Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

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

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$N = 2$ central-charge saturation is equivalent to annihilation by the null supercharge pair

Physical scope

Conditional representation-theoretic statement for a massive rest fiber of a four-dimensional rigid N=2 unitary super-Poincare representation in natural units; it asserts no interacting QFT, particle existence, or stability.

Statement

Assume the Axiom of Countable Choice. Use the four-dimensional rigid $N = 2$ super-Poincare convention with metric $η = diag (- 1, 1, 1, 1)$ , $ϵ_{12} = 1$ , and natural units $ℏ = c = 1$ . Let $K \neq {0}$ be a positive Hilbert rest-fiber representation with $P^{μ} = (M, 0, 0, 0)$ , $M > 0$ , scalar central charges $Z^{12} = z I$ and $Z^{21} = - z I$ , and a common dense invariant domain $D$ on which the supercharges, their actual Hilbert adjoints, and every product below are defined. Thus $P_{0} = - M$ , each supercharge has mass dimension one-half, and $M, z$ have mass dimension one. If $z \neq 0$ , set $ω = z / ∣ z ∣$ ; if $z = 0$ , set $ω = 1$ . Then $∣ ω ∣ = 1$ and $\overline{ω} z = ∣ z ∣$ by the central-charge theorem. On $D$ put

$q = Q_{1}^{1}, r = Q_{2}^{2}, A = q - ω r^{*}, A^{♯} = q^{*} - \overline{ω} r .$

For every $ψ \in D$ ,

$⟨ (A A^{♯} + A^{♯} A) ψ, ψ ⟩ = ∥ A ψ ∥^{2} + ∥ A^{♯} ψ ∥^{2} = 4 (M - ∣ z ∣) ∥ ψ ∥^{2} .$

Consequently, for every nonzero $ψ \in D$ , $M = ∣ z ∣$ if and only if $A ψ = A^{♯} ψ = 0$ . The operator $A^{♯}$ is the displayed sum of the individual charge adjoints; for unbounded charges it is not asserted to be the maximal-domain adjoint of the sum $A$ . This conditional statement neither asserts that such a vector or representation occurs in a physical model nor proves its dynamical stability.

Statement references

Bilal, §3.4

Tong, §2.4.2

Positive central-charge bounds on a massive rest representation fiber

Positive unitary super-Poincare representation with spectral-domain covariance

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Facts & Assumptions

Given: The stated massive rest-fiber data and $ψ \in D$ .

[A1]

Assume ${A C}_{ω}$ , as required by the exact unbounded-operator and representation suppliers (The Axiom of Countable Choice ( ${A C}_{ω}$ )): for every sequence of nonempty sets there is a choice function selecting one element from each set.

[F1]

Positive central-charge bounds on a massive rest representation fiber: in the $N = 2$ rest fiber, ${q, q^{*}} = {r, r^{*}} = 2 M I$ , ${q, r} = 2 z I$ , ${r^{*}, q^{*}} = 2 \overset{ˉ}{z} I$ , and $A A^{♯} + A^{♯} A = 4 (M - ∣ z ∣) I$ on the common product domain; at $M = ∣ z ∣$ the two displayed combinations annihilate every vector in that domain.

[F2]

Positive unitary super-Poincare representation with spectral-domain covariance: the Hilbert inner product is positive and linear in its first variable; each charge adjoint is its actual Hilbert adjoint; the common dense invariant domain is contained in the domains of all generator products in the algebra.

Proof

technique · direct

Given: [A1], [F1], [F2], and the operators in the Statement.

1.1F2givenalgebra

Expanding $∥ A ψ ∥^{2} + ∥ A^{♯} ψ ∥^{2}$ gives $∥ q ψ ∥^{2} + ∥ r^{*} ψ ∥^{2} + ∥ q^{*} ψ ∥^{2} + ∥ r ψ ∥^{2} - 2 Re (\overline{ω} ⟨ q ψ, r^{*} ψ ⟩) - 2 Re (ω ⟨ q^{*} ψ, r ψ ⟩)$ . Expanding $A A^{♯} + A^{♯} A$ gives the same four diagonal terms. Its cross terms are $- \overline{ω} (q r + r q) - ω (r^{*} q^{*} + q^{*} r^{*})$ ; the actual-adjoint identities $⟨ q r ψ, ψ ⟩ = ⟨ r ψ, q^{*} ψ ⟩$ , $⟨ r q ψ, ψ ⟩ = ⟨ q ψ, r^{*} ψ ⟩$ , $⟨ r^{*} q^{*} ψ, ψ ⟩ = ⟨ q^{*} ψ, r ψ ⟩$ , and $⟨ q^{*} r^{*} ψ, ψ ⟩ = ⟨ r^{*} ψ, q ψ ⟩$ pair these terms into exactly the two negative real parts above. The common product-domain hypothesis makes each identity legitimate, so $⟨ (A A^{♯} + A^{♯} A) ψ, ψ ⟩ = ∥ A ψ ∥^{2} + ∥ A^{♯} ψ ∥^{2}$ .

2.1A1F1step 1.1

By [F1], the left side in step 1.1 equals $4 (M - ∣ z ∣) ∥ ψ ∥^{2}$ , so $4 (M - ∣ z ∣) ∥ ψ ∥^{2} = ∥ A ψ ∥^{2} + ∥ A^{♯} ψ ∥^{2}$ for every $ψ \in D$ . This is an identity of nonnegative squared norms.

3.1givenstep 2.1algebra

If $ψ \neq 0$ and $A ψ = A^{♯} ψ = 0$ , step 2.1 gives $4 (M - ∣ z ∣) ∥ ψ ∥^{2} = 0$ . Since $∥ ψ ∥^{2} > 0$ , division by $4 ∥ ψ ∥^{2}$ yields $M = ∣ z ∣$ .

4.1F1step 3.1∎

If $M = ∣ z ∣$ , then [F1] gives $A ψ = A^{♯} ψ = 0$ for every $ψ \in D$ . Together with step 3.1 this proves both directions for every nonzero domain vector.

$N = 2$ central-charge saturation is equivalent to annihilation by the null supercharge pair

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

N=2N=2 central-charge saturation is equivalent to annihilation by the null supercharge pair

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources

$N = 2$ central-charge saturation is equivalent to annihilation by the null supercharge pair