{"schemaVersion":1,"publication":{"id":"transport-projective","title":"Transport and Projective Reduction in a Source–Readout Derivation of Einstein Gravity","subtitle":"Reconstruct the affine connection from smooth transport, remove the projective ambiguity with a volume-restored observable, and follow the complete off-shell spin-two bootstrap to Einstein–matter dynamics within the admitted realization.","description":"Reconstruct the affine connection from smooth transport, remove the projective ambiguity with a volume-restored observable, and follow the complete off-shell spin-two bootstrap to Einstein–matter dynamics within the admitted realization.","doi":"10.5281/zenodo.23954887","date":"2026-11-04","url":"/source-readout/transport-projective","pdfUrl":"/publications/spectral/transport-projective/paper.pdf","markdownUrl":"/publications/spectral/transport-projective/complete.md","manifestUrl":"/source-readout/transport-projective/manifest.json","paperSlug":"transport-projective","headline":"Transport recovers the connection used by Einstein gravity.","author":"Jeremy Rodgers","programme":"source-readout","supplementUrl":"/publications/spectral/transport-projective/verification.zip"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"transport-projective","slug":"01-the-gravitational-recovery-problem","title":"The gravitational recovery problem","description":"Recover the full Einstein–matter dynamics on one source realization, and identify which premises make that recovery possible.","order":1,"url":"/source-readout/transport-projective/01-the-gravitational-recovery-problem","sourcePath":"content/paper-integrations/transport-projective/01-the-gravitational-recovery-problem.mdx","sections":[{"id":"abstract","title":"From transport to Einstein gravity"},{"id":"section-1","title":"1. What a complete recovery has to include"},{"id":"section-1-1","title":"1.1 Conventions and units"}],"anchors":["abstract","section-1","section-1-1","eq-1","eq-2"],"words":717},{"paperId":"transport-projective","slug":"02-source-support-and-preparation","title":"Source support, preparation and common witnesses","description":"Construct the admitted fan and separate its literal finite support from the off-shell gravitational sector.","order":2,"url":"/source-readout/transport-projective/02-source-support-and-preparation","sourcePath":"content/paper-integrations/transport-projective/02-source-support-and-preparation.mdx","sections":[{"id":"section-2","title":"2. Admitted support and realized variables"},{"id":"section-2-1","title":"2.1 A native source family"},{"id":"section-2-2","title":"2.2 What the literal preparation contains"},{"id":"section-2-3","title":"2.3 Recover operations on one common witness"}],"anchors":["section-2","section-2-1","proposition-2-1","section-2-2","eq-3","eq-4","eq-5","section-2-3","eq-6"],"words":724},{"paperId":"transport-projective","slug":"03-regular-transport","title":"Regular transport reconstructs a connection","description":"Derive the local velocity-linear jet, lift all variations and bound finite-resolution errors.","order":3,"url":"/source-readout/transport-projective/03-regular-transport","sourcePath":"content/paper-integrations/transport-projective/03-regular-transport.mdx","sections":[{"id":"section-3","title":"3. Transport regularity and resolution"},{"id":"section-3-1","title":"3.1 Smoothness at constant paths forces the first jet"},{"id":"section-3-2","title":"3.2 The variations and curvature also descend"},{"id":"section-3-3","title":"3.3 Transport error and curvature error need different control"}],"anchors":["section-3","section-3-1","theorem-3-1","eq-7","eq-8","eq-9","proposition-3-2","eq-10","section-3-2","eq-11","eq-12","section-3-3","proposition-3-3","eq-13","eq-14","eq-15","eq-16"],"words":972},{"paperId":"transport-projective","slug":"04-projective-connection","title":"Projective reduction recovers metric transport","description":"Allow every affine component, derive the precise stationary kernel, and construct the observable that removes it.","order":4,"url":"/source-readout/transport-projective/04-projective-connection","sourcePath":"content/paper-integrations/transport-projective/04-projective-connection.mdx","sections":[{"id":"section-4","title":"4. The unrestricted affine connection"},{"id":"section-4-1","title":"4.1 Fix the curvature convention before varying"},{"id":"section-4-2","title":"4.2 The full affine Hessian has just the projective kernel"},{"id":"section-4-3","title":"4.3 Restore volume in the transport readout"}],"anchors":["section-4","section-4-1","eq-17","eq-18","eq-19","section-4-2","eq-20","eq-21","theorem-4-1","eq-22","eq-23","eq-24","eq-25","section-4-3","eq-26","eq-27","eq-28","theorem-4-2","eq-29"],"words":762},{"paperId":"transport-projective","slug":"05-the-cubic-bootstrap","title":"The full source stress closes the cubic bootstrap","description":"Lift the action and horizontal variation, compute the quotient inverse, and derive the exact off-shell residual.","order":5,"url":"/source-readout/transport-projective/05-the-cubic-bootstrap","sourcePath":"content/paper-integrations/transport-projective/05-the-cubic-bootstrap.mdx","sections":[{"id":"section-5","title":"5. The bootstrap on the projective quotient"},{"id":"section-5-1","title":"5.1 Recover the action and its variation domain"},{"id":"section-5-2","title":"5.2 Construct the quotient inverse explicitly"},{"id":"section-5-3","title":"5.3 Vary the auxiliary background horizontally"},{"id":"section-5-4","title":"5.4 The exact residual closes the current"}],"anchors":["section-5","section-5-1","eq-30","eq-31","section-5-2","eq-32","eq-33","eq-34","eq-35","eq-36","eq-37","section-5-3","eq-38","eq-39","theorem-5-1","eq-40","eq-41","eq-42","eq-43","section-5-4","theorem-5-2","eq-44","eq-45","eq-46"],"words":886},{"paperId":"transport-projective","slug":"06-einstein-matter","title":"The Einstein–matter endpoint","description":"Recover normalized nonlinear metric and matter equations, with the correct gauge, scalar, and boundary contracts.","order":6,"url":"/source-readout/transport-projective/06-einstein-matter","sourcePath":"content/paper-integrations/transport-projective/06-einstein-matter.mdx","sections":[{"id":"section-6","title":"6. Einstein–matter recovery, symmetry and boundary data"},{"id":"section-6-1","title":"6.1 Recover the nonlinear metric equations and their normalization"},{"id":"section-6-2","title":"6.2 Common-density coupling and its precise scalar consequence"},{"id":"section-6-3","title":"6.3 Covariance and constraints follow from the recovered law"},{"id":"section-6-4","title":"6.4 The boundary, scale and free limit"}],"anchors":["section-6","section-6-1","eq-47","eq-48","eq-49","eq-50","section-6-2","eq-51","eq-52","eq-53","eq-54","proposition-6-1","section-6-3","eq-55","eq-56","eq-57","section-6-4"],"words":980},{"paperId":"transport-projective","slug":"07-what-the-source-selects","title":"What the source architecture selects","description":"Separate derived reductions from physical premises and test the remaining freedom with explicit competing realizations.","order":7,"url":"/source-readout/transport-projective/07-what-the-source-selects","sourcePath":"content/paper-integrations/transport-projective/07-what-the-source-selects.mdx","sections":[{"id":"section-7","title":"7. Dependency reduction and physical selection"},{"id":"section-7-1","title":"7.1 Trace every consequential premise"},{"id":"section-7-2","title":"7.2 Competing realizations expose sharp obstructions"},{"id":"section-7-3","title":"7.3 What the architecture explains and what comes next"}],"anchors":["section-7","section-7-1","table-1","section-7-2","proposition-7-1","eq-58","eq-59","section-7-3"],"words":1087},{"paperId":"transport-projective","slug":"08-executable-checks","title":"Exact controls and finite transport reconstruction","description":"Inspect the complete verification contract, the nonlinear metric and matter fixtures, and the quantitative reconstruction table.","order":8,"url":"/source-readout/transport-projective/08-executable-checks","sourcePath":"content/paper-integrations/transport-projective/08-executable-checks.mdx","sections":[{"id":"section-8","title":"8. Verification and discriminating controls"},{"id":"section-8-1","title":"8.1 Test the implementations against independent routes"},{"id":"section-8-2","title":"8.2 Nonlinear metric and matter fixtures"},{"id":"section-8-3","title":"8.3 Finite-resolution reconstruction"},{"id":"section-8-4","title":"8.4 Reproduce the reported finite comparisons"}],"anchors":["section-8","section-8-1","table-2","section-8-2","eq-60","eq-61","eq-62","section-8-3","eq-63","eq-64","table-3","section-8-4"],"words":998},{"paperId":"transport-projective","slug":"09-conditional-recovery","title":"A complete conditional recovery of gravity","description":"See what has been recovered and formulate the remaining selection problem without inserting the Einstein endpoint.","order":9,"url":"/source-readout/transport-projective/09-conditional-recovery","sourcePath":"content/paper-integrations/transport-projective/09-conditional-recovery.mdx","sections":[{"id":"section-9","title":"9. What the complete construction establishes"}],"anchors":["section-9"],"words":307},{"paperId":"transport-projective","slug":"10-boundary-and-variation-appendices","title":"Appendices A and B: boundaries and variations","description":"The exact boundary current, density inversion and fixed-transport obstruction.","order":10,"url":"/source-readout/transport-projective/10-boundary-and-variation-appendices","sourcePath":"content/paper-integrations/transport-projective/10-boundary-and-variation-appendices.mdx","sections":[{"id":"appendix-a","title":"A. Boundary integrations and density inversion"},{"id":"appendix-b","title":"B. Variation domains and the fixed-transport obstruction"}],"anchors":["appendix-a","eq-65","eq-66","eq-67","eq-68","appendix-b","eq-69","eq-70"],"words":395},{"paperId":"transport-projective","slug":"11-reconstruction-and-scope-appendices","title":"Appendices C and D: reconstruction and scope","description":"Derive the reconstruction fixture and distinguish three different equivalences.","order":11,"url":"/source-readout/transport-projective/11-reconstruction-and-scope-appendices","sourcePath":"content/paper-integrations/transport-projective/11-reconstruction-and-scope-appendices.mdx","sections":[{"id":"appendix-c","title":"C. Analytic checks behind the reconstruction table"},{"id":"appendix-d","title":"D. Scope of the recovery statements"}],"anchors":["appendix-c","eq-71","appendix-d"],"words":322},{"paperId":"transport-projective","slug":"12-references","title":"References","description":"The mathematical and source-architecture references for the transport and gravity construction.","order":12,"url":"/source-readout/transport-projective/12-references","sourcePath":"content/paper-integrations/transport-projective/12-references.mdx","sections":[{"id":"references","title":"References"}],"anchors":["references","ref-dadhich-pons","ref-deser-1970","ref-deser-2010","ref-rodgers-2026a","ref-rodgers-2026b","ref-rodgers-2026c","ref-schreiber-waldorf"],"words":165}],"completeMarkdown":"/publications/spectral/transport-projective/complete.md","originalPdf":"/publications/spectral/transport-projective/paper.pdf"}