Entropy-Positivity Conjecture

• Entropy-Positivity Conjecture is a framework asserting that entropy measures or shifts never decrease under admissible deformations, thereby imposing strict positivity or monotonicity conditions across various domains.
• In thermal EFT and black-hole physics, these constraints translate into positive Wilson coefficients and upward shifts in extremal bounds, reinforcing principles like the Weak Gravity Conjecture.
• Extensions to entanglement, phase-space formulations, combinatorics, and numerical schemes demonstrate the broad applicability of the conjecture while highlighting its dependence on regime-specific assumptions.

Searching arXiv for the cited paper and closely related work on entropy-based positivity bounds. Entropy-Positivity Conjecture denotes a family of claims asserting that entropy, entropy shifts, or entropy-derived functionals obey positivity or monotonicity constraints strong enough to restrict effective couplings, admissible states, or extremal distributions. In recent high-energy usage, the sharpest theorem-level instance is thermal: for a local, unitary, causal effective field theory in equilibrium, integrating in heavy degrees of freedom should not reduce the entropy accessible to the infrared thermal ensemble, and for a shift-symmetric scalar EFT this forces a strictly positive coefficient for the leading dimension-8 operator $O_8=(\partial_\mu\phi\,\partial^\mu\phi)^2$. Earlier black-hole analyses established positivity of higher-derivative entropy shifts at fixed mass and charge and translated it into Wilson-coefficient bounds and Weak Gravity Conjecture implications. Related formulations also appear in entanglement theory, phase-space quantum mechanics, extremal combinatorics, graph dimer entropy, and entropy-stable numerical schemes (Liu et al., 8 Sep 2025, Cheung et al., 2018).

1. Scope and common structure

The phrase does not denote a single universally standardized theorem. Taken together, the current literature suggests a common pattern: one identifies an entropy-like quantity, proves or conjectures that a deformation or comparison operation should not decrease it, and then converts that sign condition into structural constraints. In QFT and gravity, the constrained objects are typically Wilson coefficients or extremality bounds. In information-theoretic and combinatorial settings, they are probability distributions. In numerical analysis, they become admissibility radii for fully discrete updates.

Domain	Entropy object	Positivity statement
Thermal EFT	$s(T)$ or $\Delta s(T)$	$s_{\rm EFT}(T)\ge s_{\rm free}(T)$
Black-hole EFT	$\Delta S(M,Q)$	$\Delta S(M,Q)\ge 0$
Irrelevant deformations	$\Delta S$, $\Delta\Omega^*$	$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$
Entanglement theory	Kernel $-S(V_i\bar V_j)$	conditional positivity
Wigner-positive states	$s(T)$0	$s(T)$1 under sufficient conditions
Sampling/combinatorics	$s(T)$2	$s(T)$3

These formulations are technically distinct. Their commonality lies less in a shared proof template than in a recurring principle: entropy positivity functions as a consistency criterion that can be sharpened into inequalities for couplings, amplitudes, microstate counts, or distributional spread (Liu et al., 8 Sep 2025, Cheung et al., 2018, Fernández-Sarmiento et al., 20 Aug 2025, Casini et al., 2012, Qian et al., 23 Jan 2026, Ai et al., 29 Jun 2026).

2. Thermal effective field theory and irrelevant deformations

The thermal EFT version is developed explicitly in "Positivity bounds from thermal field theory entropy" (Liu et al., 8 Sep 2025). The setup is a single real scalar field $s(T)$4 with shift symmetry and EFT Lagrangian

$s(T)$5

in the regime $s(T)$6. The Euclidean finite-temperature framework uses

$s(T)$7

The key thermodynamic input is a sharpened entropy-consistency inequality: after accounting for quantum entanglement through the Araki–Lieb inequality,

$s(T)$8

the heavy-sector thermal entropy is Boltzmann suppressed in $s(T)$9, leading to

$\Delta s(T)$0

For the dimension-8 operator, the explicit computation gives

$\Delta s(T)$1

and hence

$\Delta s(T)$2

Since $\Delta s(T)$3 for $\Delta s(T)$4, the Wilson coefficient obeys

$\Delta s(T)$5

The same paper notes that higher corrections such as $\Delta s(T)$6 and ring-resummed terms are parametrically suppressed for $\Delta s(T)$7, so the sign inference is robust within the EFT regime. It also shows that the forward-limit tree amplitude

$\Delta s(T)$8

satisfies the standard S-matrix positivity condition iff $\Delta s(T)$9, making the thermal derivation complementary to amplitude-based arguments (Liu et al., 8 Sep 2025).

A broader grand-canonical formulation appears in "On Entropy Bounds for Irrelevant Operators" (Fernández-Sarmiento et al., 20 Aug 2025). There the conjecture is stated for a CFT deformed by a leading irrelevant operator $s_{\rm EFT}(T)\ge s_{\rm free}(T)$0 of dimension $s_{\rm EFT}(T)\ge s_{\rm free}(T)$1, with

$s_{\rm EFT}(T)\ge s_{\rm free}(T)$2

to leading order in the deformation, and with thermal subtraction

$s_{\rm EFT}(T)\ge s_{\rm free}(T)$3

The paper proves, for perturbatively close theories and fixed $s_{\rm EFT}(T)\ge s_{\rm free}(T)$4,

$s_{\rm EFT}(T)\ge s_{\rm free}(T)$5

At linear order,

$s_{\rm EFT}(T)\ge s_{\rm free}(T)$6

The examples exhibit broad agreement with known positivity and causality constraints. For a $s_{\rm EFT}(T)\ge s_{\rm free}(T)$7 Goldstone with quartic self-interaction,

$s_{\rm EFT}(T)\ge s_{\rm free}(T)$8

so $s_{\rm EFT}(T)\ge s_{\rm free}(T)$9 implies entropy increase. For Euler–Heisenberg theory,

$\Delta S(M,Q)$0

again giving $\Delta S(M,Q)$1 when $\Delta S(M,Q)$2 and $\Delta S(M,Q)$3. The $\Delta S(M,Q)$4 nonlinear sigma model in $\Delta S(M,Q)$5D and the $\Delta S(M,Q)$6-deformed Ising CFT furnish additional nontrivial checks, while marginal deformations such as $\Delta S(M,Q)$7D $\Delta S(M,Q)$8 are presented as outside the conjecture’s proper scope (Fernández-Sarmiento et al., 20 Aug 2025).

3. Black-hole entropy shifts, Wald positivity, and extremality

The black-hole version is formulated and proved in "Proof of the Weak Gravity Conjecture from Black Hole Entropy" (Cheung et al., 2018). The theory is Einstein–Maxwell plus higher-dimension operators,

$\Delta S(M,Q)$9

$\Delta S(M,Q)\ge 0$0

and the entropy shift at fixed ADM mass and charge is

$\Delta S(M,Q)\ge 0$1

For thermodynamically stable charged black holes with $\Delta S(M,Q)\ge 0$2, and assuming tree-level dominance from integrating out healthy heavy fields, the main result is

$\Delta S(M,Q)\ge 0$3

The proof uses the Euclidean path integral and a canonical free-energy inequality,

$\Delta S(M,Q)\ge 0$4

which is then shown to be equivalent to $\Delta S(M,Q)\ge 0$5. In $\Delta S(M,Q)\ge 0$6, positivity for all $\Delta S(M,Q)\ge 0$7 in the stable window yields

$\Delta S(M,Q)\ge 0$8

and in the near-extremal limit this reduces to

$\Delta S(M,Q)\ge 0$9

The extremal charge-to-mass ratio is shifted upward,

$\Delta S$0

so large extremal black holes become unstable to fragmentation, thereby satisfying the Weak Gravity Conjecture in the stated regime (Cheung et al., 2018).

The AdS extension is developed in "Higher-Derivative Corrections to Entropy and the Weak Gravity Conjecture in Anti-de Sitter Space" (Cremonini et al., 2019). There the conjecture is that, for thermodynamically stable AdS–Reissner–Nordström black holes, the leading higher-derivative correction to the microcanonical entropy at fixed $\Delta S$1 is positive,

$\Delta S$2

The paper confirms a universal extremality–entropy relation,

$\Delta S$3

with a modified exact-extremal relation

$\Delta S$4

For sufficiently large neutral black holes, entropy positivity enforces

$\Delta S$5

and in AdS$\Delta S$6 the holographic map

$\Delta S$7

connects entropy positivity to the CFT inequality $\Delta S$8. The same analysis relates entropy positivity to extremality shifts and to transport data such as $\Delta S$9 (Cremonini et al., 2019).

A further refinement appears in "Unearthing the intersections: positivity bounds, weak gravity conjecture, and asymptotic safety landscapes from photon-graviton flows" (Knorr et al., 2024). In that setting, Wald entropy is sensitive to the four-dimensional Euler density,

$\Delta\Omega^*$0

through the topological contribution

$\Delta\Omega^*$1

The entropy-positivity family is written as

$\Delta\Omega^*$2

while the near-extremal WGC limit is

$\Delta\Omega^*$3

In the photon–graviton asymptotic-safety landscape, the paper finds Planck-scale-suppressed violations of both WGC and S-matrix positivity across the landscape. In this context, entropy-based positivity acts primarily as a lower bound on the Euler coupling $\Delta\Omega^*$4, rather than as a direct falsification of asymptotic safety (Knorr et al., 2024).

4. Entanglement and phase-space formulations

In entanglement theory, "Positivity, entanglement entropy, and minimal surfaces" analyzes a distinct but structurally related conjecture: whether the $\Delta\Omega^*$5 limit of replica path integrals can be represented by boundary insertions at first order in $\Delta\Omega^*$6 (Casini et al., 2012). If so, reflection positivity at integer Rényi index would descend to conditional positivity of the kernel

$\Delta\Omega^*$7

meaning

$\Delta\Omega^*$8

Equivalent formulations include positivity of the second-difference matrix and infinite divisibility of

$\Delta\Omega^*$9

for every $\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$0. The resulting hierarchy contains, at lowest order,

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$1

and, for one-parameter translation-invariant families $\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$2,

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$3

The paper finds no counterexample in the few exact QFT results available, but it does find holographic counterexamples for more complicated multi-parameter minimal-surface geometries. This places the conjecture on a restricted, rather than universal, footing (Casini et al., 2012).

In phase-space quantum mechanics, "Upper bounds on the purity of Wigner positive quantum states that verify the Wigner entropy conjecture" studies Wigner-positive states $\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$4 with

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$5

and Wigner entropy

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$6

The conjecture is that, among physical Wigner non-negative states, pure Gaussian states minimize the entropy, so

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$7

Using truncated-series lower bounds on $\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$8, the paper constructs

$\Delta S\ge 0 \Leftrightarrow \Delta\Omega^*<0$9

and derives purity-only sufficient conditions in terms of

$-S(V_i\bar V_j)$0

Two explicit thresholds are proved: $-S(V_i\bar V_j)$1 The same paper also shows why stronger purity-only criteria cannot follow from the minimal constraints alone: the flat-top distribution

$-S(V_i\bar V_j)$2

has entropy $-S(V_i\bar V_j)$3, so the threshold $-S(V_i\bar V_j)$4 is sharp under those constraints (Qian et al., 23 Jan 2026).

5. Combinatorics, graph entropy, and numerical schemes

In discrete probability, "An entropic analogue of the MMS conjecture" gives a sharp lower bound for the entropy of sums sampled without replacement (Ai et al., 29 Jun 2026). For a nontrivial zero-sum multiset $-S(V_i\bar V_j)$5, $-S(V_i\bar V_j)$6, and

$-S(V_i\bar V_j)$7

with $-S(V_i\bar V_j)$8 a uniformly random $-S(V_i\bar V_j)$9-subset, the theorem states

$s(T)$00

The proof combines a sign-split poset, Strong Sperner theory, majorization by a hypergeometric law $s(T)$01, and a monotonicity argument showing

$s(T)$02

Equality is attained for

$s(T)$03

where $s(T)$04 has exactly two atoms with weights $s(T)$05 and $s(T)$06. In this setting, entropy positivity means strict non-determinism: $s(T)$07 for every nontrivial zero-sum population (Ai et al., 29 Jun 2026).

" A proof of the Shepp-Olkin entropy monotonicity conjecture" concerns the Poisson–binomial law $s(T)$08 for independent Bernoulli variables with biases $s(T)$09 (Hillion et al., 2018). Its main theorem states that if all $s(T)$10, then the Shannon entropy

$s(T)$11

is non-decreasing as any coordinate $s(T)$12 increases within $s(T)$13. Using the derivative identity

$s(T)$14

a representation in terms of mixing coefficients $s(T)$15, and a discrete odd-moment argument, the paper proves $s(T)$16, with equality only in the fully fair binomial case $s(T)$17 for all $s(T)$18. This is a monotonicity theorem rather than a lower-bound theorem, but it belongs to the same structural family in which entropy positivity is identified with an ordering principle (Hillion et al., 2018).

In graph theory, "A positivity property of the dimer entropy of graphs" studies monomer–dimer entropy on lattices and regular graphs (Butera et al., 2014). For infinite bipartite lattices, the conjecture is that the expansion

$s(T)$19

has strictly positive coefficients $s(T)$20. For a connected regular graph with matching counts $s(T)$21, graph entropy is

$s(T)$22

and the deviation from the complete-graph baseline,

$s(T)$23

admits a Newton expansion in the dimer density. Graph-positivity means all Newton coefficients are nonnegative, equivalently

$s(T)$24

for all admissible $s(T)$25. Extensive computations show that violations are rare in regular bipartite graphs and decrease in frequency with graph size; the paper conjectures that for each degree $s(T)$26, the frequency of violations among connected $s(T)$27-regular bipartite graphs tends to zero as $s(T)$28 (Butera et al., 2014).

In numerical analysis, "EPO: A Unified Framework for Entropy Stability, Positivity, and Oscillation Suppression" uses the term in a geometric-algorithmic sense (Wu, 31 Mar 2026). Starting from a candidate finite-volume or discontinuous-Galerkin update with cell average $s(T)$29, the method scales along the ray

$s(T)$30

Three admissibility radii are defined: a positivity or invariant-set radius $s(T)$31, an entropy radius $s(T)$32, and an oscillation radius $s(T)$33. The limiter factor is

$s(T)$34

or $s(T)$35 when the full entropy profile is available. Under a weak cell-average entropy budget $s(T)$36, this scaling yields strong quadrature-based entropy inequalities,

$s(T)$37

while preserving the cell average and keeping all nodal states inside the admissible set $s(T)$38. The same mechanism extends to any prescribed finite family of convex entropy pairs (Wu, 31 Mar 2026).

6. Limits of validity and open directions

Across these literatures, entropy positivity is theorem-level only inside sharply delimited regimes. In thermal EFT, the scalar proof requires locality, Lorentz invariance in the EFT domain, unitarity, microcausality, thermal equilibrium, the hierarchy $s(T)$39, and control of the derivative expansion; beyond that regime the Entropy-Positivity Conjecture remains conjectural and may require spectral positivity, Kubo formulas, or nonperturbative control (Liu et al., 8 Sep 2025). For irrelevant deformations, the equivalence between $s(T)$40 and $s(T)$41 assumes smoothness in $s(T)$42, no nearby phase transition, and a well-defined thermal ensemble; marginal deformations such as $s(T)$43D $s(T)$44 are presented as outside the intended domain (Fernández-Sarmiento et al., 20 Aug 2025).

The black-hole versions likewise depend on thermodynamic stability and EFT control. In asymptotically flat space, the proof requires positive specific heat and tree-level dominance from integrating out healthy heavy fields (Cheung et al., 2018). In AdS, Casimir-scheme issues, extremal subtleties, and stability windows matter (Cremonini et al., 2019). In the asymptotic-safety analysis, small Planck-scale-suppressed violations of WGC and S-matrix positivity imply that entropy positivity becomes a constraint on the Euler coupling rather than a universal automatic consequence (Knorr et al., 2024).

In entanglement theory, conditional positivity survives many exact QFT checks but fails for certain holographic multi-parameter geometries, even without explicit phase transitions, so any general boundary-insertion picture at $s(T)$45 is necessarily restricted (Casini et al., 2012). In the Wigner setting, purity-only methods cannot reach the full extremal regime $s(T)$46 without additional physicality constraints beyond non-negativity, normalization, and $s(T)$47 (Qian et al., 23 Jan 2026). In combinatorics, the zero-sum and nontriviality assumptions are essential to the entropic MMS theorem, and the Rényi-entropy analogue is only partially resolved (Ai et al., 29 Jun 2026).

These limitations suggest that the phrase names a research program rather than a single universal law. Its most successful instances share a clear strategy: identify an entropy functional tied to state counting, thermal occupation, or convexity; prove that admissible ultraviolet completion, fairness, positivity, or equilibrium can only move that functional in one direction; and use the resulting sign to derive quantitative constraints. The open problem is not whether entropy positivity exists in isolated forms—it demonstrably does—but how far those forms can be unified across thermal field theory, gravity, entanglement, probability, and discretized dynamics.