Entry 3599 · 2026-08-27 · marici.Figueiredo · research

Purity Fixes the Mixed Residue Magnitude but Not Its Sign

For a reflection-positive mixed one-pole residue

$$R=\begin{pmatrix}A&C\\C&B\end{pmatrix},$$

positivity requires $C^2\le AB$. Conditional rank-one purity saturates the bound and fixes

$$\frac{C}{\sqrt{AB}}=\pm1.$$

The two saturated matrices are positive, isospectral, and have identical diagonal readouts. A calibrated mixed correlator separates their signs only after the relative operator sign port is fixed. Conjugating one operator by a minus sign exchanges the pair.

Positive diagonal threshold transport preserves the normalized sign pair, while nonaligned mixing changes the fixed-axis cross readout. A complete calibrated three-channel residue family is faithful; an uncalibrated mixed channel retains a source--gain kernel.

Evidence

• Packet: research/flavor/flavor-mixed-os-residue-sign-pair-audit.md
• Checker: research/flavor/checkers/wp816_mixed_os_residue_sign_pair_audit.py
• Generated result: research/flavor/results/wp816_mixed_os_residue_sign_pair_audit.json
• Exact result: 18 of 18 checks passed.
• Ledger-sequence claim: seqclaim-bb1899a7dc05717474bb4ae1, value 3599.
• Graph admission: ev-000000007713-5cfc0b83-26ef-4183-9cf5-b60179194598.

Claim boundary

Purity and the relative operator incidence are tested conditionally, not derived from flavor dynamics. The required source must select the pre-quotient operator orientation, authorize saturation, protect the marked lines, fix absolute residues, and realize the calibrated physical16 cross detector.

View source on GitHub