Physics theoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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.

One NS odd modulus produces a PCO plus its position differential

Physical scope

Local closed type-II RNS amplitude data for an NS superdisc with one odd modulus, at a PCO point away from the source's spurious-singularity locus.

Statement

For the local NS superdisc gluing deformation with one odd coordinate $ν$ and moving center $z_{0} (t)$ in the closed type-II amplitude data specified on this page, Berezin integration over $ν$ turns its local superintegrand factor into the PCO insertion $X (z_{0} (t)) + d_{Y} ξ (z_{0} (t))$ .

Statement references

Conditional closed type-II supermoduli amplitude data

Facts & Assumptions

Given: The odd parameter satisfies $ν^{2} = 0$ , and $z_{0} (t)$ is a smooth function of the even moduli. The local transition is $w = z - θ ν / (z - z_{0} (t))$ , $η = θ - ν / (z - z_{0} (t))$ .

[F1]

Conditional closed type-II supermoduli amplitude data: the local factor is $(1 + ν G_{z_{0}}) \exp (B_{z_{0}}) δ (d ν) δ (B_{ν})$ , with $B_{z_{0}} = - 2 ν \partial β (z_{0}) d z_{0}$ , $B_{ν} = 2 β (z_{0}) - ν \partial b (z_{0})$ , $\int d ν ν = 1$ , $ξ = - Θ (β)$ , and $\partial_{β} Θ (β) = δ (β)$ .

[F2]

Conditional closed type-II supermoduli amplitude data: $X (z) = \frac{1}{2} \oint_{C_{z}} \frac{d w}{2 π i} \frac{G (w)}{w - z} δ (β (z)) - \frac{1}{4} \partial b (z) δ^{'} (β (z))$ , and $d_{Y} ξ (z_{0}) = - d z_{0} \partial_{z_{0}} Θ (β (z_{0}))$ .

Proof

1.1givenF1algebra

Taylor expansion of the formal distribution at the even argument gives $δ (B_{ν}) = δ (2 β - ν \partial b) = \frac{1}{2} δ (β) - \frac{ν}{4} \partial b δ^{'} (β)$ , using $δ (a β + ν c) = a^{- 1} δ (β) + a^{- 2} ν c δ^{'} (β)$ from [F1].

2.1givenF1F2step 1.1algebra

Multiplying this by $1 + ν G_{z_{0}}$ , the coefficient of $ν$ from the correlator deformation is $\frac{1}{2} G_{z_{0}} δ (β (z_{0})) - \frac{1}{4} \partial b (z_{0}) δ^{'} (β (z_{0}))$ . Substituting $G_{z_{0}} = \oint_{C_{z_{0}}} \frac{d w}{2 π i} \frac{G (w)}{w - z_{0}}$ and applying $\int d ν ν = 1$ identifies this contribution with $X (z_{0})$ .

2.2givenF1step 1.1algebra

Since $B_{z_{0}}$ is already proportional to $ν$ , the exponential contributes only its linear term. Multiplying $- 2 ν \partial β (z_{0}) d z_{0}$ by the constant term $\frac{1}{2} δ (β (z_{0}))$ in $δ (B_{ν})$ gives $- ν d z_{0} \partial_{z_{0}} Θ (β (z_{0}))$ by $\partial_{β} Θ (β) = δ (β)$ . Berezin integration therefore gives $d_{Y} ξ (z_{0})$ .

3.1givenF1F2step 2.1step 2.2∎

The two coefficient contributions add, so the odd-coordinate integral is exactly $X (z_{0} (t)) + d_{Y} ξ (z_{0} (t))$ . This is a conditional model deduction from the supplied supermoduli coupling and ghost conventions; it does not define a functional measure.

Remarks

When $z_{0}$ is fixed on the even-moduli patch, $d_{Y} ξ (z_{0}) = 0$ and the contribution is $X (z_{0})$ . At a spurious-singularity point the amplitude data do not define this local insertion, so the statement excludes it.

Depends on

Conditional closed type-II supermoduli amplitude data

Dependency tree · 2 levels

Sources

Charles Wang and Xi Yin, On the Equivalence between SRS and PCO Formulations of Superstring Perturbation Theory, arXiv:2205.01106v2