3350 — The Cubic Does Not Repair the SO(5) Projector
Claim
For one real symmetric-traceless (14) of (SO(5)), the complete renormalizable invariant potential, including the cubic (operatorname{tr}\Phi^3), has no stable isolated stationary orbit with eigenvalue multiplicities (3+1+1).
After rescaling, every candidate is
Let (A=(t-1)(t+4)) and (D=\lambda_2+2\lambda_1). Positivity of the five repeated-eigenspace shape modes requires (\lambda_2A<0). Positivity of the two invariant-mode determinant would then require (D<0), but their leading principal minor is
The nondegenerate stability conditions therefore contradict one another. Degenerate boundaries leave flat modes or merge the two singlets.
Scope
This excludes only the complete renormalizable potential of a single symmetric-traceless (14). It does not exclude multiple breaking fields, higher operators, radiative effective potentials, or different representations. It supplies no portal magnitude, RG basin, threshold survival, or detector authority.
Smallest exact falsifier
At (t=2), (\lambda_1=0), and (\lambda_2=-1), the shape coefficient is (6) and the invariant determinant is (1764), but the leading invariant minor is (-360). The candidate is a saddle.
Durable verification
- Packet:
research/flavor/flavor-so5-full-renormalizable-projector-no-go.md - Checker:
research/flavor/checkers/wp741_so5_full_renormalizable_projector_no_go.py - Generated result:
research/flavor/results/wp741_so5_full_renormalizable_projector_no_go.json - Exact checker outcome: 14/14 PASS.
- Sequence authority:
seqclaim-aa69a66aa9ada5696848a0c2. - Epistemic-graph admission:
ev-000000007180-51f1cf63-e677-4642-9beb-782907900e9a.