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 Certificate: thermodynamic preflight verifier
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
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
with a universal density, a local boost-energy variation and the correct conservation identity.
Why relabelling does not derive gravity
One can impose
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:
- a local operator or subregion algebra;
- a state and reduced density operator;
- vacuum entanglement rather than only global level degeneracy;
- a map from algebraic cuts to geometric area;
- a first-law or Clausius identity for all local null directions;
- universal normalization fixing \(G\);
- 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.