# Thermodynamic-gravity preflight: state entropy is not yet area entropy

**Date:** 24 October 2026  
**Status:** exact entropy-type separation; local geometric realization remains open  
**Evidence:** [machine-readable result](../../../evidence/data/lattice-subroutine-thermodynamic-gravity-preflight-v1.json)  
**Certificate:** [thermodynamic preflight verifier](../../../verification/verify/verify_lattice_subroutine_thermodynamic_gravity_preflight.mjs)

## The attractive route

Thermodynamic and entanglement-equilibrium derivations show that Einstein's equation can arise when every local causal horizon carries entropy proportional to area and satisfies a local first law. A modular state-counting subroutine therefore looks relevant: it supplies precise degeneracies and entropy.

The entropy is initially the wrong type.

For a chiral \(c=24\) theory, Cardy growth gives

\[
\log d(n)\sim
2\pi\sqrt{\frac{cn}{6}}
=4\pi\sqrt n.
\]

This is global microcanonical entropy at fixed conformal grade. It counts how many states occupy an energy/grade shell of the two-dimensional theory.

The entropy needed for the Einstein equation is attached to a local spacetime cut or causal horizon. It must have the form

\[
S_{\rm geom}=\frac{A}{4G}
\]

with a universal density, a local boost-energy variation and the correct conservation identity.

## Why relabelling does not derive gravity

One can impose

\[
A(n)=4G\log d(n).
\]

But this defines an area variable from the answer. The character then determines only \(A(n)/G\). It determines neither area nor Newton's constant separately, and supplies no spacetime region whose boundary has that area.

The first Monster coefficients illustrate the distinction:

| grade \(n\) | \(d(n)\) | \(\log d(n)\) | \(4\pi\sqrt n\) |
|---:|---:|---:|---:|
| 1 | 196884 | about 12.19 | about 12.57 |
| 2 | 21493760 | about 16.88 | about 17.77 |
| 3 | 864299970 | about 20.58 | about 21.77 |

The approach to Cardy growth is real. Nothing in the table specifies a subregion, codimension-two surface, causal diamond, null generator or local energy flux.

## Exact missing interface

The thermodynamic route requires:

1. a local operator or subregion algebra;
2. a state and reduced density operator;
3. vacuum entanglement rather than only global level degeneracy;
4. a map from algebraic cuts to geometric area;
5. a first-law or Clausius identity for all local null directions;
6. universal normalization fixing \(G\);
7. conservation and controlled higher-curvature corrections.

The genus-two sewing data may eventually help with factorization and entanglement between handles. But sewing a worldsheet is not yet partitioning four-dimensional spacetime.

## Consequence

The thermodynamic route remains conceptually plausible but cannot begin from \(J\) alone. The next admissible construction is a net of subregion algebras or tensor-factor/defect model derived from the lattice CFT, followed by an entropy calculation. Only if that entropy admits a geometric area interpretation should the Jacobson/entanglement-equilibrium step be attempted.

This also clarifies the “universe as a computer” picture. A global state counter is not enough. Gravity would require the subroutine to know which information belongs to which local boundary and how information flow changes when that boundary is deformed.