DSame local eigenvalues, different spectrum
A matrix-valued potential has eigenvalues and eigen-directions at every site. Multiplicative regularization of such fields often talks about the eigenvalues. Here both sites keep eigenvalues 3 and 1 throughout, and only the direction at site 2 turns. The energy levels of the coupled system still move.
So a matrix field cannot be regularized by its eigenvalues alone: the frames carry spectrum too.
The NNC anchor named below, Florack's 2012 matrix-field paper, was not available for review: the catalog's file holds only the SSVM 2011 proceedings front matter. Nothing on this page depends on what that paper says.
Top: the two site potentials as ellipses (axes 3 and 1), joined by the kinetic coupling K. Bottom: hollow rings are the spectrum with frames aligned, red dots the spectrum now, brass bands the Weyl radius about each aligned eigenvalue.
- spectrum (closed form)
- -2.7321, -2, -1.1102e-16, 0.73205
- Jacobi eigensolve agrees to
- 8.9e-16
- largest shift from aligned
- 1 (Weyl bound 1)
- sum mu, sum mu^2 (frame-blind)
- -4, 12
- sum mu^4 = 84 - 16 sin^2 t
- 72 = 72
- local eigenvalues at both sites
- 3, 1 (unchanged by t)
The precise statement
Two sites, K = [[I, -I], [-I, I]], V_i = R(th_i) diag(3, 1) R(th_i)^T, H = K - V. With site 2 rotated by t, spec(H) = {-1 -+ 2 cos(t/2), -1 -+ 2 sin(t/2)} exactly. tr H, tr H^2 and tr H^3 do not depend on the frames; tr H^4 = 84 - 16 sin^2 t does.
Each eigenvalue moves by at most 2 |sin(t'/2)|, t' = t reduced mod pi into [-pi/2, pi/2], and the bound is attained on two sites: at t = pi/2 the shift is 1.41421356, the spectrum -2.41421, -2.41421, 0.414214, 0.414214.
Where it breaks
The effect vanishes for isotropic potentials (equal local eigenvalues), and the spectrum returns when the frames agree mod pi (t = pi). On graphs with cycles the holonomy cannot be gauged away, so the mod-pi reduction is exact only on trees.
Positive definiteness does not single out log-Euclidean averaging: convex Euclidean averaging is also positive definite but inflates the determinant. What log-Euclidean keeps is the determinant (no swelling): two orthogonal frames fully averaged give eigenvalues 1.73205081 and 1.73205081, that is sqrt(3) I. It also changes spec(H), so it is not frame-neutral either.
Prior art and sources
Classical results it reduces to
- Connection Laplacians (Singer-Wu; Kenyon)
- Log-Euclidean metrics (Arsigny, Fillard, Pennec, Ayache 2006)
- Weyl and Hoffman-Wielandt perturbation bounds
NNC anchors
- Regularization of Positive Definite Matrix Fields Based on Multiplicative Calculus (florack_2012_matrixfields)not reviewed: the catalog file holds only the SSVM 2011 proceedings front matter, not the paper (X2/FINDINGS.md)
OpenAI families named (context only)
- Sharp finite-matrix Lieb-Thirring inequalities and all equality cases (family 262)
Model-written and unreviewed. No result on this page depends on one.
Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X2/FINDINGS.md. Every number on this page is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.