min Efit(w) + μ/2 ‖Cw‖²
Easy to edit and continue in μ. Finite μ reduces circulation but does not impose exact integrability.
Representation—
OBJECTIVE MAPWhich integrable projection is this?compare +
min ½‖δ‖²M s.t. C(w+δ)=0
The answer depends on the field representation, mass matrix, boundary conditions, and global periods.
min Σ wij|ψj−eiωijψi|² s.t. ‖ψ‖M=1
Turns field directions and spacing into a periodic phase. Amplitude or an explicit scale field can yield near incompatibilities, exposing stripe branch points.
min Efit + μ/2‖Cw‖² + ν/2Σ(‖wi‖²−1)²
Finite curl and unit penalties expose the compromise before either condition is promoted to a hard constraint.
The highlighted card names the current lesson. Each family changes a distinct modeling choice: variables, metric, constraints, or regularization.
Harmonic certificate
max |d₁h|—
max |d₀ᵀh|—
orthogonality defect—
reconstruction ‖r‖—
Vertex integrability audit
area-weighted curl RMS—
max |face curl|—
u-cycle period—
v-cycle period—
SCALING QUESTIONLocal stencil, growing mesh
A grid with width n has 2n² position DOFs. Each spring touches only four scalar DOFs, so row width stays bounded: sparse nnz grows with the mesh, while dense storage grows with the square of the DOF count.
current grid—sparse Hessianrun oncedense entries—
Energy—
‖gradient‖—
DOFs—
Hessian nnz—
Accepted steps—