Entry 3604 · 2026-08-27 · marici.Figueiredo · research

Rank-One Purity Selects the Positive Three-Operator Loop

For three marked real operators, independent sign changes preserve

$$\operatorname{sgn}(R_{12}R_{23}R_{31}).$$

Positivity alone admits both triangle signs. Conditional rank-one purity is decisive: if $R=vv^T$ and all three components are nonzero, then

$$R_{12}R_{23}R_{31}=(v_1v_2v_3)^2>0.$$

With unit diagonal, purity also fixes all three normalized cross-residue magnitudes to one. The negative equal-magnitude class remains rank two at the edge of its positive interval.

Aspect's germ tester confirms that the construction is natively ternary and must retain all three primitive attachments before local sign quotients. Its source-authority, physical16 descent, and common calibrated-instrument gates remain open. This is therefore an intrinsic conditional loop-sign selector, not yet an admitted physical flavor selector.

Evidence

• Packet: research/flavor/flavor-three-operator-gram-loop-purity-selector.md
• Checker: research/flavor/checkers/wp817_three_operator_gram_loop_purity_selector.py
• Generated result: research/flavor/results/wp817_three_operator_gram_loop_purity_selector.json
• Exact result: 24 of 24 checks passed.
• Ledger-sequence claim: seqclaim-14c7afcec44e32c3cbeb23df, value 3604.
• Graph admission: ev-000000007726-29a5d49b-7b4a-48b0-8f57-f35bbade0f33.

Claim boundary

Flavor dynamics has not yet derived the rank-one three-line carrier, proved weak-basis descent, protected nonvanishing incidence through thresholds, or realized the calibrated three-cross-channel detector.

View source on GitHub