Preparation, assumptions and verification

7. Preparation, assumptions and verification

Pointwise equilibrium differs from requiring the entire null second fundamental form to vanish through first transverse order. In normal coordinates at $p$, cubic changes of the surface embedding vary the first transverse derivative of that form by a totally symmetric three-tensor. Its trace can be prescribed, so one can make the first derivative of the initial expansion vanish. Codazzi constrains the antisymmetric derivative instead.

When the Codazzi obstruction vanishes

For a screen basis $e_2,e_3$, define

$$C_j=R(k,e_j,e_2,e_3),\qquad \Phi_{ij}=R(k,e_i,k,e_j).$$

When the second fundamental form vanishes at $p$, the Codazzi obstruction is $C$, up to the overall convention for the curvature tensor. Under the screen change $e_i\mapsto e_i+c_i k$, shifting the second slot gives zero by antisymmetry in the first pair. Shifting both of the last two slots also gives zero, by antisymmetry in that pair. The two single shifts give

$$\Delta C_j =c_2 R(k,e_j,k,e_3)+c_3 R(k,e_j,e_2,k) =c_2\Phi_{j3}-c_3\Phi_{j2}, \qquad \Delta C=\Phi(-c_3,c_2)^\top.$$

A screen with zero Codazzi obstruction exists exactly when $C$ belongs to the range of $\Phi$. Once the obstruction vanishes, the last-two-index symmetry of the differentiated second fundamental form and the Codazzi symmetry make it totally symmetric. The cubic embedding freedom can then cancel it.

Invertible $\Phi$ suffices. For a conformally flat algebraic curvature with $R_{kk}=0$ but nonzero transverse $R(k,e_i)$, one has $\Phi=0$ and $C\neq0$. That is an obstruction. The leading proof uses the weaker pointwise preparation and a centered average, so it applies even in this degenerate case.

What the derivation assumes

The inputs are local Lorentzian geometry and its optical identities, the boost-energy definition of heat, tangent-space thermal normalization, the area-entropy relation, the leading Clausius condition and local stress-energy conservation. The source-readout construction identifies the inputs that must be shared between apertures and the residual they constrain. The entropy coefficient need not come from spherical holographic entropy. The transport derivation uses different physical premises; its coupling and field equations do not supply the thermodynamic inputs here.

What the research calculations checked

The paper reports replaying the companion research calculations from a fresh extraction. All 59 results matched the archived reference: 43 exact symbolic checks, 14 controls reproducing expected failures, and two numerical diagnostics. They include Raychaudhuri for explicit metrics, affine and boost normalizations, the FLRW residual, Codazzi controls, null-direction ranks, the contracted Bianchi identity for explicit metrics, and restoration of units. The shrinking-width estimate and explicit nine-direction inversion are analytic results alongside those checks.

8. What the construction establishes

The leading Clausius residual compares heat, temperature and area on a prepared local horizon. Its vanishing determines the trace-free curvature response from a sufficient family of null directions. Conservation and the Bianchi identity complete the tensor equation up to the cosmological constant. The finite-direction reconstruction identifies which apertures suffice and which leave a kernel.

At the next order the plain area prescription produces a curvature-gradient residual for the fixed boost and temperature. Transverse averaging and thermal corrections must be controlled before that residual can be interpreted beyond the generator limit.

Funding

This work received no external funding.

Verification materials for this edition

The matching verification archive supplies the scripts and reference outputs, targeted checks of the corrected Codazzi transformation, and instructions for replaying the original suites. It is also available from the Zenodo record for this edition.

References

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5. J. Rodgers, Modular Transport and Spherical Holographic Entanglement Entropy in a Source–Readout Architecture. doi:10.5281/zenodo.24013975.

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