounded kernel, not the source of extraction formulas), and a follow-on 2020 lossy medium paper added with a note that it's the foundation for the deferred Sommerfeld work. Found this connection through ieeexplore as well.
Remaining blocker before Phase 2 implementation: OPEN ITEM 7.4-A (perpendicular frame computation for each curve type).