A Three-Domain Modular Correspondence with Covariant Small-Diamond Reconstruction
Quantum Information to Dynamic Semiclassical Geometry develops a falsifiable correspondence between quantum relative information, modular dynamics, local stress-energy, and semiclassical spacetime geometry. The framework is deliberately divided into two sectors so that quantum-information tests are not confused with gravitational backreaction.
The kinematic sector, LMIEGC-K, begins with quantum relative entropy and modular theory. For regulated systems it uses the exact identity D(ρ||σ)=Δ⟨K⟩−Δs, while the continuum formulation uses Araki relative entropy appropriate to type-III local algebras. The modular generator is then tested against a geometry-weighted local stress-energy form. Three domains are distinguished: D0 contains exact local-modular benchmarks such as Rindler and conformal-ball constructions; D1 contains controlled time-dependent curved backgrounds; and D2 proposes an RG-aware ultraviolet-locality hypothesis that permits masses, relevant operators, running couplings, and logarithmic corrections.
A finite-regulator probe-completeness criterion prevents apparent agreement from being inferred from a trivial set of test states. The manuscript further extends this criterion to continuum QFT by introducing an operational truncation remainder. A continuum-locality claim is allowed only when both the retained-sector response error and the omitted-operator contribution vanish along the regulator and basis-enlargement sequence.
The gravitational sector, LMIEGC-G, is logically separate. It uses a matched renormalized semiclassical effective action, generalized entropy, and the off-shell Iyer-Wald causal-diamond identity. Modular nonlocality is propagated into a regional gravitational residual. The principal closure result is local rather than global: sufficiently small causal diamonds inside a convex normal neighborhood recover quadratic projections of the local field-equation residual. A finite, assignable family of timelike orientations reconstructs the complete symmetric residual tensor. The finite-radius reconstruction error is bounded explicitly by the local Riemann tensor, its first covariant derivative, and first and second derivatives of the field-equation residual.
This local construction avoids assuming a globally invertible Lorentzian Radon transform. Global caustics and conjugate points therefore do not automatically invalidate the pointwise result, provided each causal diamond lies within a controlled convex normal neighborhood and the required local scale separation exists. The framework remains first order in the gravitational propagation step and does not claim a derivation of nonlinear quantum gravity.
The paper also imposes a semiclassical source condition based on stress-energy fluctuations and requires matched renormalization of the matter stress tensor, gravitational effective action, generalized entropy, and allowed stress-tensor improvement terms. This prevents renormalization conventions or boundary terms from being mistaken for new information-gravity effects.
The paper does not claim that analog quantum simulators measure real gravitational backreaction. A proposed laboratory program using entanglement-Hamiltonian tomography, Gaussian covariance reconstruction, and curved-spacetime Bose-Einstein-condensate simulation tests only LMIEGC-K.
The associated reproducibility package provides executable finite-dimensional and numerical checks of the relative-entropy identity, probe-frame completeness, continuum truncation logic, small-diamond remainder scaling, and the explicit four-dimensional orientation-frame reconstruction. Negative controls demonstrate failure under rank-deficient probes and non-vanishing truncation tails.
The framework is falsifiable at each stage: modular ultraviolet locality, continuum truncation control, probe completeness, renormalization matching, semiclassical source validity, generalized-entropy closure, and the small-diamond approximation can fail independently. The result is a bounded proposal for linking quantum information to semiclassical geometry without introducing an arbitrary information-gravity coupling.