{"schemaVersion":1,"publication":{"id":"modular-transport","title":"Modular Transport and Spherical Holographic Entanglement Entropy in a Source–Readout Architecture","subtitle":"One vacuum preparation supplies a restricted diamond state and a thermal state on the hyperbolic cylinder. Modular transport fixes the temperature; the regulated entropy calculation adds the holographic dictionary.","description":"One vacuum preparation supplies a restricted diamond state and a thermal state on the hyperbolic cylinder. Modular transport fixes the temperature; the regulated entropy calculation adds the holographic dictionary.","doi":"10.5281/zenodo.24013975","date":"2026-11-06","url":"/source-readout/modular-transport","pdfUrl":"/publications/gravity/modular-transport/paper.pdf","markdownUrl":"/publications/gravity/modular-transport/complete.md","manifestUrl":"/source-readout/modular-transport/manifest.json","paperSlug":"modular-transport","headline":"One vacuum, two descriptions.","author":"Jeremy Rodgers","programme":"source-readout","series":"gravity-holography"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"modular-transport","slug":"01-vacuum-and-apertures","title":"One vacuum, two local descriptions","description":"Set up the common source, the algebraic aperture and the transport that relates vacuum restriction to hyperbolic thermality.","order":1,"url":"/source-readout/modular-transport/01-vacuum-and-apertures","sourcePath":"content/paper-integrations/modular-transport/01-vacuum-and-apertures.mdx","sections":[{"id":"section-1","title":"1. The question and the assumptions"},{"id":"section-2","title":"2. Source data, apertures and return maps"}],"anchors":["abstract","section-1","section-2","source-datum","aperture-and-return","modular-commuting-diagram"],"words":530},{"paperId":"modular-transport","slug":"02-modular-thermality","title":"From vacuum restriction to thermal time","description":"Construct the wedge, diamond and cylinder maps and derive the physical KMS temperature without a continuum density matrix.","order":2,"url":"/source-readout/modular-transport/02-modular-thermality","sourcePath":"content/paper-integrations/modular-transport/02-modular-thermality.mdx","sections":[{"id":"section-3","title":"3. Vacuum restriction and geometric thermality"},{"id":"wedge-to-diamond","title":"The wedge-to-diamond map"},{"id":"diamond-to-hyperbolic-cylinder","title":"The hyperbolic cylinder"},{"id":"geometric-flow","title":"The modular flow and charge"},{"id":"theorem-3-1","title":"Theorem 3.1. Physical inverse temperature"},{"id":"type-iii-and-split","title":"What thermality does not supply"}],"anchors":["section-3","wedge-to-diamond","diamond-to-hyperbolic-cylinder","geometric-flow","theorem-3-1","proof-3-1","type-iii-and-split"],"words":535},{"paperId":"modular-transport","slug":"03-bulk-and-cutoffs","title":"Bulk horizon geometry and the counting cutoff","description":"Recover the same time normalization in the AdS wedge, then distinguish the stationary entropy density and regulator from the algebraic aperture.","order":3,"url":"/source-readout/modular-transport/03-bulk-and-cutoffs","sourcePath":"content/paper-integrations/modular-transport/03-bulk-and-cutoffs.mdx","sections":[{"id":"section-4","title":"4. Bulk realization and horizon entropy"},{"id":"wald-density","title":"Stationary entropy density"},{"id":"section-5","title":"5. Counting and matching the cutoffs"},{"id":"tube-and-aperture","title":"The tube and the algebra it generates"}],"anchors":["section-4","bulk-coordinate-map","wald-density","section-5","bulk-and-cft-cutoffs","tube-and-aperture"],"words":579},{"paperId":"modular-transport","slug":"04-entropy-coefficients","title":"Which entropy coefficients survive a cutoff change?","description":"Derive the spherical logarithm and constant, and calculate exactly when a change of counting cutoff moves either result.","order":4,"url":"/source-readout/modular-transport/04-entropy-coefficients","sourcePath":"content/paper-integrations/modular-transport/04-entropy-coefficients.mdx","sections":[{"id":"section-6","title":"6. Universal coefficients and a finite-jet criterion"},{"id":"universal-log-and-constant","title":"The even logarithm and the odd constant"},{"id":"theorem-6-1","title":"Theorem 6.1: What survives a cutoff substitution"},{"id":"cutoff-cancellation-examples","title":"What the criterion permits"},{"id":"three-dimensional-substitution","title":"The two sharp cutoffs in three dimensions"}],"anchors":["section-6","entropy-integral","universal-log-and-constant","theorem-6-1","proof-6-1","cutoff-cancellation-examples","three-dimensional-substitution"],"words":509},{"paperId":"modular-transport","slug":"05-geometric-checks-and-references","title":"Anomaly check, geometric appendix and references","description":"Check the logarithm independently on the sphere, recover the bulk and boundary flows directly, and trace the assumptions and references.","order":5,"url":"/source-readout/modular-transport/05-geometric-checks-and-references","sourcePath":"content/paper-integrations/modular-transport/05-geometric-checks-and-references.mdx","sections":[{"id":"section-7","title":"7. The anomaly check and verification"},{"id":"archived-verification","title":"What the paper checks independently"},{"id":"section-8","title":"8. What the construction recovers"},{"id":"appendix-a","title":"Appendix A. Geometric checks and normalization"},{"id":"chm-cylinder-conventions","title":"Conventions for the maps"},{"id":"references","title":"References"}],"anchors":["section-7","sphere-anomaly-check","four-dimensional-boundary-term","archived-verification","section-8","appendix-a","null-coordinate-flow","bulk-killing-and-period","chm-cylinder-conventions","references","ref-chm","ref-architecture","ref-transport","ref-cscf","ref-bw","ref-hl","ref-tube","ref-wald","ref-rt","ref-jm","ref-ms"],"words":824}],"completeMarkdown":"/publications/gravity/modular-transport/complete.md","originalPdf":"/publications/gravity/modular-transport/paper.pdf"}