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Shadow Theory

Shadow Theory / Complete paper article

One vacuum, two descriptions.

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.

Jeremy Rodgers · Independent Researcher · 2026-11-06
doi:10.5281/zenodo.24013975

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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.

Conformal transport and KMS thermality identify the transported state and fix its temperature. The regulated entropy calculation also requires the holographic dictionary and matched cutoff. The universal coefficient does not fix regulator-dependent terms.

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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. One vacuum, two local descriptions

    Set up the common source, the algebraic aperture and the transport that relates vacuum restriction to hyperbolic thermality.

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  2. From vacuum restriction to thermal time

    Construct the wedge, diamond and cylinder maps and derive the physical KMS temperature without a continuum density matrix.

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  3. Bulk horizon geometry and the counting cutoff

    Recover the same time normalization in the AdS wedge, then distinguish the stationary entropy density and regulator from the algebraic aperture.

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  4. Which entropy coefficients survive a cutoff change?

    Derive the spherical logarithm and constant, and calculate exactly when a change of counting cutoff moves either result.

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  5. Anomaly check, geometric appendix and references

    Check the logarithm independently on the sphere, recover the bulk and boundary flows directly, and trace the assumptions and references.

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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.

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.

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.

Published source: Modular Transport and Spherical Holographic Entanglement Entropy in a Source–Readout Architecture. Read the original PDF alongside the web treatment.