Emergent Spacetime from Informational Recursion

This paper develops the spacetime-emergence component of Unified Recursion Theory (URT), showing that time, space, Lorentz symmetry, causal structure, and curvature arise from the recursion law that governs physical evolution. No geometric or spacetime assumptions are introduced. The geometry of the world appears as the smooth limit of informational updates.

URT contains exactly three independent recursion operators:

1. Psi_cons — reversible updates (Delta H_cons = 0)

2. Psi_comp — irreversible compressive updates (Delta E = lambda * k_B * T_loc * Delta H)

3. ORM — the admissibility and selection operator

These three operators define three independent informational directions. In the continuum limit, these directions generate the three spatial dimensions. Temporal structure arises from the recursion index n, which increases only when an admissible update occurs. The intrinsic recursion cadence Delta t_0 provides the operational definition of physical time.

Spatial geometry emerges from informational separation, quantified by the Fisher-information metric. Differences between nearby recursion states define spatial distances through:
ds^2 = g_ij(theta) dtheta^i dtheta^j
The metric reflects the sensitivity of informational configurations to parameter changes. Regions of high informational curvature correspond to regions of geometric curvature.

Spacetime arises when the temporal increments associated with recursion (dt = Delta t_0) are combined with these spatial increments, yielding the emergent interval:
d tau^2 = (dt)^2 - ds_space^2
This interval is not imposed; it is the only quadratic form consistent with reversible recursion and the Fisher metric. Causality follows from the rule that no system can execute more than one recursion update per cadence. This constraint produces the emergent light cone and yields the bound:
ds_space / dt <= 1
Identifying this maximal rate with the physical constant c gives:
c = 1 / Delta t_0

Lorentz symmetry appears as the invariance of the informational interval under reversible recursion. Transformations preserving (dt)^2 - ds_space^2 form the Lorentz group. Irreversible compression events break Lorentz symmetry, mirroring the division between reversible and irreversible dynamics.

Curvature arises from gradients in the Fisher-information metric. Informational Field Theory in Strong Curvature (IFT-SC) showed that stiffness acquires a gravitational component sigma_grav(K), which regulates recursion efficiency in curved regions. In the emergent spacetime picture, geometry is the macroscopic shadow of informational gradients, while gravity reflects how recursion efficiency responds to curvature.

This paper unifies:
• time as recursion index
• space as Fisher-information geometry
• light cones as admissibility constraints
• Lorentz symmetry as reversible-recursion invariance
• curvature as informational sensitivity
• gravity as stiffness-regulated recursion behavior

No background manifold or geometric axioms are assumed. Spacetime emerges from informational recursion alone.

Dependencies within the URT paper family:
This work relies directly on the foundational framework established in:
(1) Unified Recursion Theory: Core Proportionality Law
(2) Informational Field Theory in Strong Curvature (IFT-SC)
(3) Stiffness Wave Theory (δσ-Waves)
(4) ORM and the Measurement Problem

URT PAPER FAMILY

This work forms part of the Unified Recursion Theory (URT) research program, which develops a cross-domain framework for physical evolution based on constrained informational recursion and an energy–entropy proportionality law. Each paper in the series is self-contained, while collectively establishing the theoretical structure across quantum, geometric, biological, cosmological, and particle-level domains.

Related URT works available on Zenodo:

FOUNDATIONAL PAPERS

1. Unified Recursion Theory — Core Framework (URT Core)

DOI: 10.5281/zenodo.17642761
Record: https://zenodo.org/records/17642761

2. Discrete Admissible Regimes in Unified Recursion Theory: Operator Closure, Constraint Topology, and the Necessity of Five Operators

DOI: 10.5281/zenodo.18148192
Record: https://zenodo.org/records/18148193

3. Informational Field Theory in Strong Curvature (IFT-SC)

DOI: 10.5281/zenodo.17850379
Record: https://zenodo.org/records/17850379

4. Dynamical Evolution of the Informational Stiffness Field (ISW Theory)

DOI: 10.5281/zenodo.17860533
Record: https://zenodo.org/records/17860533

RESOLUTION PAPERS (PHYSICAL PARADOXES)

5. Informational Recursion and the Dissolution of the Black Hole Information Paradox

DOI: 10.5281/zenodo.17868662
Record: https://zenodo.org/records/17868662

6. ORM and the Quantum Measurement Problem (ORM)

DOI: 10.5281/zenodo.17881944
Record: https://zenodo.org/records/17881944

BRIDGING / CONSTRAINT PAPER

7. Distinguishability Geometry in Informational State Space

DOI: 10.5281/zenodo.17957062
Record: https://zenodo.org/records/17957062

Provides the geometric foundation for informational state space.
Underpins the emergence of spacetime, efficiency universality, and landscape geometry.

THEORETICAL EXPANSION PAPERS

8. Emergent Spacetime from Informational Recursion

DOI: 10.5281/zenodo.17885555
Record: https://zenodo.org/records/17885555

9. λ-Universality Across Scales (λ-UAS)

DOI: 10.5281/zenodo.17934065
Record: https://zenodo.org/records/17934065

10. Free-Energy Landscape Geometry in Unified Recursion Theory

DOI: 10.5281/zenodo.17940995
Record: https://zenodo.org/records/17940995

BIOLOGY / COMPLEXITY PAPER

11. URT in Biology: Efficiency, Folding Funnels, Replication Fidelity, and Molecular Motor Dynamics

DOI: 10.5281/zenodo.17945209
Record: https://zenodo.org/records/17945209

COSMOLOGICAL EXTENSIONS

12. Cyclic Cosmology from Informational Recursion

DOI: 10.5281/zenodo.17955043
Record: https://zenodo.org/records/17955043

13. Antimatter as Inverse Recursion: Temporal Operator Asymmetry and Matter–Antimatter Imbalance in Unified Recursion Theory

DOI: 10.5281/zenodo.17955043
Record: https://zenodo.org/records/17955625