Exact transverse-symmetry acceleration

Decision

Until 2026-08-16, reflection symmetry reduced only the topology variables and their gradients. Every Maxwell solve still used the complete x/y/z domain. The solver now has an opt-in exact y/z parity backend for the centered, z-polarized atomic dipole. It reduces the fine-grid linear system from 68,640 to 16,960 unknowns (4.047x) and the periodic-lead surface plane from 1,716 to 424 unknowns.

This is a new transverse boundary discretization, not a reinterpretation of an old result. A checkpoint transfer carries only raw and the rebuilt density; it never carries a pole or Q. Spectral, PML, and grid convergence must be repeated before any physical claim.

Why a naive reflection is not exact

The legacy transverse finite-difference matrices omit one member of each outer staggered-edge pair. Reflecting that square-grid operator does not commute with it: the relative commutator is about 0.4 on a small test. Merely averaging reflected field values would therefore solve a Galerkin approximation rather than an invariant symmetry sector.

The new backend closes the y and z Yee differences periodically at the outer boundary, behind the transverse PML. The resulting cell and lower-edge index sets are exactly closed under reflection:

For each component, an orthonormal basis is assembled from the appropriate even or odd reflected pairs. For the centered z dipole, the component parities are

If Q is the resulting basis, the reduced driven problem is

The transpose is intentional. The Maxwell matrix remains complex symmetric, and reciprocity and topology backpropagation continue to use bilinear transpose products rather than Hermitian products. The transverse closure is inside the absorber, so it is still subject to the full PML-convergence gate.

Exactness checks

On the smallest complete defect-plus-two-lead regression problem, the reduced and unreduced reflection-closed formulations give:

The fine-grid reduced backend separately passes the boundary identities: complex symmetry 3.28e-14, two-period translation 5.85e-12, surface-Dyson residual 3.93e-14, lead reciprocity 1.08e-17, and condensation reciprocity 1.63e-14. The ordinary-cladding clearances remain 0.24 lambda0 in y and 0.40 lambda0 in z, and the protected air-strip density remains exactly zero.

Measured acceleration

At frequency 1.0 on the 40 x 26 x 22 fine grid:

The full fresh reacquisition used two frequency workers: 41 wide points in 170.1 s, 113 dense-local points in 524.3 s, and 28 independent confirmatory points in 114.5 s. The largest recorded worker RSS was 2.084 GiB. This is a measured throughput improvement, not a guarantee that every topology-gradient or validation phase scales by the same factor.

Current scientific status

Because the boundary discretization changed, the old fine-grid pole was not carried. A fresh wide/local/confirmatory acquisition found a stable candidate near

The original 113-point merged-window validator failed because square-root periodic-lead branch edges bracket the target pole. Fresh repeated solves had relative linear residuals below 3.9e-14 and reproduced all seven responses, which ruled out solver or parallel noise. A 449-point acquisition then demonstrated that the predeclared plus/minus-two-halfwidth interval around the target is meromorphic to numerical precision while the wide sweep still retains both lead-edge features for mode identity.

The versioned sheet-local-v2 validator is now accepted: worst interleaved NRMSE is 2.678e-10, separately solved confirmatory NRMSE is 4.346e-12, and linewidth-Q difference is 1.360%. All six transverse-PML perturbations also pass fresh combined gates; the maximum complex pole change is 0.005713 halfwidths and Q-fit span is 0.7526%. A subsequent bounded common-ascent atom step improved both normalized atom LDOS and fitted Q and passed a fresh validator. That updated topology must repeat PML qualification before a second step.

Why x is not reduced yet

The design variables remain projected over all eight x/y/z reflected copies, but the electromagnetic solve currently exploits only y/z symmetry. An exact x reduction must exchange the independently oriented left and right retarded surface Green functions while preserving the mirror phase at both interfaces. It must also close the longitudinal staggered-edge indexing under reflection. The present x operator fails that commutation test by about 0.4, so projecting it would alter the physics. A safe x backend needs a paired-lead even/odd formulation and its own condensed/full-period and boundary- translation regressions.

Use

Create a topology-only checkpoint for the new backend with:

python prepare_symmetry_checkpoint.py \
  --warmstart outputs_refined_fine_workstation64_checkpoint/run_data.npz \
  --output-dir outputs_refined_fine_yz_symmetry_checkpoint

For a new profile, vector3d_optimizer.py --transverse-symmetry-reduction selects the reflection-closed transverse boundary and exact y/z reduction. Do not attach old spectral or PML metrics to the resulting configuration.