bibkey: cardy2016annulus authors: John Cardy and Erik Tonni year: 2016 title: Entanglement hamiltonians in two-dimensional conformal field theory doi: 10.1088/1742-5468/2016/12/123103 url: https://arxiv.org/abs/1608.01283 claim: In the specified annulus geometries, reduced-state moments admit open- and closed-channel partition functions with fixed boundary conditions. strata_touched: [] license: citation-only triage: anchor

Annulus replicas and the entropy derivative

Primary mathematical input

J. Stat. Mech. (2016), 123103, arXiv:1608.01283v1. Section 2, equations (16), (19), (21) and (22), gives the normalized replica identity, the annulus transfer generator, and the open/closed channel spectra. Sections 3.1–3.3 specify the width for the vacuum interval, finite temperature and a spatial circle. These representations and the familiar leading single-interval entropy are literature-attested inputs, not new results in the RT owner.

When the two conformal boundary states are the same, the closed-channel coefficients are squared overlaps and hence nonnegative. Different boundary states need not give nonnegative coefficients. An n-independent common spectrum follows here from the actual annulus transfer representation; it is not supplied by knowing finitely many normalized moments alone. Boundary conditions, exact annulus width and normalization must remain common as n varies.

Use in the RC theorem

The RC addition to the existing arithmetic RT owner optimizes the entropy-derivative correction over positive spectral measures with a specified upper bound on the nonvacuum remainder at one real replica index r>1 and an optional gap. It gives sharp extremal envelopes and a uniform family-of-intervals certificate. Those formulas are separately derived using a tilted measure and a one-variable extremization; they are not attributed to this paper.

The benchmark input Q is an upper bound on Z(r)/Z0(r)-1 in the unnormalized closed channel. A purity measurement without the necessary normalization does not supply Q. A bound for an annulus does not prove multi-interval block dominance, a changing-genus replica representation, a CFT/gravity dual or the original gravitational RT formula.

Source and priority boundary

The primary arXiv abstract, journal metadata and parsed section 2 equations were read. Requests to view PDF pages 9 and 10 as images failed; no figure or visual-page conclusion is used. A bounded primary-source search did not establish prior occurrence or global novelty of the RC extremal formula. The annulus representation itself is explicitly credited to the source. No Lean theorem or independent review is recorded here.