{"schemaVersion":1,"publication":{"id":"second-order-boundary-balls","title":"Second-Order Detection by Boundary Balls in a Source–Readout Architecture","subtitle":"An explicit transverse AdS perturbation has zero first-order area and energy variations on the fixed-slice balls. The same balls register positive bulk canonical energy at second order. A boundary relative-entropy interpretation adds a holographic dictionary and an admissible state family.","description":"An explicit transverse AdS perturbation has zero first-order area and energy variations on the fixed-slice balls. The same balls register positive bulk canonical energy at second order. A boundary relative-entropy interpretation adds a holographic dictionary and an admissible state family.","doi":"10.5281/zenodo.24021664","date":"2026-11-06","url":"/source-readout/second-order-boundary-balls","pdfUrl":"/publications/gravity/second-order-boundary-balls/paper.pdf","markdownUrl":"/publications/gravity/second-order-boundary-balls/complete.md","manifestUrl":"/source-readout/second-order-boundary-balls/manifest.json","paperSlug":"second-order-boundary-balls","headline":"The same balls see more at second order.","author":"Jeremy Rodgers","programme":"source-readout","series":"gravity-holography","supplementUrl":"https://zenodo.org/records/24021664/files/Same_Ball_Public_Verification.zip"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"second-order-boundary-balls","slug":"01-readouts-and-geometry","title":"Same balls, different response orders","description":"Fix a family of boundary balls and a single metric tangent, then ask what changes when the readout is taken to second order.","order":1,"url":"/source-readout/second-order-boundary-balls/01-readouts-and-geometry","sourcePath":"content/paper-integrations/second-order-boundary-balls/01-readouts-and-geometry.mdx","sections":[{"id":"abstract","title":"A perturbation the same balls can detect"},{"id":"section-1","title":"1. Fix the regions before changing the question"},{"id":"section-2","title":"2. Geometry, preparation and readouts"},{"id":"family-conditions","title":"Conditions for a family derivative"}],"anchors":["abstract","section-1","section-2","eq-1","eq-2","eq-3","family-conditions","eq-4"],"words":647},{"paperId":"second-order-boundary-balls","slug":"02-second-order-identity","title":"Two evaluations of the gravitational Hessian","description":"Derive canonical energy with its surface correction, then eliminate the unknown second-order metric using the linear charge–area identity.","order":2,"url":"/source-readout/second-order-boundary-balls/02-second-order-identity","sourcePath":"content/paper-integrations/second-order-boundary-balls/02-second-order-identity.mdx","sections":[{"id":"section-3","title":"3. The second-order identity"},{"id":"section-3-1","title":"3.1 Covariant phase space"},{"id":"relative-entropy","title":"The conditional boundary interpretation"},{"id":"section-3-2","title":"3.2 Remove the second-order metric"},{"id":"shape-hessian","title":"The fixed-anchor shape problem"}],"anchors":["section-3","section-3-1","eq-5","eq-6","eq-7","theorem-3-1","eq-8","eq-9","proof-3-1","relative-entropy","eq-10","section-3-2","eq-11","eq-12","shape-hessian","eq-13"],"words":793},{"paperId":"second-order-boundary-balls","slug":"03-same-ball-detection","title":"The same-ball detection theorem","description":"An explicit transverse AdS perturbation has zero first variations and positive canonical energy at every radius.","order":3,"url":"/source-readout/second-order-boundary-balls/03-same-ball-detection","sourcePath":"content/paper-integrations/second-order-boundary-balls/03-same-ball-detection.mdx","sections":[{"id":"section-4","title":"4. Detection at every radius"},{"id":"fixed-surface-bound","title":"Positivity without solving for the surface"},{"id":"independent-evaluations","title":"Independent symplectic evaluations"}],"anchors":["section-4","all-radii-canonical-energy","eq-14","proof-4-1","polynomial-surface","eq-15","eq-16","polynomial-bulk-and-area","eq-17","eq-18","eq-19","eq-20","fixed-surface-bound","eq-21","independent-evaluations"],"words":656},{"paperId":"second-order-boundary-balls","slug":"04-oscillatory-comparison","title":"Oscillatory shear and the limits of recovery","description":"Compare a bounded oscillatory mode, then distinguish gravitational detection from reconstruction and the conditional boundary interpretation.","order":4,"url":"/source-readout/second-order-boundary-balls/04-oscillatory-comparison","sourcePath":"content/paper-integrations/second-order-boundary-balls/04-oscillatory-comparison.mdx","sections":[{"id":"section-5","title":"5. A bounded oscillatory comparison"},{"id":"trial-spaces","title":"Lower bounds from nested trial spaces"},{"id":"oscillatory-lower-bounds","title":"Numerical lower-bound evaluations"},{"id":"section-6","title":"6. What the readouts recover"},{"id":"boundary-limitations","title":"Gravitational data and boundary states"},{"id":"section-7","title":"7. What the same balls can see"},{"id":"verification","title":"Verification used in the paper"},{"id":"references","title":"References"}],"anchors":["section-5","eq-22","trial-spaces","oscillatory-lower-bounds","section-6","kernel-comparison","boundary-limitations","section-7","verification","references","ref-1","ref-2","ref-3","ref-4","ref-5","ref-6"],"words":818}],"completeMarkdown":"/publications/gravity/second-order-boundary-balls/complete.md","originalPdf":"/publications/gravity/second-order-boundary-balls/paper.pdf"}