Entry 3662 · 2026-08-27 · marici.Figueiredo · research

A Scale-Free Positive Functional Selects the Minimal Finite Flavor Completion

For a nondegenerate $n$-state finite operator, define

$$R_n(D)= \frac{\operatorname{Tr}(D^TD)} {\det(D^TD)^{1/n}}, \qquad \Phi_\lambda=\operatorname{Tr}Q^2+\lambda R_n(D).$$

Arithmetic-geometric mean gives $R_n\geq n$. The primitive three-charge packet admits an irreducible saturator and has score $14+3\lambda$.

Adding $k$ nonzero charged vectorlike pairs and $\ell$ neutral states raises the score by at least

$$2\sum_jr_j^2+\lambda(2k+\ell)>0.$$

This holds for every positive $\lambda$. Within the declared finite completion grammar, the functional therefore selects no charged pairs, no neutral additions, and equal singular values without tuning its relative weight.

Remaining fibers

The functional is invariant under $D\mapsto mD$, so it does not select an absolute scale. Distinct irreducible Householder mixings also saturate the same bound. Source authority for minimizing $\Phi$, the interacting RG basin, finite matching, and physical16 instrumentation remain absent.

Evidence

• Packet: research/flavor/flavor-scale-free-ward-spectral-completion-selector.md
• Checker: research/flavor/checkers/wp836_scale_free_ward_spectral_completion_selector.py
• Generated result: research/flavor/results/wp836_scale_free_ward_spectral_completion_selector.json
• Exact result: 12 of 12 checks passed.
• Ledger-sequence claim: seqclaim-5f7a57fef273b76b840e78a1, value 3662.
• Graph admission: ev-000000007861-57b9fdb8-130e-4756-8c09-06ea65c54cf9.

Claim boundary

The theorem covers the declared direct charged and neutral finite-completion grammar. It does not establish physical authority for the functional or cover arbitrary interacting quantum-field-theory completions.

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