{"schemaVersion":1,"publication":{"id":"einstein-1916","title":"The Foundations of General Relativity: A Source–Readout Reconstruction of Einstein’s 1916 Theory","headline":"Recover Einstein’s complete chain.","description":"Calibrated measurements recover geometry. Compatible comparison recovers curvature. The same realization carries gravity, matter and the observations that test them. Follow the full source–readout reconstruction of Einstein’s 1916 theory.","subtitle":"Calibrated measurements recover geometry. Compatible comparison recovers curvature. The same realization carries gravity, matter and the observations that test them. Follow the full source–readout reconstruction of Einstein’s 1916 theory.","doi":"10.5281/zenodo.24064553","date":"2026-11-09","url":"/source-readout/einstein-1916","pdfUrl":"/publications/source-lift/einstein-1916/paper.pdf","markdownUrl":"/publications/source-lift/einstein-1916/complete.md","manifestUrl":"/source-readout/einstein-1916/manifest.json","paperSlug":"einstein-1916","author":"Jeremy Rodgers","programme":"source-readout","series":"source-lift","supplementUrl":"/publications/source-lift/einstein-1916/verification.zip","supplementDescription":"Public verification package"},"kind":"complete-translated-web-edition","chapters":[{"paperId":"einstein-1916","slug":"01-the-reconstruction","title":"Recover the complete gravitational chain","description":"Shadow Theory makes the path from local measurements to gravitational dynamics explicit. This edition follows the full construction: the supporting preparation, the geometric and matter realization, the readouts, and the operations that connect them.","order":1,"url":"/source-readout/einstein-1916/01-the-reconstruction","sourcePath":"content/paper-integrations/einstein-1916/01-the-reconstruction.mdx","sections":[{"id":"sec-intro","title":"1. Introduction"},{"id":"organization-and-source-fidelity","title":"Organization and source fidelity"}],"anchors":["sec-intro","organization-and-source-fidelity"],"words":622},{"paperId":"einstein-1916","slug":"02-source-and-recovery","title":"What source–readout recovery requires","description":"A source support, a realized physical sector and an observable output play different roles. Here are the exact maps and the composition result that let the subsequent derivation pass through those layers.","order":2,"url":"/source-readout/einstein-1916/02-source-and-recovery","sourcePath":"content/paper-integrations/einstein-1916/02-source-and-recovery.mdx","sections":[{"id":"sec-architecture","title":"2. Source architecture and the reconstruction problem"},{"id":"an-admitted-support-and-its-literal-preparation","title":"An admitted support and its literal preparation"},{"id":"what-recovery-means","title":"What recovery means"},{"id":"thm-composition","title":"Lemma 2.1 (Composition and legitimate information loss)"},{"id":"proof-thm-composition","title":"Proof."},{"id":"sec-conventions","title":"Conventions"}],"anchors":["sec-architecture","an-admitted-support-and-its-literal-preparation","eq-2-1","eq-native","eq-2-2","eq-preparation","eq-2-3","eq-sector","what-recovery-means","eq-2-4","eq-intertwiner","thm-composition","proof-thm-composition","sec-conventions","eq-2-5","eq-conventions","eq-2-6","eq-signs","eq-2-7","eq-ricci-sign"],"words":840},{"paperId":"einstein-1916","slug":"03-calibrated-spacetime","title":"Intervals determine the metric","description":"A calibration turns coordinate directions into measured components. Its quadratic readout recovers every local Lorentz metric, and its complete fibre identifies precisely the frame freedom left unresolved.","order":3,"url":"/source-readout/einstein-1916/03-calibrated-spacetime","sourcePath":"content/paper-integrations/einstein-1916/03-calibrated-spacetime.mdx","sections":[{"id":"sec-metric","title":"3. Local measurements and metric recovery"},{"id":"thm-metric","title":"Theorem 3.1 (Local section and complete frame fibre)"},{"id":"proof-thm-metric","title":"Proof."},{"id":"variations-coordinate-changes-and-compatible-patches","title":"Variations, coordinate changes and compatible patches"},{"id":"what-a-coarser-readout-would-lose","title":"What a coarser readout would lose"},{"id":"the-rotating-measurement-example","title":"The rotating measurement example"}],"anchors":["sec-metric","eq-3-1","eq-calibration","eq-3-2","eq-metricdet","eq-3-3","eq-polarization","thm-metric","eq-3-4","eq-lorentz-fibre","proof-thm-metric","eq-3-5","eq-metric-section","variations-coordinate-changes-and-compatible-patches","eq-3-6","eq-metric-variation","eq-3-7","eq-coordinate-metric","eq-3-8","eq-overlap-defects","what-a-coarser-readout-would-lose","eq-3-9","eq-conformal-control","eq-3-10","eq-rindler","the-rotating-measurement-example","eq-3-11","eq-rotation","eq-3-12","eq-rotmetric","eq-3-13","eq-rotrest"],"words":1015},{"paperId":"einstein-1916","slug":"04-comparison-and-tensors","title":"Comparison recovers covariant differentiation","description":"Tensor slots can be read out reversibly. Differentiation needs more: a compatible comparison rule. Its construction recovers the full metric connection while keeping raw torsion and nonmetricity visible.","order":4,"url":"/source-readout/einstein-1916/04-comparison-and-tensors","sourcePath":"content/paper-integrations/einstein-1916/04-comparison-and-tensors.mdx","sections":[{"id":"sec-calculus","title":"4. Tensor operations and compatible comparison"},{"id":"tensor-readouts-and-their-inverses","title":"Tensor readouts and their inverses"},{"id":"thm-tensor","title":"Proposition 4.1 (Tensor-algebra recovery)"},{"id":"proof-thm-tensor","title":"Proof."},{"id":"why-a-comparison-connection-is-necessary","title":"Why a comparison connection is necessary"},{"id":"thm-omega","title":"Theorem 4.2 (Unique compatible comparison)"},{"id":"proof-thm-omega","title":"Proof."},{"id":"connection-readout-and-complete-defects","title":"Connection readout and complete defects"},{"id":"thm-derivative","title":"Theorem 4.3 (Differentiation intertwines with readout)"},{"id":"proof-thm-derivative","title":"Proof."}],"anchors":["sec-calculus","tensor-readouts-and-their-inverses","eq-4-1","eq-tensor-map","thm-tensor","proof-thm-tensor","why-a-comparison-connection-is-necessary","eq-4-2","eq-raw-D","eq-4-3","eq-omega-transform","eq-4-4","eq-MT","eq-4-5","eq-structure","thm-omega","eq-4-6","eq-omega-formula","proof-thm-omega","connection-readout-and-complete-defects","eq-4-7","eq-C-map","eq-4-8","eq-4-9","eq-nonmetricity","eq-torsion","eq-4-10","eq-christoffel","eq-4-11","eq-C-coordinate","thm-derivative","eq-4-12","eq-D-intertwine","proof-thm-derivative"],"words":838},{"paperId":"einstein-1916","slug":"05-geodesics-and-divergences","title":"From stationary intervals to differential identities","description":"The same calibrated interval supplies geodesic motion, Hessians, volume derivatives and divergence identities. Boundary terms and tensor symmetries identify where each recovered operation applies.","order":5,"url":"/source-readout/einstein-1916/05-geodesics-and-divergences","sourcePath":"content/paper-integrations/einstein-1916/05-geodesics-and-divergences.mdx","sections":[{"id":"sec-geodesic","title":"5. Geodesic variation and differential identities"},{"id":"fixed-endpoint-interval-variation","title":"Fixed-endpoint interval variation"},{"id":"the-original-scalar-to-tensor-progression","title":"The original scalar-to-tensor progression"},{"id":"determinants-divergences-and-antisymmetric-derivatives","title":"Determinants, divergences and antisymmetric derivatives"}],"anchors":["sec-geodesic","fixed-endpoint-interval-variation","eq-5-1","eq-length-action","eq-5-2","eq-length-variation","eq-5-3","eq-geodesic-residual","eq-5-4","eq-source-geodesic","the-original-scalar-to-tensor-progression","eq-5-5","eq-hessian","eq-5-6","eq-ranktwo-derivative","determinants-divergences-and-antisymmetric-derivatives","eq-5-7","eq-5-8","eq-inverse-det","eq-volume-trace","eq-5-9","eq-metriccompatcomponents","eq-5-10","eq-5-11","eq-5-12","eq-5-13","eq-vector-div","eq-curl","eq-cyclic","eq-antisym-div","eq-5-14","eq-mixed-div","eq-5-15","eq-symmetric-div"],"words":636},{"paperId":"einstein-1916","slug":"06-curvature-and-variations","title":"Keep the nonlinear curvature","description":"The comparison commutator recovers both derivative and quadratic curvature terms. Local flatness, coordinate-volume choices and variation coverage are then different consequences of that complete operation.","order":6,"url":"/source-readout/einstein-1916/06-curvature-and-variations","sourcePath":"content/paper-integrations/einstein-1916/06-curvature-and-variations.mdx","sections":[{"id":"sec-curvature","title":"6. Curvature, local flatness and variation coverage"},{"id":"thm-curvature","title":"Theorem 6.1 (Curvature recovery on the raw domain)"},{"id":"proof-thm-curvature","title":"Proof."},{"id":"flatness-and-coordinate-volume-specialization","title":"Flatness and coordinate-volume specialization"},{"id":"variations-of-the-differential-operation","title":"Variations of the differential operation"},{"id":"controls-on-the-nonlinear-comparison","title":"Controls on the nonlinear comparison"}],"anchors":["sec-curvature","eq-6-1","eq-framecurvature","thm-curvature","eq-6-2","eq-curvature-intertwine","proof-thm-curvature","eq-6-3","eq-einstein-commutator","eq-6-4","eq-6-5","eq-WS","eq-W-S-def","flatness-and-coordinate-volume-specialization","eq-6-6","eq-volume-chart","variations-of-the-differential-operation","eq-6-7","eq-omega-variation","eq-6-8","eq-C-variation","controls-on-the-nonlinear-comparison","eq-6-9","eq-lapse-control","eq-6-10","eq-quadratic-control"],"words":727},{"paperId":"einstein-1916","slug":"07-vacuum-and-action","title":"Recover the action and its variation","description":"Recovering the value of a gravitational action is only the beginning. The construction also covers its test directions and boundary response, exposing why coordinate gauge and constrained variation cannot be interchanged.","order":7,"url":"/source-readout/einstein-1916/07-vacuum-and-action","sourcePath":"content/paper-integrations/einstein-1916/07-vacuum-and-action.mdx","sections":[{"id":"sec-vacuum","title":"7. Vacuum dynamics and the variational operation"},{"id":"a-source-comparison-action","title":"A source comparison action"},{"id":"thm-action","title":"Theorem 7.1 (Complete action-variation recovery)"},{"id":"proof-thm-action","title":"Proof."},{"id":"coordinate-gauge-versus-a-constrained-variational-domain","title":"Coordinate gauge versus a constrained variational domain"},{"id":"sec-boundary-action","title":"A boundary-complete covariant comparison"}],"anchors":["sec-vacuum","eq-7-1","eq-physical-geodesic","eq-7-2","eq-source-ricci","eq-7-3","eq-vacuum-W","a-source-comparison-action","eq-7-4","eq-source-action","eq-7-5","eq-H","thm-action","eq-7-6","eq-H-variation","eq-7-7","eq-S-variation","proof-thm-action","eq-7-8","eq-inverse-variation-lift","eq-7-9","eq-H-partials","eq-7-10","eq-source-EL","coordinate-gauge-versus-a-constrained-variational-domain","eq-7-11","eq-trace-loss","eq-7-12","eq-trace-witness","sec-boundary-action","eq-7-13","eq-7-14","eq-L1","eq-7-15","eq-boundary-identity","eq-7-16","eq-7-17","eq-EHvariation","eq-7-18","eq-transverse-action","eq-7-19","eq-boundary-witness"],"words":1140},{"paperId":"einstein-1916","slug":"08-gravitational-energy","title":"The gravitational current keeps its coordinate tag","description":"The action supplies Einstein’s canonical gravitational current and its exact off-shell divergence. An explicit flat example shows why internal frame invariance does not make that current a coordinate tensor.","order":8,"url":"/source-readout/einstein-1916/08-gravitational-energy","sourcePath":"content/paper-integrations/einstein-1916/08-gravitational-energy.mdx","sections":[{"id":"sec-gravenergy","title":"8. Gravitational energy and coordinate dependence"},{"id":"a-retained-non-tensor-readout","title":"A retained non-tensor readout"}],"anchors":["sec-gravenergy","eq-8-1","eq-canonical-source-current","eq-8-2","eq-tcanonical","eq-8-3","eq-energy-offshell","eq-8-4","eq-tquad","eq-8-5","eq-mixed-vacuum","eq-8-6","eq-vacuum-flux","a-retained-non-tensor-readout","eq-8-7","eq-energy-chart","eq-8-8","eq-nontensor-witness","eq-8-9","eq-nonflatvacuum"],"words":469},{"paperId":"einstein-1916","slug":"09-matter-and-field-equations","title":"Put matter on the same realization","description":"Geometry and stress must share one calibration, comparison and normalization. Joint sections and variations recover their coupling, with trace reversal preserving information that trace deletion would lose.","order":9,"url":"/source-readout/einstein-1916/09-matter-and-field-equations","sourcePath":"content/paper-integrations/einstein-1916/09-matter-and-field-equations.mdx","sections":[{"id":"sec-matter","title":"9. Matter on the same realization and the general field equation"},{"id":"trace-reversal-not-trace-deletion","title":"Trace reversal, not trace deletion"}],"anchors":["sec-matter","eq-9-1","eq-stress-map","eq-9-2","eq-joint-frame","eq-9-3","eq-stress-variation","trace-reversal-not-trace-deletion","eq-9-4","eq-trace-reversal","eq-9-5","eq-trace-intertwiner","eq-9-6","eq-total-current","eq-9-7","eq-coupled-residuals","eq-9-8","eq-mixed-coupled","eq-9-9","eq-einstein-equation","eq-9-10","eq-source-einstein"],"words":504},{"paperId":"einstein-1916","slug":"10-conservation-and-flux","title":"Follow exchange and boundary flux","description":"Conservation follows through an explicit double-divergence chain, with its volume defect retained. Matter and gravitational contributions exchange equal and opposite amounts on the selected sector.","order":10,"url":"/source-readout/einstein-1916/10-conservation-and-flux","sourcePath":"content/paper-integrations/einstein-1916/10-conservation-and-flux.mdx","sections":[{"id":"sec-conservation","title":"10. Total conservation, exchange and boundary flux"},{"id":"the-original-double-divergence-chain","title":"The original double-divergence chain"},{"id":"thm-doublediv","title":"Theorem 10.1 (Double divergence with its volume defect)"},{"id":"proof-thm-doublediv","title":"Proof."},{"id":"opposite-matter-and-gravitational-exchange","title":"Opposite matter and gravitational exchange"},{"id":"a-separate-curvature-identity-proof","title":"A separate curvature-identity proof"},{"id":"finite-regions-and-explicit-coupled-checks","title":"Finite regions and explicit coupled checks"}],"anchors":["sec-conservation","the-original-double-divergence-chain","eq-10-1","eq-P-potential","eq-10-2","eq-P-residual","thm-doublediv","eq-10-3","eq-10-4","eq-double-first","eq-double-full","proof-thm-doublediv","eq-10-5","eq-antisymmetric-L","eq-10-6","eq-total-offshell","eq-10-7","eq-total-conservation","eq-10-8","eq-antisymmetric-potential","eq-10-9","eq-potential-divergence","opposite-matter-and-gravitational-exchange","eq-10-10","eq-exchange","eq-10-11","eq-gravexchange-residual","eq-10-12","eq-opposite-exchange","eq-10-13","eq-einstein57a","eq-10-14","eq-source-stress-div","eq-10-15","eq-matter-divergence","a-separate-curvature-identity-proof","eq-10-16","eq-bianchi-source","eq-10-17","eq-contracted-bianchi","eq-10-18","eq-bianchi-residual","finite-regions-and-explicit-coupled-checks","eq-10-19","eq-10-20","eq-10-21","eq-total-flux","eq-matter-flux","eq-10-22","eq-dust-fixture","eq-10-23","eq-dust-exchange","eq-10-24","eq-dust-finite-flux"],"words":1021},{"paperId":"einstein-1916","slug":"11-fluids","title":"Recover the fluid equations","description":"A constitutive stress readout turns matter primitives into the coupled stress. Its conservation law yields the fluid equations only with the normalization and constitutive inputs that make those operations valid.","order":11,"url":"/source-readout/einstein-1916/11-fluids","sourcePath":"content/paper-integrations/einstein-1916/11-fluids.mdx","sections":[{"id":"sec-fluid","title":"11. Fluid constitutive readouts and Euler equations"},{"id":"thm-fluid","title":"Theorem 11.1 (Reversible energy and Euler decomposition)"},{"id":"proof-thm-fluid","title":"Proof."},{"id":"closure-degeneracy-and-verification-examples","title":"Closure, degeneracy and verification examples"}],"anchors":["sec-fluid","eq-11-1","eq-fluid-source","eq-11-2","eq-11-3","eq-fluid-stress","eq-fluid-mixed","eq-11-4","eq-fluid-epsilon","eq-11-5","eq-fluid-kinematics","eq-11-6","eq-fluid-divergence","eq-11-7","eq-fluid-div-readout","thm-fluid","eq-11-8","eq-fluid-projections","eq-11-9","eq-fluid-split","eq-11-10","eq-euler","proof-thm-fluid","eq-11-11","eq-coordinate-euler","eq-11-12","eq-11-13","eq-fluid-norm-defect","closure-degeneracy-and-verification-examples","eq-11-14","eq-radiation-fixture"],"words":769},{"paperId":"einstein-1916","slug":"12-electromagnetism","title":"Recover Maxwell operations and stress","description":"The electromagnetic potential, field, current and stress occupy distinct levels. The same geometry carries their derivative operations, gauge fibre and transfer identities, including the information that stress alone cannot recover.","order":12,"url":"/source-readout/einstein-1916/12-electromagnetism","sourcePath":"content/paper-integrations/einstein-1916/12-electromagnetism.mdx","sections":[{"id":"sec-em","title":"12. Electromagnetic fields, Maxwell operations and stress"},{"id":"primitive-coverage-and-potential-contact","title":"Primitive coverage and potential contact"},{"id":"thm-potential","title":"Proposition 12.1 (Local potential section and gauge fibre)"},{"id":"proof-thm-potential","title":"Proof."},{"id":"index-raising-sourced-divergence-and-charge-conservation","title":"Index raising, sourced divergence and charge conservation"},{"id":"the-original-electric-and-magnetic-component-systems","title":"The original electric and magnetic component systems"},{"id":"transfer-and-the-electromagnetic-stress-calculation","title":"Transfer and the electromagnetic stress calculation"},{"id":"thm-em-energy","title":"Theorem 12.2 (Complete electromagnetic transfer identity)"},{"id":"proof-thm-em-energy","title":"Proof."},{"id":"the-zero-transfer-branch-and-material-joins","title":"The zero-transfer branch and material joins"},{"id":"sec-em-example","title":"A complete same-realization Einstein–Maxwell check"}],"anchors":["sec-em","primitive-coverage-and-potential-contact","eq-12-1","eq-em-readouts","eq-12-2","eq-em-residuals","eq-12-3","eq-potential-operation","eq-12-4","eq-contact-full","eq-12-5","eq-homogeneous-maxwell","thm-potential","eq-12-6","eq-potential-section","proof-thm-potential","index-raising-sourced-divergence-and-charge-conservation","eq-12-7","eq-em-raising","eq-12-8","eq-maxwell-divergence","eq-12-9","eq-charge-residual","the-original-electric-and-magnetic-component-systems","eq-12-10","eq-em-components","eq-12-11","eq-12-12","eq-maxwell-vector-hom","eq-maxwell-vector-src","eq-12-13","eq-em-component-control","transfer-and-the-electromagnetic-stress-calculation","eq-12-14","eq-em-transfer","eq-12-15","eq-12-16","eq-12-17","eq-em-stress-source","eq-em-stress-cov","eq-em-stress-target","thm-em-energy","eq-12-18","eq-em-full-residual","proof-thm-em-energy","eq-12-19","eq-em-cyclic-product","eq-12-20","eq-em-transfer-on-shell","eq-12-21","eq-12-22","eq-poynting","eq-force-components","the-zero-transfer-branch-and-material-joins","eq-12-23","eq-joined-matter-residual","sec-em-example","eq-12-24","eq-em-wave","eq-12-25","eq-wave-calibration","eq-12-26","eq-wave-connection","eq-12-27","eq-wave-field-equation","eq-12-28","eq-wave-flux"],"words":1631},{"paperId":"einstein-1916","slug":"13-newtonian-limit","title":"Fix the Newtonian normalization","description":"Controlled weak-field and slow-motion reductions connect the covariant theory to Newtonian acceleration. Derivative norms, matter scaling and the chosen quasi-static regime keep the approximation quantitative.","order":13,"url":"/source-readout/einstein-1916/13-newtonian-limit","sourcePath":"content/paper-integrations/einstein-1916/13-newtonian-limit.mdx","sections":[{"id":"sec-newton","title":"13. Finite-order reconstruction and the Newtonian limit"},{"id":"thm-jets","title":"Theorem 13.1 (Finite-jet recovery)"},{"id":"proof-thm-jets","title":"Proof."},{"id":"coordinate-time-and-the-slow-motion-operation","title":"Coordinate time and the slow-motion operation"},{"id":"the-field-reduction-and-material-assumptions","title":"The field reduction and material assumptions"}],"anchors":["sec-newton","thm-jets","eq-13-1","eq-linear-calibration","proof-thm-jets","coordinate-time-and-the-slow-motion-operation","eq-13-2","eq-coordinate-time","eq-13-3","eq-newton-motion","eq-13-4","eq-newton-error","the-field-reduction-and-material-assumptions","eq-13-5","eq-linear-Ric44","eq-13-6","eq-poisson-g44","eq-13-7","eq-green-potential","eq-13-8","eq-poisson-ball","eq-13-9","eq-kappa-units"],"words":786},{"paperId":"einstein-1916","slug":"14-spherical-exterior","title":"Complete the central exterior","description":"The leading weak-field data do not supply every higher-order orbital term. A separately identified static spherical completion provides that missing dependency and the exact family used in the optical and orbital estimates.","order":14,"url":"/source-readout/einstein-1916/14-spherical-exterior","sourcePath":"content/paper-integrations/einstein-1916/14-spherical-exterior.mdx","sections":[{"id":"sec-exterior","title":"14. The central exterior and its higher-order completion"},{"id":"a-labelled-static-spherical-completion","title":"A labelled static spherical completion"},{"id":"information-needed-beyond-a-first-order-metric","title":"Information needed beyond a first-order metric"}],"anchors":["sec-exterior","eq-14-1","eq-central-first","eq-14-2","eq-mass-alpha","a-labelled-static-spherical-completion","eq-14-3","eq-raw-radial","eq-14-4","eq-radial-omega","eq-14-5","eq-radial-Ricci","eq-14-6","eq-radial-reduction","eq-14-7","eq-exterior-calibration","eq-14-8","eq-cartesian-exterior","information-needed-beyond-a-first-order-metric","eq-14-9","eq-Alambda"],"words":629},{"paperId":"einstein-1916","slug":"15-clocks-and-frequency","title":"Read clocks and frequency transfer","description":"Static observers read proper time and compare frequencies through the same interval law. Exact lapse factors and explicit remainder estimates show what the leading redshift formula retains.","order":15,"url":"/source-readout/einstein-1916/15-clocks-and-frequency","sourcePath":"content/paper-integrations/einstein-1916/15-clocks-and-frequency.mdx","sections":[{"id":"sec-measurements","title":"15. Rods, clocks and static frequency transfer"}],"anchors":["sec-measurements","eq-15-1","eq-radial-rod","eq-15-2","eq-clock","eq-15-3","eq-clock-potential","eq-15-4","eq-clock-bounds","eq-15-5","eq-redshift"],"words":382},{"paperId":"einstein-1916","slug":"16-light-and-bending","title":"Read null rays and optical bending","description":"Null propagation requires an operation separate from normalized timelike length. Optical variation and regularized endpoint integrals recover bending with the boundary geometry and error bounds stated explicitly.","order":16,"url":"/source-readout/einstein-1916/16-light-and-bending","sourcePath":"content/paper-integrations/einstein-1916/16-light-and-bending.mdx","sections":[{"id":"sec-light","title":"16. Null rays, optical variation and bending"},{"id":"the-original-asymptotic-integral-and-its-finite-endpoint-qualification","title":"The original asymptotic integral and its finite-endpoint qualification"},{"id":"thm-optical","title":"Theorem 16.1 (Boundary-sensitive optical recovery)"},{"id":"proof-thm-optical","title":"Proof."},{"id":"a-controlled-infinite-path-comparison","title":"A controlled infinite-path comparison"}],"anchors":["sec-light","eq-16-1","eq-null","eq-16-2","eq-optical-functional","the-original-asymptotic-integral-and-its-finite-endpoint-qualification","eq-16-3","eq-light-speed","eq-16-4","eq-bending-integrand","eq-16-5","eq-bending-leading","thm-optical","eq-16-6","eq-optical-L","eq-16-7","eq-optical-boundary","eq-16-8","eq-finite-deflection","proof-thm-optical","a-controlled-infinite-path-comparison","eq-16-9","eq-null-impact","eq-16-10","eq-null-factor","eq-16-11","eq-exact-bending","eq-16-12","eq-light-bound"],"words":708},{"paperId":"einstein-1916","slug":"17-perihelion","title":"Recover the perihelion dependency","description":"The orbit calculation uses the declared higher-order static completion and the cited orbital dependency. Its controlled expansion makes the perihelion result part of the complete recovered chain.","order":17,"url":"/source-readout/einstein-1916/17-perihelion","sourcePath":"content/paper-integrations/einstein-1916/17-perihelion.mdx","sections":[{"id":"sec-orbit","title":"17. Perihelion advance and the cited dependency"},{"id":"turning-roots-before-expansion","title":"Turning roots before expansion"},{"id":"thm-perihelion","title":"Theorem 17.1 (Perihelion coefficient and remainder)"},{"id":"proof-thm-perihelion","title":"Proof."},{"id":"scope-of-the-numerical-statements","title":"Scope of the numerical statements"}],"anchors":["sec-orbit","eq-17-1","eq-orbit-constants","eq-17-2","eq-radial-orbit","eq-17-3","eq-cubic-orbit","eq-17-4","eq-binnet","turning-roots-before-expansion","eq-17-5","eq-orbit-root-constants","eq-17-6","eq-orbit-factor","eq-17-7","eq-orbit-elements","eq-17-8","eq-orbit-integral","thm-perihelion","eq-17-9","eq-perihelion-leading","eq-17-10","eq-perihelion-bound","proof-thm-perihelion","eq-17-11","eq-einstein75","scope-of-the-numerical-statements"],"words":651},{"paperId":"einstein-1916","slug":"18-cumulative-recovery","title":"Assemble the recovery theorem","description":"The local sections, comparison identities, matter maps and observational reductions now form one compatible reconstruction. The cumulative theorem states exactly which operations and target classes are covered.","order":18,"url":"/source-readout/einstein-1916/18-cumulative-recovery","sourcePath":"content/paper-integrations/einstein-1916/18-cumulative-recovery.mdx","sections":[{"id":"sec-recovery","title":"18. The cumulative reconstruction theorem"},{"id":"thm-main","title":"Theorem 18.1 (Conditional source–readout recovery of the 1916 chain)"},{"id":"proof-thm-main","title":"Proof."}],"anchors":["sec-recovery","thm-main","proof-thm-main"],"words":737},{"paperId":"einstein-1916","slug":"19-relation-and-contribution","title":"What the reconstruction adds","description":"The construction uses established coframe and connection mathematics. Its distinctive result is the integrated source–readout account of the historical chain, with explicit coverage, discarded information and boundary-sensitive contracts.","order":19,"url":"/source-readout/einstein-1916/19-relation-and-contribution","sourcePath":"content/paper-integrations/einstein-1916/19-relation-and-contribution.mdx","sections":[{"id":"sec-comparison","title":"19. Relation to established formulations and contribution"}],"anchors":["sec-comparison"],"words":499},{"paperId":"einstein-1916","slug":"20-verification-and-limits","title":"Inspect the verification and its domain","description":"Executable checks accompany the proofs. Their exact, numerical and negative-control roles remain separate, as do the local reconstruction theorem and questions about global geometry or physical selection.","order":20,"url":"/source-readout/einstein-1916/20-verification-and-limits","sourcePath":"content/paper-integrations/einstein-1916/20-verification-and-limits.mdx","sections":[{"id":"sec-verification","title":"20. Verification, limitations and reproducibility"},{"id":"limits-of-the-reconstruction","title":"Limits of the reconstruction"}],"anchors":["sec-verification","table-1","limits-of-the-reconstruction"],"words":738},{"paperId":"einstein-1916","slug":"21-conclusion","title":"One connected reconstruction","description":"The result is a complete conditional realization of Einstein’s chain: preparation, joint geometry and matter, recovered operations, and measured consequences. Its equations and qualification registers remain available in the following chapters.","order":21,"url":"/source-readout/einstein-1916/21-conclusion","sourcePath":"content/paper-integrations/einstein-1916/21-conclusion.mdx","sections":[{"id":"sec-conclusion","title":"21. Conclusion"},{"id":"declarations","title":"Declarations"}],"anchors":["sec-conclusion","declarations"],"words":280},{"paperId":"einstein-1916","slug":"22-printed-source-register","title":"Read the printed-source corrections","description":"The supplied historical copy contains expressions that cannot all hold with their surrounding equations. This register preserves the literal forms and the separately derived consistent replacements.","order":22,"url":"/source-readout/einstein-1916/22-printed-source-register","sourcePath":"content/paper-integrations/einstein-1916/22-printed-source-register.mdx","sections":[{"id":"app-source","title":"A. Printed-source qualifications and correction register"},{"id":"the-rank-statement-on-page-784","title":"The rank statement on page 784"},{"id":"the-trace-shift-footnote-on-page-804","title":"The trace-shift footnote on page 804"},{"id":"the-unnumbered-variation-sign-on-page-805","title":"The unnumbered variation sign on page 805"},{"id":"the-gravitational-exchange-line-on-page-809","title":"The gravitational exchange line on page 809"},{"id":"the-mass-coefficient-on-page-819-and-its-propagation","title":"The mass coefficient on page 819 and its propagation"},{"id":"the-coordinate-light-speed-sign-on-page-822","title":"The coordinate light-speed sign on page 822"},{"id":"statements-requiring-hypotheses-rather-than-typographical-correction","title":"Statements requiring hypotheses rather than typographical correction"},{"id":"consolidation-changes-relative-to-the-checkpoints","title":"Consolidation changes relative to the checkpoints"}],"anchors":["app-source","the-rank-statement-on-page-784","the-trace-shift-footnote-on-page-804","the-unnumbered-variation-sign-on-page-805","eq-A-1","eq-printed-action-comparison","the-gravitational-exchange-line-on-page-809","eq-A-2","eq-exchange-print-correction","the-mass-coefficient-on-page-819-and-its-propagation","eq-A-3","eq-printed-mass-comparison","the-coordinate-light-speed-sign-on-page-822","eq-A-4","eq-printed-light-comparison","statements-requiring-hypotheses-rather-than-typographical-correction","consolidation-changes-relative-to-the-checkpoints"],"words":931},{"paperId":"einstein-1916","slug":"23-equation-correspondence","title":"Trace every original equation","description":"The correspondence register maps all 75 principal numbers and 26 lettered variants to their recovered operations. It preserves the original pagination and the exact scope of each correspondence.","order":23,"url":"/source-readout/einstein-1916/23-equation-correspondence","sourcePath":"content/paper-integrations/einstein-1916/23-equation-correspondence.mdx","sections":[{"id":"app-correspondence","title":"B. Complete original-equation correspondence"}],"anchors":["app-correspondence","table-1"],"words":1488},{"paperId":"einstein-1916","slug":"24-assumption-register","title":"Inspect the assumptions and theorem status","description":"Every reconstruction rests on a declared target domain and physical inputs. This register separates those inputs, mathematical consequences, approximation statements and unresolved global questions.","order":24,"url":"/source-readout/einstein-1916/24-assumption-register","sourcePath":"content/paper-integrations/einstein-1916/24-assumption-register.mdx","sections":[{"id":"app-assumptions","title":"C. Assumption and theorem-status register"},{"id":"evidence-and-theorem-status","title":"Evidence and theorem status"},{"id":"what-the-public-verification-binds","title":"What the public verification binds"}],"anchors":["app-assumptions","table-1","evidence-and-theorem-status","table-2","what-the-public-verification-binds"],"words":866},{"paperId":"einstein-1916","slug":"25-references","title":"References and publication source","description":"The complete bibliography of the Einstein source–readout reconstruction, with original publication and DOI links.","order":25,"url":"/source-readout/einstein-1916/25-references","sourcePath":"content/paper-integrations/einstein-1916/25-references.mdx","sections":[{"id":"ref-1","title":"[1]"},{"id":"ref-2","title":"[2]"},{"id":"ref-3","title":"[3]"},{"id":"ref-4","title":"[4]"},{"id":"ref-5","title":"[5]"},{"id":"ref-6","title":"[6]"},{"id":"ref-7","title":"[7]"},{"id":"ref-8","title":"[8]"},{"id":"ref-9","title":"[9]"},{"id":"ref-10","title":"[10]"},{"id":"ref-11","title":"[11]"},{"id":"ref-12","title":"[12]"}],"anchors":["ref-1","ref-2","ref-3","ref-4","ref-5","ref-6","ref-7","ref-8","ref-9","ref-10","ref-11","ref-12"],"words":371}],"completeMarkdown":"/publications/source-lift/einstein-1916/complete.md","originalPdf":"/publications/source-lift/einstein-1916/paper.pdf"}