ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (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.

M2 Spectral Cut Crossing And Vanishing Pfaffian

Physical scope

Finite-dimensional mathematical spectral model only. No geometric M2 operator or physical membrane is asserted.

Statement

For a finite quaternionic eigenvalue crossing, calculate the reciprocal Pfaffian/electric phase changes and the zero of the Pfaffian section. This shows why dividing by the amplitude is not a global phase construction. The matrix family below is not a geometric M2 Dirac family.

Construction

Let $V = C^{2}$ with its standard Hermitian metric and quaternionic structure $J (z_{1}, z_{2}) = (- \overline{z_{2}}, \overline{z_{1}}) .$ Use the fixed complex-bilinear skew form $ω_{0} (v, w) = v_{1} w_{2} - v_{2} w_{1}$ . For $t \in R$ , set $D_{t} = t I_{V}$ and $K_{t} = t ω_{0}$ , whose matrix in the standard basis is $K_{t} = (\begin{array}{cc} 0 & t \\ - t & 0 \end{array}) .$ The eta family $D_{t}$ and Pfaffian form $K_{t}$ are explicitly paired finite data with the same one-pair crossing parameter; no infinite-dimensional identification is asserted. The finite Pfaffian line is the fixed one-dimensional space $L = Λ^{2} V^{*}$ with unit basis $ω_{0}$ , and its section is $K_{t} = t ω_{0}$ . The electric line in this model is $L^{*}$ with dual unit basis $ω_{0}^{*}$ ; its coefficient for $t \neq 0$ is defined to be the reciprocal eta-quarter phase. Thus both line fibres exist also at $t = 0$ , independently of whether a section vanishes. The horizontal coordinate in the sketch is the real parameter $t$ ; the vertical coordinate is an eigenvalue. The two coincident eigenvalues cross zero together. No proof below relies on the drawing.

Facts & Assumptions

Given: The finite-dimensional family in the Construction and the source convention $τ_{D}^{1 / 4} = \exp (2 π i η_{D} (0) / 8)$ on invertible cut charts.

Proof

1.1givenalgebra

The map $J$ is antiunitary and $J^{2} = - I$ , while $D_{t}$ commutes with $J$ . The matrix $K_{t}$ is skew-symmetric, and its Pfaffian is its upper-right entry $t$ . For $t \neq 0$ , $D_{t}$ has two eigenvalues equal to $t$ , so $η_{t} (s) = 2 sgn (t) ∣ t ∣^{- s}$ and $η_{t} (0) = 2 sgn (t)$ . Here $∣ t ∣^{- s} = \exp (- s \log ∣ t ∣)$ is entire in $s$ for each fixed $t \neq 0$ , so evaluation at zero requires no infinite-dimensional analytic continuation. The electric eta-quarter phase is therefore $\exp (- 2 π i η_{t} (0) / 8) = - i sgn (t)$ .

2.1givenstep 1.1algebra∎

For $t \neq 0$ , the normalized Pfaffian section is $sgn (t)$ ; multiplying by the eta-quarter phase gives $- i$ on both sides. Thus each phase changes by $- 1$ across the one-pair crossing and the tensor transition is $+ 1$ , in the fixed finite tensor product $L \otimes L^{*} ≅ C$ . At $t = 0$ the Pfaffian section equals zero, so its magnitude-normalized phase is undefined although the determinant/Pfaffian line itself remains defined. Division by the section therefore cannot define a global phase through the crossing.

Scope and context

The finite calculation uses only the displayed matrices and eta convention. The separate smooth Euclidean family pairing theorem remains an unresolved analytic and gauge-descent obligation; no part of that theorem is assumed or proved here. The two constant finite lines in this example do not supply a geometric family construction, connection-compatible trivialization, or physical operator.

Sources

Freed–Moore, Setting the Quantum Integrand of M-Theory, §1.3: source sign and eta-quarter conventions; the finite matrix calculation is supplied locally.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Daniel S. Freed and Gregory W. Moore, Setting the Quantum Integrand of M-Theory