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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Positive $N = 4$ scalar potential for two noncommuting SU(2) fields

Physical scope

A finite-dimensional classical field-value calculation in the adopted N=4 SU(2) scalar potential; it is not a field solution or empirical claim.

Example

For real $a, b \in R$ with units $m^{- 1}$ , take canonical adjoint scalar values $X_{c a n}^{1} = a T_{1}$ , $X_{c a n}^{2} = b T_{2}$ , and $X_{c a n}^{I} = 0$ for $I = 3, 4, 5, 6$ , where $T_{a} = - i σ_{a} / 2$ and $σ_{1} = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), σ_{2} = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array}), σ_{3} = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$ If $a, b \neq 0$ , the adopted $N = 4$ scalar potential is $g^{2} a^{2} b^{2} / 2 > 0$ ; if either coefficient is zero, it is zero.

Example references

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model

Classical pure SU(2) Yang–Mills model

Facts & Assumptions

Given: The $N = 4$ canonical scalar potential, the SU(2) basis, and its normalized Lie bracket and invariant form.

[A1]

Four-dimensional $N = 4$ SU(2) super-Yang-Mills model: $V = \frac{g^{2}}{2} \sum_{I < J} κ ([X_{c a n}^{I}, X_{c a n}^{J}], [X_{c a n}^{I}, X_{c a n}^{J}])$ .

[F1]

Classical pure SU(2) Yang–Mills model: $κ (T_{3}, T_{3}) = 1$ .

Verification

Given: [A1], [F1], and $g > 0$ .

1.1givenA1algebra

Direct multiplication gives $σ_{1} σ_{2} = (\begin{array}{cc} i & 0 \\ 0 & - i \end{array}) = i σ_{3}$ and $σ_{2} σ_{1} = - i σ_{3}$ . Therefore $[T_{1}, T_{2}] = (- i / 2)^{2} (σ_{1} σ_{2} - σ_{2} σ_{1}) = T_{3}$ , so $[X_{c a n}^{1}, X_{c a n}^{2}] = a b T_{3}$ ; every other commutator in the sum contains a zero field.

2.1givenA1F1step 1.1algebra∎

Substitution into [A1] yields $V = \frac{g^{2}}{2} a^{2} b^{2} κ (T_{3}, T_{3}) = \frac{g^{2} a^{2} b^{2}}{2}$ . Since $g > 0$ , this is positive exactly when both $a$ and $b$ are nonzero, and the formula gives zero on either zero-coefficient boundary.

Positive $N = 4$ scalar potential for two noncommuting SU(2) fields

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Positive N=4N=4 scalar potential for two noncommuting SU(2) fields

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Positive $N = 4$ scalar potential for two noncommuting SU(2) fields