Gravity from entropy (GfE) identifies its Lagrangian with the geometric quantum relative entropy (GQRE) between the spacetime metric and a metric induced by matter and curvature, treating the metric itself as a quantum operator. We extend this construction to the Tsallis, Rényi, and Sharma-Mittal families, which we call gravity from generalized entropy (GfGE), and determine their common Einstein limit, weak-field spectra, auxiliary-field formulation, and horizon entropy. The entropy choice controls the leading higher-curvature terms and additional modes. We confirm that the linearized GfE spectrum contains a healthy scalar and a Planck-scale tachyonic spin-2 ghost. We identify the conditions that remove this ghost: curvature must enter through its scalar trace, or the entropy generator must be flat to quadratic order at the identity. The latter gives exact general relativity for Tsallis at
q=0 and ghost-free
f(R,Lm) theories for Rényi and Sharma-Mittal. On shell, the auxiliary G-field is proportional to the tensor in Wald's entropy formula. For analytic entropy functions, horizons with a single curvature scale have an area term followed by constant and inverse-area corrections. We also extend the volume-constraint cosmological constant to the full family; its leading entropy dependence is set by the scalar mass. A finite set of entropy-function derivatives controls the leading weak-curvature departures from general relativity.
This paper establishes a local structural bridge between two recently developed information-geometric frameworks for observer-dependent probability representations on finite state spaces. Background. On the parametric side, the partition-adapted framework of Khomyakov (KLEO v3.24.1) defines observer entropy as the Kullback–Leibler divergence between a parametric distribution and the uniform lift of its coarse-graining, and shows that it scales quadratically with the resolution threshold, with a coefficient determined by the Fisher information matrix. On the empirical side, the Interference-First Reality (IFR) framework of Bianchi (2026) identifies multi-observer objectivity with the Fréchet barycenter of observer-induced distributions in the Jensen–Shannon metric, with almost-sure convergence guaranteed by metric barycenter theory. Principal results. The paper derives four main results: 1. Local Metric Bridge Lemma (Section 3): On the open probability simplex, the Jensen–Shannon divergence satisfies the local quadratic expansion D_JS(p ‖ p + εδ) = (ε²/8) I_p(δ) + O(ε³), where the explicit coefficient 1/8 arises from the midpoint structure of the symmetric divergence. 2. Multi-Observer Quadratic Aggregation Theorem (Section 4): For a partition-adapted parametric family satisfying conditions (A1)–(A3) of Khomyakov (KLEO v3.24.1), the empirical Jensen–Shannon Fréchet barycenter of an N-observer ensemble, centered on the within-class neutral locus, exists uniquely for small ε and coincides to leading order with the weighted Fisher–Rao barycenter of the within-class perturbation directions. The minimal Jensen–Shannon dispersion satisfies a quadratic scaling law governed by the within-class block I_ww(θ₀*) of the Fisher information matrix. 3. Leading-order coincidence of estimators (Corollary 6.1): The variational Fréchet minimizer p_F and the closed-form geometric overlap estimator p_G of Bianchi (2026) agree to order O(ε²) in Jensen–Shannon distance within the local regime. 4. Multi-observer Landauer corollary (Corollary 6.6): Under a per-channel irreversibility assumption and an additivity postulate for independent observer channels, the aggregate energetic cost of barycenter formation is bounded below by k_B T times the Jensen–Shannon dispersion. Scope. The bridge is local in ε (the within-class perturbation amplitude) and finite in N (the number of observers). It does not subsume Bianchi's almost-sure N → ∞ consistency theorem nor Khomyakov's ε → 0⁺ quadratic scaling result; it establishes their structural compatibility in the overlapping regime where both apply. The results are specialized to balanced exponential families (softmax), with explicit worked-example coefficients for four- and five-point state spaces. Domain-specific applicability is assessed against the cross-domain distinguishability hierarchy of Bianchi (2026), Table 14, identifying the low-dispersion regime (d_JS ≲ 10⁻² nats) as the primary domain of validity. Keywords: Jensen–Shannon divergence, Kullback–Leibler divergence, Fisher information, Fisher–Rao metric, multi-observer aggregation, Fréchet barycenter, probability simplex, partition-adapted parameterization, observer entropy, information geometry. MSC 2020: 62B10, 94A17, 53B12, 62F12.
Observer-information entropy production quantifies the loss of recoverable information about a physical source when an already formed observation undergoes source-independent irreversible degradation. The statistical second-law statement
Σobs[s,t]=I(X;Rs)−I(X;Rt)=I(X;Rs∣Rt)≥0 follows from data processing, but by itself it does not identify a physical energetic or thermodynamic cost. This paper establishes such a connection within an explicit and deliberately restricted stochastic-thermodynamic regime. We consider a finite observer memory
Rt coupled to a single heat bath at temperature
T. A source label
X is fixed during the passive relaxation, the transition generator of
Rt is independent of
X, and the memory dynamics satisfies detailed balance with respect to a strictly positive equilibrium law
q. For the joint source--memory ensemble and for the unconditioned memory marginal, we define the standard stochastic-thermodynamic entropy-production rates. Using the relative-entropy chain rule and the detailed-balance
H-theorem, we prove the exact correlation-dissipation decomposition
kBS˙totXR=σobs+kBS˙totR. Consequently,
S˙totXR≥kBσobs,ΔStotXR[s,t]≥kBΣobs[s,t]. At fixed temperature the same result is an excess-free-energy dissipation bound, \[ \Fex_{XR}(s)-\Fex_{XR}(t) \ge k_BT\Sigma_{\mathrm{obs}}[s,t]. \] Equality holds when the memory marginal remains at equilibrium, in which case all thermodynamic irreversibility is attributable to the destruction of source--record correlation. A symmetric two-state memory provides an exactly solvable saturation example. The result is not a universal Landauer principle. It constrains total entropy production and loss of nonequilibrium free energy during passive decorrelation; it does not assert a universal heat or work cost for arbitrary information loss. Driven memories, source-dependent transition rates, active measurement, feedback, multiple reservoirs, and nonequilibrium steady states require the broader information-thermodynamic bookkeeping developed in stochastic thermodynamics. We then place the thermodynamic result beside two Fisher--Shannon bridges. Under Gaussian observation heat flow
Yt=X+Bt, observer-information dissipation obeys the exact identity
σobs(t)=21(td−J(Yt))=2t21mmse(X∣Yt), while pairwise statistical distinguishability obeys \[ \frac{d}{dt}\KL(p_t\|q_t) =-\frac12\Jrel(p_t\|q_t). \] Recent temporal-Fisher speed-limit results provide a complementary route from physical entropy production to the speed of probability evolution. Together these results separate three objects that are often conflated: thermodynamic irreversibility, Shannon information loss about the physical source, and Fisher-geometric statistical motion. The paper thereby supplies a physical-cost layer for observer information theory without identifying information with energy or thermodynamic entropy.
The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of
π/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is
ε-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.
Entropies are fundamental measures of uncertainty with central importance in information theory and statistics and applications across all the quantitative sciences. Under a natural set of operational axioms, the most general form of entropy is captured by the family of Rényi entropies, parameterized by a real number
α. Conditional entropy extends the notion of entropy by quantifying uncertainty from the viewpoint of an observer with access to potentially correlated side information. However, despite their significance and the emergence of various useful definitions, a complete characterisation of measures of conditional entropy that satisfy a natural set of operational axioms has remained elusive. In this work, we provide a complete characterisation of conditional entropy, defined through a set of axioms that are essential for any operationally meaningful definition: additivity for independent random variables, invariance under relabeling, and monotonicity under conditional mixing channels. We show that the most general conditional entropy is a generalized convex combination of conditional entropies defined as exponential averages of Rényi entropies of the conditioned distribution, parameterized by a real number and a probability measure on the positive reals.
Most quantum divergences derive their structure from classical f-divergences or Renyi-type constructions, a dependence that obscures several quantum geometric effects. We introduce a quantum relative-alpha-entropy that extends Umegaki's relative entropy while falling outside the f-divergence class. The proposed divergence exhibits a nonlinear convexity property, which yields a generalized convexity result for the Petz-Renyi divergence for alpha greater than one, complementing the known convexity for alpha less than one. It is additive under tensor products, invariant under unitary transformations, and depends only on the relative geometry of quantum states rather than their absolute magnitudes. Using Nussbaum-Szkola-type distributions, we also establish an exact correspondence of this divergence with classical relative-alpha-entropy. This reveals relative-alpha-entropy as a fundamentally geometric notion of quantum distinguishability not captured by existing divergence frameworks.