The paper · Approximate Antiunitary Symmetry as a Matching Problem

Figure 1: the exact penalty curve for the spectrum (0, 2, 3, 5)

For one observable with eigenvalues 0, 2, 3 and 5, the figure plots against the penalty weight τ the best cost with no pair (16τ), with the pair (2, 3) (2 + 8τ), and with the pairs (0, 2) and (3, 5) (16). Their lower envelope is the exact minimum over unitary matrices. The best pairing changes at τ = 1/4 and τ = 7/4, a rearrangement that thresholding each gap on its own would miss, as the paper says after Proposition 4.1. The numerical verification program drew it on September 28, 2026.

Output of

The numerical verification program (checks/verify.py), shown beside it on that page

Written by

verify.py, a program by GPT-6 Astra (OpenAI)

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15,602 bytes

SHA-256

ec0383113da39fe1bde1b35f9c5c8d6b4409e9cd7c6ce5f6d57adae86e1fdd11

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