TheoremStatement: Literature-sourcedProof: Literature-sourced

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.

External spectral pairings and the integer sphere-interchange sign

PDF p. 2: Motivic projective bundle property; PDF p. 8: Orientation for the splitting principle

Statement

Assume the Axiom of Choice and the fixed size conventions of the stable spectra interface. For spectra $E, F$ and integers $u, v$ , smashing representatives gives natural bilinear pairings $π_{u} (E) \times π_{v} (F) ⟶ π_{u + v} (E \land F) .$ They respect associativity and the sphere unit. Under symmetry $τ : E \land F \to F \land E$ , one has $τ_{*} (x \land y) = (- 1)^{u v} (y \land x)$ . Equivalently, interchanging $S [u]$ and $S [v]$ acts by $(- 1)^{u v}$ on $S [u + v]$ , also for negative integers.

External input

Reference 0 imports the pairings and symmetry law. Stable spectra and exact triangles supplies spectra, homotopy groups, integer suspension and smash in the same category. Retain The Axiom of Choice and the interface's size assumptions. No local proof is claimed.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Bastiaan Cnossen, Stable Homotopy Theory and Higher Algebra · Section 16.6.2, Definition 16.6.6 and Observation 16.6.7 with its explanation; HTML revision 6 November 2026. Linked basic calculation: Warning 4.2.11.

Integer homotopy groups have natural bilinear smash pairings, compatible with associativity, units and Koszul symmetry, including negative degrees.

Hypotheses: The ordinary symmetric monoidal infinity-category of spectra under smash, arbitrary spectra and all integer shifts. Retain the fixed universe and ZFC assumptions of the declared spectra interface; no connectivity or sequential-prespectrum multiplication hypothesis.

Reading limits: Freshly read Section 16.6.2, including the full explanation of Observation 16.6.7, and linked Warning 4.2.11 on the primary author website. Checked that Section 16.6 uses reduced excisive functors and the sphere-unit separately colimit-preserving smash, as does the declared interface. Import these results; earlier monoidal and stabilization foundations are not reconstructed. No monoidal comparison with sequential prespectra is claimed.

Sources

Bastiaan Cnossen, Stable Homotopy Theory and Higher Algebra · Section 16.6.2, Definition 16.6.6 and Observation 16.6.7 with its explanation; HTML revision 6 November 2026. Linked basic calculation: Warning 4.2.11.