Lemma 5 ([Energy cost of a multiplier]). Let Ψ∈ Q_n be a normalized eigenstate with eigenvalue e_Ψ . If F is a bounded real Lipschitz function of the spatial configuration, invariant under particle permutations, then q_n[FΨ]-e_Ψ\|FΨ\|₂² …

Lemma 4 ([Reduction to a global sector minimum]). For every fixed nuclear system there is a largest integer N such that E_N<E_N-1 . It satisfies N≤3Z , has a normalized minimizer Ψ , and E:=E_N=_n≥0 E_n. Any bound on this N bounds every …

Lemma 3 ([Reciprocal-potential bound]). If E_n<E_n-1 and every z_m≥1 , then n≤3Z .

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Both FΨ and F^2Ψ belong to Q_n . Test the weak eigen-equation with F^2Ψ and expand the gradients. Subtraction from q_n[FΨ] cancels all multiplication potentials and mixed-gradient terms, leaving.

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Lemma 5 ([Energy cost of a multiplier]). Let Ψ∈ Qn be a normalized eigenstate with eigenvalue eΨ . If F is a bounded real Lipschitz function of the spatial configuration, invariant under particle permutations, then qn[FΨ]-eΨ\|FΨ\|22 …

Lemma 4 ([Reduction to a global sector minimum]). For every fixed nuclear system there is a largest integer N such that EN<EN-1 . It satisfies N≤3Z , has a normalized minimizer Ψ , and E:=EN=n≥0 En. Any bound on this N bounds every …

Lemma 3 ([Reciprocal-potential bound]). If En<En-1 and every zm≥1 , then n≤3Z .

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Lemma 5 ([Energy cost of a multiplier]). Let Ψ∈ Q_n be a normalized eigenstate with eigenvalue e_Ψ . If F is a bounded real Lipschitz function of the spatial configuration, invariant under particle permutations, then q_n[FΨ]-e_Ψ\|FΨ\|₂² …

Lemma 4 ([Reduction to a global sector minimum]). For every fixed nuclear system there is a largest integer N such that E_N<E_N-1 . It satisfies N≤3Z , has a normalized minimizer Ψ , and E:=E_N=_n≥0 E_n. Any bound on this N bounds every …

Lemma 3 ([Reciprocal-potential bound]). If E_n<E_n-1 and every z_m≥1 , then n≤3Z .