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.

M2 M5 tension product identity

Physical scope

Conditional arithmetic identity for the selected M2/M5 normalization; a global electric/magnetic Dirac interpretation requires a separate charge-lattice convention and is not derived.

Statement

Under the adopted positive Planck length $ℓ_{p}$ and normalizations $2 κ_{11}^{2} = (2 π)^{8} ℓ_{p}^{9}, T_{M 2} = (2 π)^{- 2} ℓ_{p}^{- 3}, T_{M 5} = (2 π)^{- 5} ℓ_{p}^{- 6},$ one has the exact arithmetic identity $2 κ_{11}^{2} T_{M 2} T_{M 5} = 2 π .$ The same inputs also give the R4 CJS M2 source coefficient $2 κ_{11}^{2} T_{M 2} = (2 π)^{6} ℓ_{p}^{6} .$ This identity is a normalization check. Interpreting $2 π$ as a global electric/magnetic Dirac unit also assumes the selected $C_{3} / C_{6}$ charge convention and an independently specified charge lattice; neither follows from this multiplication or from the shifted $G_{4}$ condition alone.

Statement References

Witten, §2.2 (membrane coupling and shifted-flux context)

Polchinski, Eqs. (12)–(15)

Facts & Assumptions

Given: $ℓ_{p} > 0$ and the exact normalization inputs in the statement.

[F1]

The tension-normalization postulate adopts all three displayed formulas and warns that absolute quantum units are physical inputs (M2 M5 tension normalizations).

[F2]

The shifted-flux postulate constrains the real periods of $G_{4}$ but does not itself identify a complete electric/magnetic charge lattice (Shifted G4 quantization in M theory).

Proof

Given: [F1], with $ℓ_{p} > 0$ .

1.1givenF1algebra

Multiply the two brane tensions: $T_{M 2} T_{M 5} = (2 π)^{- 2 - 5} ℓ_{p}^{- 3 - 6} = (2 π)^{- 7} ℓ_{p}^{- 9} .$

1.2F1algebra

Multiplying $2 κ_{11}^{2} = (2 π)^{8} ℓ_{p}^{9}$ by $T_{M 2} = (2 π)^{- 2} ℓ_{p}^{- 3}$ gives $2 κ_{11}^{2} T_{M 2} = (2 π)^{8 - 2} ℓ_{p}^{9 - 3} = (2 π)^{6} ℓ_{p}^{6} .$

2.1F1step 1.1algebra

Multiplying by the gravitational normalization gives $2 κ_{11}^{2} T_{M 2} T_{M 5} = (2 π)^{8} ℓ_{p}^{9} (2 π)^{- 7} ℓ_{p}^{- 9} = (2 π)^{1} ℓ_{p}^{0} = 2 π .$ The $ℓ_{p}$ factors cancel because $ℓ_{p} > 0$ , and the powers of $2 π$ add to one.

3.1F2step 1.1step 1.2step 2.1∎

The shifted-flux input [F2] supplies a real-period condition, not a charge-lattice theorem. The equalities proved in steps 1.1, 1.2, and 2.1 are normalization identities only.

M2 M5 tension product identity

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources