ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedjudge pass (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.

The zero-dimensional Clifford cokernel is Z

Example

For $C_{0} = R$ with the trivial grading, a finite-dimensional graded module is $M = M^{0} \oplus M^{1}$ . Its Grothendieck group is $M_{0} ≅ Z \oplus Z$ by $(\dim M^{0}, \dim M^{1})$ . For $C_{1} = C l (R, - ∥ \cdot ∥^{2})$ the odd generator $e$ satisfies $e^{2} = - 1$ and identifies the even and odd components of every graded module. Conversely, for every $k \geq 0$ , $R^{k} \oplus R^{k}$ with $e (x, y) = (- y, x)$ is a graded $C_{1}$ -module. Therefore the restriction image is the diagonal subgroup and $A_{0} ≅ Z$ by even dimension minus odd dimension.

Facts & Assumptions

Given: Finite-dimensional real graded modules and the restriction homomorphism $r_{0} : M_{1} \to M_{0}$ .

[F1]

$M_{0}$ retains even and odd dimensions as separate summands, and $A_{0} = coker (r_{0})$ (Clifford-module restriction cokernels A_n).

[F2]

A graded Clifford action sends even generators to parity-preserving maps and odd generators to parity-reversing maps (Real graded Clifford-module bundles and their Grothendieck groups).

Verification

1.1givenF1F2constructalgebra

In a graded $C_{1}$ -module, the odd generator $e$ maps $M^{0}$ to $M^{1}$ and $M^{1}$ to $M^{0}$ by [F2]. Since $e^{2} = - I$ , these restrictions are inverse up to sign, so $\dim M^{0} = \dim M^{1}$ . Thus every restricted class has equal two coordinates. The explicit module $R \oplus R$ with $e (x, y) = (- y, x)$ satisfies $e^{2} = - I$ , so its restriction is $(1, 1)$ ; hence the image in the Grothendieck group is exactly $Z (1, 1)$ .

2.1F1step 1.1algebra∎

The homomorphism $Z^{2} \to Z$ , $(a, b) \mapsto a - b$ , is surjective and has kernel $Z (1, 1)$ . By [F1], its quotient is $A_{0}$ , so the induced map $A_{0} \to Z$ is an isomorphism and sends a graded module to its even dimension minus its odd dimension.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. F. Atiyah, R. Bott, and A. Shapiro, Clifford Modules, Topology 3, Supplement 1 (1964) (standard reference, not scraped)
Jost-Hinrich Eschenburg and Bernhard Hanke, Bott–Thom isomorphism, Hopf bundles and Morse theory (2021) (standard reference, not scraped)
M. F. Atiyah, K-Theory (lectures, notes by D. W. Anderson, 1967), with K-Theory and Reality reprint (standard reference, not scraped)