Proto-Area Entropy: Concepts & Prototypes

• Proto-Area Entropy is a structural framework where entropy-like observables or area measures are governed by codimension-one data before the standard gravitational area law emerges.
• It unifies semiclassical black-hole results, non-gravitational bosonic state counting, and operational entropies from particle scattering and classical measurements.
• The concept extends to holographic codes and noncommutative geometries, highlighting how pre-geometric area data can underlie conventional entropy relations.

“Proto-Area Entropy” (Editor's term) denotes a family of constructions in which an entropy, an entropy-like observable, or an area-like datum is controlled by a codimension-one measure or functions as a precursor to a standard gravitational area law. Across the cited works, the relevant objects range from the one-loop entropy of a Schwarzschild–AdS black hole in four-dimensional critical gravity, to microstate-counting entropies in non-gravitational bosonic systems, to cross-section-controlled entanglement in particle scattering, to state-dependent proto-area data obtained from approximate recovery in holographic codes (Son et al., 2011, Dvali, 2017, Low et al., 2024, Li, 21 Jun 2026). In some settings the quantity is an actual entropy proportional to horizon area or to the area of a $(d-1)$-sphere; in others it is an area-like remainder or boundary-length datum whose compatibility with a single local bulk metric is itself the nontrivial issue. This suggests that proto-area entropy is best understood as a structural category rather than as a single invariant.

1. Conceptual range

The cited literature supports at least three distinct but related uses of area-based entropy structure. First, there are semiclassical black-hole results in which the entropy itself obeys an area law, even when the tree-level gravitational contribution is trivial. Second, there are non-gravitational systems in which microstate entropy, entanglement entropy, or vacuum-subtracted classical entropy scales with a codimension-one area rather than with volume. Third, there are pre-geometric or operational quantities that are only “area-like,” such as averaged holographic entropy data or state-dependent proto-area remainders, whose geometric interpretation requires additional compatibility conditions (Son et al., 2011, Dvali, 2017, Czech et al., 2021, Li, 21 Jun 2026).

A common thread is that the area law is not treated as exclusively tied to the classical Bekenstein–Hawking formula. In four-dimensional critical gravity, the tree-level Euclidean entropy vanishes at the critical point, yet the one-loop matter contribution restores an area law (Son et al., 2011). In a bosonic field theory on $S^d$, a critical shell of gapless modes has degeneracy scaling as $k^{d-1}$, leading to microstate entropy proportional to the area of a $(d-1)$-sphere (Dvali, 2017). In particle scattering, the outgoing linear entropy is controlled by the elastic cross section in units of the transverse packet area, $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$, so the entropy is governed by an operational area ratio rather than by a spatial entangling surface (Low et al., 2024). In vacuum-subtracted classical measurement entropies, area-law behavior reappears after removing the extensive uncertainty floor fixed by the vacuum (Haas, 2024).

These differences matter. Some of the objects under discussion are thermodynamic entropies, some are counting entropies of low-energy sectors, some are entanglement measures, and some are state-dependent remainders in an entropy decomposition. The literature therefore does not describe a single universal “proto-area entropy”; rather, it presents several routes by which area scaling arises before, beyond, or underneath standard horizon entropy.

2. Semiclassical black-hole area entropy in critical gravity

A precise black-hole realization appears in four-dimensional critical gravity, whose action is

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$

with $\Lambda<0$ and critical point

$\alpha=\frac{3}{2\Lambda}.$

For the Schwarzschild–AdS background with horizon radius $r_h$, Hawking temperature

$$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$

and horizon area $S^d$0, the tree-level Euclidean free energy is multiplied by the factor $S^d$1. Consequently,

$S^d$2

and both vanish at criticality: $S^d$3, $S^d$4 (Son et al., 2011).

The same paper resolves the apparent contradiction between this trivial tree-level result and the nontrivial entropy obtained from the brick wall method. A minimally coupled scalar field,

$S^d$5

is quantized on the fixed black-hole background. The one-loop partition function

$S^d$6

yields a finite-temperature free energy that can be rewritten as

$S^d$7

with $S^d$8 the number of modes below energy $S^d$9. This is exactly the brick wall formula, so the brick wall free energy is identified with the one-loop Euclidean free energy of matter fields rather than with an unrelated ad hoc construction (Son et al., 2011).

After regulating the near-horizon divergence with a proper-distance cutoff $k^{d-1}$0, the entropy becomes

$k^{d-1}$1

or, restoring dimensions,

$k^{d-1}$2

Thus $k^{d-1}$3, and with the canonical choice

$k^{d-1}$4

one obtains $k^{d-1}$5 (Son et al., 2011).

The significance of this result is limited but sharp. The area law survives only semiclassically: the geometry is fixed, gravitational fluctuations are omitted, and back reaction is neglected. The same paper notes that after renormalization one generally cannot maintain both the scalar-graviton decoupling condition $k^{d-1}$6 and the critical relation $k^{d-1}$7. A plausible implication is that proto-area entropy in this setting is robust at one loop but not yet established as a full quantum property of critical gravity.

3. Non-gravitational prototypes of area scaling

A non-gravitational microstate realization is provided by a non-relativistic bosonic field $k^{d-1}$8 living on a $k^{d-1}$9-dimensional sphere $(d-1)$0 of radius $(d-1)$1, with attractive momentum-dependent interaction,

$(d-1)$2

Because the Laplacian eigenvalue depends only on the highest angular momentum quantum number $(d-1)$3, each shell is highly degenerate, with asymptotic degeneracy $(d-1)$4. In the Bogoliubov double-scaling limit,

$(d-1)$5

the effective gaps become $(d-1)$6, and by tuning $(d-1)$7, one shell becomes gapless. The number of emergent gapless modes then scales as $(d-1)$8, and the microstate entropy scales as

$(d-1)$9

The paper emphasizes that this is microstate counting entropy of a degenerate low-energy sector, not entanglement entropy and not ordinary thermal entropy (Dvali, 2017).

A second non-gravitational realization appears in $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$0 particle scattering described with localized wave packets. For initially unentangled incoming particles, the outgoing linear entropy satisfies

$$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$1

and for the chosen packet profile

$$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$2

so that

$$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$3

Here $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$4 is an effective collision area, and $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$5 is also the elastic scattering probability for two wave packets of transverse size $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$6. The entropy therefore has simultaneous area and probability interpretations. The same work notes that since $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$7 typically grows with $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$8 at high energy, the entropy is correspondingly bounded by the Froissart–Martin behavior of total cross sections (Low et al., 2024).

A third prototype is furnished by classical measurement entropies. For local distributions $$\mathcal{E}^{\text{f}}_2 = 2\,\frac{\sigma_{\rm el}}{L^2} + O(\delta_p^5/|\vec{k}|^5)$$9 derived from the Wigner function, the field marginal, the momentum marginal, or the Husimi $$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$0-distribution, the central quantity is the vacuum-subtracted Rényi entropy

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$1

In Gaussian states,

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$2

and

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$3

For the $$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$4-dimensional ground state,

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$5

while for thermal Gaussian states,

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$6

These results show that central charge and temperature can be encoded in classical observables once the vacuum uncertainty contribution is removed. This suggests a proto-area entropy in the form of a classical, operational precursor to quantum area-law entanglement (Haas, 2024).

4. Kinematical and holographic proto-area data

A more structural precursor to the area law appears in Yang’s quantized spacetime algebra. There the central relation is the kinematical holographic relation

$$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$7

where $$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$8 is the number of spatial degrees of freedom in a $$I_{\rm CG}[g] = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\, \left[ R-2\Lambda+\alpha\left(R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^2\right) \right],$$9-dimensional bounded spherical region $\Lambda<0$0, $\Lambda<0$1 is its boundary area, and $\Lambda<0$2 is a dimension-dependent constant. The count $\Lambda<0$3 is obtained not by naive lattice counting but from the dimension of an irreducible $\Lambda<0$4 representation. Tanaka then builds a D0-brane gas model on these noncommutative “sites,” obtaining

$\Lambda<0$5

For $\Lambda<0$6,

$\Lambda<0$7

The claim is that black-hole entropy can be treated as bulk entropy in a noncommutative bulk whose independent degrees of freedom already scale like boundary area (Tanaka, 2013).

A different holographic reduction is obtained by averaging entropies over all subsystems of equal size. For an $\Lambda<0$8-region mixed state with purifier $\Lambda<0$9, the averaged $\alpha=\frac{3}{2\Lambda}.$0-partite entropies $\alpha=\frac{3}{2\Lambda}.$1 span a permutation-invariant sector of the holographic entropy cone. The conjectured facet family consists of

$\alpha=\frac{3}{2\Lambda}.$2

and, for $\alpha=\frac{3}{2\Lambda}.$3,

$\alpha=\frac{3}{2\Lambda}.$4

Its extreme rays are realized by flower graphs and interpolate between perfect-tensor-like and EPR-like patterns. Since these entropies are realized by RT/HRT minimal areas or graph cuts, this construction isolates a symmetry-reduced sector of area/entropy data rather than introducing a new thermodynamic entropy (Czech et al., 2021).

The most explicit use of proto-area language appears in approximate holographic codes. There the code is an isometric encoding $\alpha=\frac{3}{2\Lambda}.$5, and fixed calibrated recovery maps define

$\alpha=\frac{3}{2\Lambda}.$6

This state-dependent proto-area is an operational remainder in a boundary entropy decomposition. The main question is whether the entire interval family $\alpha=\frac{3}{2\Lambda}.$7 comes from a single local bulk metric. The answer can fail already at first order or only at second order. In finite networks, geometrizability becomes a polyhedral realization problem with primal and dual certificates. In the continuum near the hyperbolic disk, the geometric tangent space is the range of the rank-two geodesic X-ray transform, and a regular proto-area two-jet $\alpha=\frac{3}{2\Lambda}.$8 is realizable by a metric two-jet iff

$\alpha=\frac{3}{2\Lambda}.$9

The paper also exhibits Hamiltonian-skewed codes with first-order nongeometry and with quadratic-only nongeometry. Here proto-area is not itself a geometric area; it is area-like data whose common-source compatibility with one local metric is the central issue (Li, 21 Jun 2026).

5. Reinterpretations of the area law and invariant relations

One strand of the literature modifies the area law directly. In an entropic-gravity setting, the standard relation

$r_h$0

is replaced by

$r_h$1

and then reinterpreted information-theoretically through

$r_h$2

This modified bit-area relation is then inserted into equipartition and entropic-force arguments, producing corrections to Newton’s law, Einstein’s equation, and the Friedmann equation. In this construction, the “proto” layer is the bare area term $r_h$3, while the full entropy/bit law is its logarithmically deformed version (Liu et al., 2010).

A more radical reinterpretation is the proposal of self-relative or Machian information. There the horizon is tiled by Planck cells of area $r_h$4, but only relationally distinguishable configurations are counted. The argument is that a lone bit has no meaningful information; only relative placements matter. On this basis the paper claims that the effective number of independent cells is reduced by a factor of four, yielding the Bekenstein–Hawking prefactor while retaining $r_h$5 as the fundamental area carrier. It further proposes the logarithmic correction

$r_h$6

in Planck units. The paper itself presents this as a heuristic counting principle rather than as a rigorous microscopic derivation (Khosravi, 2010).

A different issue arises in multi-horizon thermodynamics. For the Ayon-Beato–Garcia regular black hole, the direct products $r_h$7 and $r_h$8 depend on the ADM mass and are therefore not universal in the paper’s sense. Nevertheless, by using the quartic horizon equation and Vieta relations, the author derives a more complicated mass-independent relation involving the event and Cauchy horizon areas. The lesson is that a simple area product need not be the correct invariant. This suggests that proto-area structure may reside in higher algebraic relations determined by the horizon polynomial rather than in a bare two-horizon product (Pradhan, 2015).

6. Limits, caveats, and persistent disputes

The literature repeatedly stresses that area-like behavior does not by itself settle the underlying microscopic interpretation. In critical gravity, the recovered area law is semiclassical and one-loop: gravitational fluctuations and back reaction are omitted, and renormalization tends to spoil the simultaneous maintenance of the critical relation and scalar-graviton decoupling (Son et al., 2011). In the spherical boson model, the area law requires critical tuning, a macroscopically occupied condensate, spherical symmetry, and a double-scaling limit; the model is explicitly non-relativistic and is presented as a prototype rather than as a derivation of black-hole entropy (Dvali, 2017).

The operational prototypes are likewise qualified. The scattering result is derived for localized wave packets with $r_h$9, and the clean formula is sharpest for initially unentangled states and for the bipartition into outgoing particles $$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$0 and $$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$1 (Low et al., 2024). The classical-entropy construction depends on vacuum subtraction and on the chosen representation; the strongest exact statements concern the Wigner distribution and Rényi-2 entropy, whereas Husimi entropies require additional subtraction and yield less universal prefactors (Haas, 2024). In approximate holographic codes, the existence of a state-dependent proto-area function does not imply a geometric bulk interpretation, because all interval data must come from a single local metric; the nongeometric component can appear already linearly or only at quadratic order (Li, 21 Jun 2026).

There are also methodological asymmetries. The noncommutative-spacetime approach derives area scaling from a kinematical reduction of bulk degrees of freedom rather than from horizon dynamics (Tanaka, 2013). The averaged holographic cone retains only permutation-invariant data and therefore discards almost all nonsymmetric structure of the full entropy vector (Czech et al., 2021). The self-relative information proposal is explicitly speculative, and the exact factor $$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$2 is extracted from a relational indistinguishability argument rather than from a full microscopic theory (Khosravi, 2010). The modified bit-area relation in entropic gravity introduces model-dependent parameters such as $$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$3 and $$T_H=\beta_H^{-1} = \frac{1-\Lambda r_h^2}{4\pi r_h},$$4, so the correction scheme is not universal in the strict sense (Liu et al., 2010).

Taken together, these works indicate that proto-area entropy is a family resemblance concept. It includes actual semiclassical area entropies, non-gravitational area-scaling counting laws, operational entropies controlled by collision or boundary area, and pre-geometric data whose metric interpretation is contingent. The shared content is the emergence of codimension-one control over information, state counting, or entropy-like observables before one arrives at the fully standard form of horizon or RT/HRT entropy.