ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 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.

Open bosonic four-tachyon Veneziano partial amplitudes

Physical scope

Disk four-point integral for flat critical open bosonic tachyons, conditional on the stated R1 spectrum and local Gaussian correlator; the meromorphic continuation and common external normalization are distinguished from the separately adopted local pole boundary convention.

Statement

Adopt the flat critical bosonic open string on one D-brane with mostly-plus target metric, $α^{'} > 0$ , four on-shell tachyons $k_{i}^{2} = 1 / α^{'}$ , $\sum_{i} k_{i} = 0$ , and normalized abelian Chan–Paton trace $1$ . Fix the disk boundary correlator convention $⟨ X^{μ} (x) X^{ν} (y) ⟩ = - 2 α^{'} η^{μ ν} \log ∣ x - y ∣$ and write $s = - (k_{1} + k_{2})^{2}$ , $t = - (k_{2} + k_{3})^{2}$ , $u = - (k_{1} + k_{3})^{2}$ . After fixing the three $P S L (2, R)$ positions, the ordering $(1, 2, 3, 4)$ gives, up to its common source-declared external-state normalization, $A (1, 2, 3, 4) = B (- 1 - α^{'} s, - 1 - α^{'} t) .$ The three inequivalent cyclic orderings give the corresponding $(s, t)$ , $(t, u)$ , and $(u, s)$ partial amplitudes. In the stated abelian sector their Chan–Paton weights are all one, so the full disk result is their unweighted sum. First evaluate in the beta-integral convergence domain and then use the meromorphic Gamma quotient. At an isolated real $s$ -channel pole $s = M^{2}$ , with the other channels away from poles, separately adopt the boundary denominator $(s - M^{2} + i 0)^{- 1}$ : the limit of $(s - M^{2} + i ε)^{- 1}$ as $ε ↓ 0$ against smooth compactly supported real- $s$ test functions. Apply the same convention after relabeling an isolated $t$ or $u$ pole. This is an additional pole convention, not a consequence of the beta integral or an attribution to the cited disk passages; no global contour at intersecting channel poles or positivity of residues is proved.

Facts & Assumptions

Given: The adopted flat tensile disk model, critical dimension and intercept, on-shell tachyon states, normalized identity Chan–Paton matrices, the displayed boundary Gaussian correlator convention, and a common external-state normalization. Complex kinematics are used first in a domain where the integral converges.

[F1]

Bosonic Polyakov Model adopts the flat tensile Polyakov model. For this disk calculation, use Tong's explicitly declared Euclidean worldsheet convention, $(σ^{1}, σ^{2}) = (σ^{1}, i σ^{0})$ (§4.0.1, printed p. 62; PDF p. 69), as used in the upper-half-plane amplitude calculation of §6.3.1. This convention is attributed to Tong, not to the Lorentzian action postulate. String Tension Alpha Prime fixes $α^{'} > 0$ and $T = (2 π α^{'})^{- 1}$ .

[F2]

Critical Dimension Flat Bosonic String supplies $D = 26, a = 1$ only conditionally: its current Step 1 record escalates the unproved quantum light-cone Lorentz-closure input.

[F3]

Open Bosonic Lightcone Spectrum gives $α^{'} M^{2} = N - 1$ under the same criticality input; its current readiness is escalated. The chosen tachyon therefore has $N = 0$ and $M^{2} = - 1 / α^{'}$ conditionally.

[F4]

Exact chiral free-boson stress tensor and vertex OPE supplies the formal algebraic Gaussian exponential correlator with its exact charge-neutrality conventions. Its current readiness is escalated because a stress/primary-field supplier record is stale; it is not treated as a path-integral measure.

[F5]

Bosonic bc ghost and Beltrami moduli pairing fixes the local $b$ pairing and conformal-Killing treatment; Bosonic Bc Ghost System gives the plane mode expansion $c (z) = \sum_{n} c_{n} z^{- n + 1}$ and the vector-field transformation law for $c$ . The global sphere and boundary zero-mode selection is supplied by [F6].

[F6]

Sphere and boundary conformal-Killing modes for tree gauge fixing: the sphere has three complex conformal-Killing vector modes and the chosen boundary disk gauge has three real $P S L (2, R)$ modes.

[F7]

Euler's Beta function on the right half-planes defines $B (p, q) = \int_{0}^{1} x^{p - 1} (1 - x)^{q - 1} d x$ for $Re p, Re q > 0$ ; The Beta-Gamma identity gives its Gamma quotient there.

[F8]

Meromorphic continuation of Gamma and Gamma has no zeros: Gamma is meromorphic on $C$ and its reciprocal is entire, so the beta quotient has a meromorphic continuation.

[A1]

The boundary correlator and physical-state normalization are the stated model/source conventions. In particular adopt Wick's rule for normal-ordered exponentials: their neutral correlation coefficient is $\prod_{i < j} ∣ x_{i} - x_{j} ∣^{2 α^{'} k_{i} \cdot k_{j}}$ , continued in the momentum components. The rank-one chiral example [F4] is algebraic context; it does not supply this Lorentzian boundary normalization or a target-time integration measure. Any disk coupling and external-state normalization are separate inputs and are not derived here. Adopt additionally the isolated-channel +i0 convention of the Statement; the cited disk passages supply the meromorphic formula, not this boundary-value choice.

Proof

1.1givenF1F2F3F4F5F6A1algebra

For ordered real boundary points $x_{1} < x_{2} < x_{3} < x_{4}$ , the fractional-linear map $f (z) = \frac{(z - x_{1}) (x_{3} - x_{4})}{(z - x_{4}) (x_{3} - x_{1})}$ sends $(x_{1}, x_{3}, x_{4})$ to $(0, 1, \infty)$ . Its determinant is $(x_{3} - x_{4}) (x_{3} - x_{1}) (x_{1} - x_{4}) > 0$ , so after positive rescaling it lies in $P S L (2, R)$ ; its derivative is positive between the poles, giving $x = f (x_{2}) \in (0, 1)$ . A fractional-linear map fixing $0, 1, \infty$ is the identity, so this gauge fixing is unique. The three fixed $c$ insertions use the vector-field type in [F5] and the global boundary zero-mode basis from [F6]: their coefficient is the $3 \times 3$ determinant of the rows $(1, x_{j}, x_{j}^{2})$ for $j = 1, 3, 4$ , equal to $(x_{3} - x_{1}) (x_{4} - x_{1}) (x_{4} - x_{3})$ . After the gauge choice $(x_{1}, x_{3}, x_{4}) = (0, 1, R)$ , this is $R (R - 1)$ ; the primary limit $c (\infty) = \lim_{R \to \infty} R^{- 2} c (R)$ makes it tend to $1$ , independent of $x_{2}$ . Thus only a common normalization is suppressed. The adopted boundary Gaussian convention in [A1], with normal-ordered exponentials evaluated by Wick's pair-contraction rule, contributes $x^{2 α^{'} k_{1} \cdot k_{2}} (1 - x)^{2 α^{'} k_{2} \cdot k_{3}}$ ; the tachyon mass shell gives $2 α^{'} k_{1} \cdot k_{2} = - 2 - α^{'} s$ and $2 α^{'} k_{2} \cdot k_{3} = - 2 - α^{'} t$ . Thus the remaining integral is $\int_{0}^{1} x^{- 2 - α^{'} s} (1 - x)^{- 2 - α^{'} t} d x$ .

2.1F7step 1.1algebra

Put $p = - 1 - α^{'} s$ and $q = - 1 - α^{'} t$ . In the nonempty complex convergence region $Re p, Re q > 0$ , the integrand in 1.1 is exactly $x^{p - 1} (1 - x)^{q - 1}$ , so [F7] identifies it as $B (p, q)$ .

3.1F7F8A1step 2.1algebra∎

The beta-gamma identity [F7] gives $B (p, q) = Γ (p) Γ (q) / Γ (p + q)$ in the convergence region, and [F8] makes that quotient meromorphic. The local pole boundary sign is the additional convention in the Statement and [A1], independent of this analytic identity. There are $(4 - 1)! = 6$ cyclic orderings after fixing leg $1$ first; reversal pairs them, and $B (p, q) = B (q, p)$ by the substitution $x \mapsto 1 - x$ . Representatives $(1, 2, 3, 4)$ , $(1, 2, 4, 3)$ , and $(1, 3, 2, 4)$ give the channel pairs $(s, t)$ , $(s, u)$ , and $(u, t)$ , respectively, which as unordered pairs are $(s, t), (u, s), (t, u)$ . In the identity Chan–Paton sector each trace is $1$ , so their sum is unweighted. This does not prove positivity of residues or a global causal contour.

Remarks

For nonabelian Chan–Paton data, each ordered partial amplitude keeps its own trace $Tr (T_{1} T_{2} T_{3} T_{4})$ ; they cannot be replaced by an unweighted sum. The conditional criticality and spectrum suppliers prevent current Step 1 readiness, but the beta-integral deduction is complete given their exact stated interface. The closed-surface coupling formula Closed string genus-n-point coupling power is not applied to this disk amplitude.

Open bosonic four-tachyon Veneziano partial amplitudes

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources