ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-11-05 (gpt-6.1-sol)

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Wrapped-tension check at string coupling three

Physical scope

Formal positive-parameter illustration of the conditional wrapped-tension identity at g_s=3; not a measurement, vacuum, or existence claim.

Statement

Choose formal parameters $g_{s} = 3$ , $ℓ_{s} = L > 0$ , and $α^{'} = L^{2}$ . Under the conditional circle dictionary, $R_{11} = 3 L$ and $ℓ_{p}^{3} = 3 L^{3}$ . With the adopted leading M2 and M5 tensions, one wrapping gives $2 π R_{11} T_{M 2} = \frac{1}{2 π L^{2}} = T_{F}, 2 π R_{11} T_{M 5} = \frac{1}{3 (2 π)^{4} L^{5}} = T_{D 4} .$ This is a symbolic parameter check under the stated postulates, not an observed value or a claim that a string/M-theory vacuum with these parameters exists.

Statement References

Witten, String Theory Dynamics in Various Dimensions

Facts & Assumptions

Given: The parameter assignment in the statement and the conditional theorem below.

[F1]

The current circle dictionary gives $R_{11} = g_{s} ℓ_{s}$ and $ℓ_{p}^{3} = g_{s} ℓ_{s}^{3}$ (M Theory Type IIA Circle Limit Dictionary).

[F2]

The tension postulate gives $T_{M 2} = (2 π)^{- 2} ℓ_{p}^{- 3}$ and $T_{M 5} = (2 π)^{- 5} ℓ_{p}^{- 6}$ (M2 M5 tension normalizations).

[F3]

The Dp action gives $T_{D 4} = μ_{4} / g_{s}$ with $μ_{4} = ((2 π)^{4} α^{' 5 / 2})^{- 1}$ at constant dilaton (Type II Dp Brane DBI WZ Action); this example uses only its DBI tension clause.

[F4]

The string-tension definition gives $T_{F} = 1 / (2 π α^{'})$ and $α^{'} = ℓ_{s}^{2}$ (String Tension Alpha Prime).

[F5]

The A-page theorem states the two wrapped-tension identities under the same conditional dictionary and one-wrap hypotheses (Wrapped M2 and M5 tensions match the IIA string and D4 tensions).

[A1]

The values $g_{s} = 3$ and $ℓ_{s} = L > 0$ are chosen formal parameters, not measured data or a specified solution.

Proof

Given: [F1]–[F5], [A1], and the definitions $α^{'} = L^{2}$ , $g_{s} = 3$ .

1.1givenA1F1algebra

Substitution into [F1] gives $R_{11} = g_{s} ℓ_{s} = 3 L$ and $ℓ_{p}^{3} = g_{s} ℓ_{s}^{3} = 3 L^{3}$ , hence $ℓ_{p}^{6} = 9 L^{6}$ .

2.1F2F4step 1.1algebra

Using [F2], $2 π R_{11} T_{M 2} = (6 π L) (2 π)^{- 2} (3 L^{3})^{- 1} = \frac{1}{2 π L^{2}} = T_{F} .$

2.2F1F2F3A1step 1.1algebra

Using [F2], $2 π R_{11} T_{M 5} = (6 π L) (2 π)^{- 5} (9 L^{6})^{- 1} = \frac{1}{3 (2 π)^{4} L^{5}} .$ By [F3] and $α^{'} = L^{2}$ , $T_{D 4} = ((2 π)^{4} g_{s} ℓ_{s}^{5})^{- 1} = 1 / (3 (2 π)^{4} L^{5})$ , so the second equality follows.

3.1A1F5step 2.1step 2.2∎

The two direct calculations agree with the conditional identities in [F5]. Positivity $L > 0$ makes every inverse finite; this arithmetic supplies no existence statement for a physical background.

Remarks

This example checks one non-unit coupling value and displays the cancellation of $g_{s}$ for the wrapped M2 and the surviving $g_{s}^{- 1}$ in the D4 tension. It remains conditional on the source-reported dictionary.

Wrapped-tension check at string coupling three

Statement

Statement References

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources