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Shadow Theory

Shadow Theory / Complete paper article

The same balls see more at second order.

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.

Jeremy Rodgers · Independent Researcher · 2026-11-06
doi:10.5281/zenodo.24021664

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Second-Order Detection by Boundary Balls in a Source–Readout Architecture

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.

The positive all-radii canonical energy is a bulk gravitational result for the displayed transverse mode. Identifying it with a boundary relative-entropy Hessian requires the holographic dictionary and a corresponding admissible state family.

Start with what the same balls miss

Boundary Ball Readouts and Linearized Einstein Gravity → establishes the first-order fixed-frame kernel. This edition takes the same family of balls and measures the next order.

Explore the related Atlas structure →

The argument in full

Follow the chapters in order or enter at a particular question. Definitions, mathematics, proofs, constructions and substantive appendices belong to this reading edition.

  1. Same balls, different response orders

    Fix a family of boundary balls and a single metric tangent, then ask what changes when the readout is taken to second order.

    Read chapter →

  2. Two evaluations of the gravitational Hessian

    Derive canonical energy with its surface correction, then eliminate the unknown second-order metric using the linear charge–area identity.

    Read chapter →

  3. The same-ball detection theorem

    An explicit transverse AdS perturbation has zero first variations and positive canonical energy at every radius.

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  4. Oscillatory shear and the limits of recovery

    Compare a bounded oscillatory mode, then distinguish gravitational detection from reconstruction and the conditional boundary interpretation.

    Read chapter →

Where the arguments meet

Transport and Projective Reduction in a Source–Readout Derivation of Einstein Gravity →
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.

Modular Transport and Spherical Holographic Entanglement Entropy in a Source–Readout Architecture →
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.

Local Horizon Thermodynamics and Einstein Gravity in a Source–Readout Architecture →
Heat, temperature and area readouts of a shrinking null pencil recover the Einstein equation under the stated thermodynamic and conservation assumptions. A finite generator segment and finite transverse width pose different questions.

Boundary Ball Readouts and Linearized Einstein Gravity in a Source–Readout Architecture →
First-order ball data on a fixed boundary slice recover a projected Einstein constraint. The fixed-frame kernel shows why recovering the full field equation needs more frame data and boundary assumptions, and why that recovery is distinct from reconstructing the metric.

Published source: Second-Order Detection by Boundary Balls in a Source–Readout Architecture. Read the original PDF alongside the web treatment.