Spectral correspondence O(3) axiom
The Spectral correspondence O(3) axiom (III.T18, surfaced as A02 in this glossary) is the second custom axiom in TauLib. It asserts that the τ-spectral correspondence — the τ-internal mapping between spectral data and the B↔I↔S triangle structure on the lemniscate boundary — extends to third-order constructions: degree-3 polynomial spectral sub-structures preserve the triangle in the same way degree-1 and degree-2 sub-structures do (which are proved). The finite envelope is checked; the universal extension to all O(3) configurations is the axiom.
τ-Definition
The Spectral correspondence O(3) axiom (III.T18, surfaced as A02 in this glossary) is the second custom axiom in TauLib. It asserts that the τ-spectral correspondence — the τ-internal mapping between spectral data and the B↔I↔S triangle structure on the lemniscate boundary — extends to third-order constructions: degree-3 polynomial spectral sub-structures preserve the triangle in the same way degree-1 and degree-2 sub-structures do (which are proved). The finite envelope is checked; the universal extension to all O(3) configurations is the axiom.
Categorical invariant. An order-3 extension of the spectral correspondence operator, asserted to commute with the B↔I↔S triangle on the τ-lemniscate boundary; the universal claim is the axiom, the finite envelope is provable.
Primary registry anchor:
III.T18
τ-Derivation Chain
-
I.K0— Universe Postulate -
II.T40— Central theorem at rank (3, 15) -
III.T19— Critical Line theorem — degree-1 + degree-2 spectral correspondences proved -
III.T18— Spectral correspondence O(3) axiom — degree-3 extension (this entry)
Lean modules referenced:
TauLib.BookIII.Doors.SpectralCorrespondence,
TauLib.BookIII.Spectral.Trichotomy
Mathematical content
The τ-spectral correspondence operator extends to third-order constructions: for every degree-3 polynomial spectral sub-structure on the τ-lemniscate boundary, the B↔I↔S triangle relation is preserved under the correspondence — in the same way it is preserved at degree 1 (proved in III.T19) and degree 2 (proved as a corollary of III.T19).
What is finite-checked. The correspondence has been verified for all O(3) τ-admissible configurations up to a stated finite bound. Specifically: for every degree-3 polynomial spectral sub-structure with coefficient bounds within the bounded enumeration, the triangle preservation is verified via native_decide.
What is axiomatized. The universal extension to all O(3) spectral data (degree-3 polynomial sub-structures with arbitrary, not just bounded, coefficients).
What closes the gap. A direct structural proof using the τ-CR operator and the Hartogs-extension principle for order-3 generalization. The Hartogs-extension principle (proved at degree 1 + 2 in TauLib BookII Hartogs.HartogsExtension) provides the template; extending to degree 3 is open.
Load-bearing for. Book III spectral dictionary extensions (higher-order Millennium reformulations). Grand GRH results for higher-rank automorphic L-functions (cf. MathG-A03) inherit the dependency on this axiom.
Lean Coverage
Status: Formalized
Module: TauLib.BookIII.Doors.SpectralCorrespondence
Lean kind: axiom
Lean symbol: Tau.BookIII.Doors.spectralCorrespondenceO3