The Invisible Ladders Theorem (TILT): a radical-theoretic criterion for θ-periodicity

A particle hopping on a ring of n sites, threaded by a flux θ, sees the flux in its spectrum through a single term, −2r cos θ, carried by one bond of amplitude r: cut that bond, and θ disappears from every observable, and from the thermal time flow of the ring’s Gibbs state. We show that this elementary fact is the Hermitian face of an exact algebraic criterion governing the θ-vacua of non-abelian gauge theories — the vacua of the gluon field. Let Jλ be the transition operator on a finite ladder of n topological sectors, whose only variable rung λ closes the last sector onto the first. The algebra generated by Jλ is semisimple if and only if λ ≠ 0, and its eigenstates are then the discrete θ-vacua; for λ = 0 it acquires a Jacobson radical of dimension n − 1, invisible to every trace and excluded from the observables. θ-periodicity is thus equivalent to the closure of the ladder. The associated Cartan form sees more than the radical: as the closing amplitude crosses λ = 1, it passes from definite, through a degenerate point whose kernel is the θ = 0 vacuum, to a non-degenerate signature (n − 1, 1). Under a twisted closure, the θ-vacua are replaced by irreducible multiplets. The same definite–degenerate–hyperbolic passage, realised by E8 → E9 → E10, has been conjectured to govern gravity near a spacelike singularity. Whether it is more than an analogy, we leave open.