What does this research mean for the field?

Under the Bost-Connes quotient at the trace level, fractional-winding orbits are annihilated, and assuming the vanishing of composite non-prime-power orbits, the quotient trace support is confined exactly to the identity orbit and prime-power translation orbits, matching the index set of the Weil explicit formula. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To explore the implications of trace-level reductions under the Bost–Connes quotient in algebraic structures.

April 10, 2026Open Access

Trace-Level Support Reduction under the Bost–Connes Quotient

Puntos clave

To explore the implications of trace-level reductions under the Bost–Connes quotient in algebraic structures.
Initial analysis of algebraic quotients relating to RN43.
Proof of a support filter concerning fractional-winding orbits.
Isolation of a trace-level hypothesis on vanishing composite non-prime-power orbits.
Fractional-winding orbits are annihilated by the quotient.
The support is limited to identity and prime-power translation orbits under the trace support assumption.
Findings connect with the index set from the Weil explicit formula.

Resumen

Research Note 48 in the "Geometry of the Critical Line" programme. This note lifts the algebraic quotient picture of RN43 to the trace level. One support filter is proved: fractional-winding orbits are annihilated by the quotient (Proposition 2.1), because any operator reaching a fractional-winding state factors through the obstruction ideal. A second filter — vanishing of composite non-prime-power orbits — is isolated as a trace-level hypothesis (Hypothesis 2.2), supported by the orthogonality of p-adic components in the GNS representation of the KMS state at β = 1/2 but not reproved as a standalone theorem here. Assuming this hypothesis, the quotient trace support is confined to the identity orbit and the prime-power translation orbits — exactly the index set of the Weil explicit formula.

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