Spacetime as Entropic Capacity: An Information-Theoretic Framework for Emergent Gravity with Empirical Validation

Spacetime as Entropic Capacity: An Information-Theoretic Framework for Emergent Gravity with Empirical Investigation Wayne A. Satz, MD * 1 and Ryan M. Gow 2 1 Temple University Health System, Philadelphia, PA, USA 2 Independent Researcher, USA December 2025 Space expands to accommodate History. Time slows to accommodate Complexity. Abstract We develop an information-theoretic framework in which spacetime, mass, gravity, and motion emerge from a nite-capacity discrete causal network. Space is identied with the addressable storage of a causal information structure, time with the monotonic growth of committed history measured by description length, mass with the minimum description length of local subsystems, and gravity with the throttling of local update rates when processing demand approaches holographic capacity bounds. We establish a Law of Entropic Capacity : the maximum information-processing rate available to any bounded region scales with boundary area and is constrained by the MargolusLevitin quantum speed limit. Using the quantum Fisher information metric to ground emergent geometry, we derive the Einstein eld equations as a capacity-response relation via Jacobson's thermodynamic approach. A 1+1D toy model demonstrates how capacity constraints yield metric structure with time-dilation behavior matching Schwarzschild geometry to leading order. We investigate empirical applicability through LogVAMS, an anomaly detection system applying entropic capacity mathematics to software systems. Initial results on synthetic data (162 injected failures) showed promising lead time ( 35.9±12.3 observations). However, large-scale validation on production data (BGL supercomputer logs, 4.7M entries) yielded F1 = 3.05%, indicating that physics-only detection does not generalize without integration with conventional machine learning methods. A hybrid architecture (ML detection + physics features) achieves F1 = 7580%, suggesting physics-inspired methods excel at interpretability enhancement rather than standalone detection. This negative result is scientically valuable and reported in full. Cross-domain validation through the 4MSB evolution experiment remains ongoing. We present falsiable predictions for holographic noise searches ( S1/2 h∼10−22 Hz−1/2 at f∼1 MHz ), complexity-sensitive gravitation ( ∆g/g ≲10−15 ), and gamma-ray burst dispersion bounds ( |∆v|/c < 10−15 at TeV energies). Keywords: Entropic gravity, holographic principle, quantum Fisher information, emergent spacetime, minimum description length, critical slowing down, computational bounds * Corresponding author. ORCID: 0000-0003-3090-3852 1 1 Introduction Modern theoretical physics rests on two extraordinarily successful yet conceptually disjoint pillars: quantum theory and general relativity. Despite decades of eortincluding loop quantum gravity, string theory, and causal dynamical triangulationsno fully accepted, experimentally anchored unication exists. Several independent lines of research suggest that information , rather than elds or geometry, may be the most fundamental quantity:  Black hole thermodynamics and the BekensteinHawking relation [1,2] indicate that entropy scales with area , not volume, implying a holographic information bound.  The holographic principle [3,4] and AdS/CFT correspondence [5] suggest that bulk gravitational physics is encoded on lower-dimensional boundaries.  Entanglement-geometry correspondences [68] and the ER=EPR conjecture [9] propose that spacetime connectivity emerges from quantum entanglement.  Entropic gravity approaches [10,11] derive gravitational dynamics from thermodynamic principles on holographic screens.  Fundamental computational bounds [12,13] establish that quantum mechanics imposes speed limits on information processing tied to energy.  Complexityaction dualities [14] relate computational complexity growth to gravitational action, suggesting black holes saturate computational speed limits. This paper synthesizes these threads into a coherent entropic capacity framework with three distinguishing features: (i) mathematical precision through quantum Fisher information geometry and operational complexity measures; (ii) explicit derivations connecting capacity constraints to metric structure; and (iii) empirical investigation through a predictive anomaly detection system, including honest reporting of both successes and failures. Recent developments in statistical mechanics have converged on a striking reconceptualization: entropy measures not intrinsic disorder but observer-relative ignorance [27]. Jaynes' resolution of the Gibbs mixing paradox demonstrated that entropy change upon mixing two gases depends not on any objective dierence between them, but on whether an observer possesses the means to distinguish themdisorder is in the eye of the beholder. This insight has matured into formal frameworks such as Aguirre's observational entropy, which species how an observer's accessible properties coarse-grain their view of reality [28]. The present work takes this observer-dependence as foundational rather than incidental: if what exists depends on what can be distinguished, and distinguishability is bounded by nite processing capacity, then physical law itself emerges from information-theoretic constraints on observers. Crucially, we distinguish entropy (uncertainty about microstates) from complexity (minimum description length required for specication). The DMT neuroimaging studies of Irrmischer et al. [26] make this distinction empirically vivid: under conditions of ego dissolution, neural entropy increases while complexity decreases the self does not dissolve into noise but loses the structured information that constitutes it. What survives, we propose, is not what maximizes entropy but what maintains complexity within capacity bounds. The central insight can be stated concisely: Core Principle: Space expands to accommodate History. Time slows to accommodate Complexity. This principle explains both cosmological phenomena (dark energy, gravitational time dilation) and complex system behavior (critical slowing down, pre-failure dynamics). 2 1.1 Computational Ontology Summary To orient readers, we present the ontological identications established in this framework: Table 1: Entropic Capacity Framework: Ontological Mappings Physical Concept Entropic Ledger Complex Systems Space Addressable storage Memory, state space Time History accumulation Transaction log Mass Algorithmic complexity ( mDL ) State complexity Gravity Capacity throttling System lag Speed of light Causal update bound ( χ ) Propagation limit Energy Computational potential Processing capacity Dark energy Storage expansion Auto-scaling Dark matter Structural metadata Non-rendered state Event horizon Capacity saturation System deadlock These are not analogies but formal identications. The framework claims that physical laws emerge as optimization strategies in a nite-capacity information-processing system, and that this interpretation is both mathematically coherent and empirically testable. 1.2 Structure of This Paper Section 2establishes the mathematical foundations through six postulates, including quantum Fisher information geometry and operational complexity measures. Section 3derives gravitational dynamics from capacity constraints with explicit 1+1D demonstration. Sections 45 develop emergent phenomena and cosmological implications. Section 6investigates the framework through LogVAMS, reporting both initial success on synthetic data and subsequent failure on production data (F1 = 3.05%), with analysis of failure modes and the viable hybrid architecture. Section 7presents the 4MSB evolution experiment testing cross-domain universality. Section 8presents quantitative experimental predictions. Section 9addresses classical objections. Section 10 concludes with research directions. 2 Mathematical Foundations We replace the continuum manifold with a minimal set of information-theoretic structures, carefully distinguishing denitions, postulates, and derived results. 2.1 Discrete Causal Structure Postulate 2.1 (Causal Set) . The fundamental substrate is a discrete causal network C whose elements are events ei with directed edges ei→ej encoding causal precedence. There is no pre-given manifold; spacetime emerges from the structure of C . This aligns with causal set theory [15,16], where Lorentzian geometry emerges from discrete partial orders via Poisson sprinkling and order-interval volume counting. The causal structure preserves Lorentz invariance statistically rather than exactly at each point. 2.2 Emergent Geometry via Quantum Fisher Information To ground distance as distinguishability, we employ the quantum Fisher information (QFI) metric rather than the classical Fisher metric, which applies only to commuting families of states. 3 Denition 2.1 (Quantum Fisher Information Metric) . For a dierentiable family of density operators ρ(θ) parameterized by coordinates θµ , the quantum Fisher information metric is dened via the symmetric logarithmic derivative (SLD): Gµν(θ) = 1 2Tr [ρ{Lµ, Lν}] (1) where Lµ satises ∂µρ=1 2{ρ, Lµ} (2) and {A, B}=AB +BA denotes the anticommutator. In the quasi-classical regime where [ρ(θ), ρ(θ′)] →0 for nearby parameters, the QFI reduces to the classical Fisher information: Gµν =X x p(x|θ)∂µln p(x|θ)∂νln p(x|θ) (3) Postulate 2.2 (Emergent Spatial Metric) . The emergent spacetime metric gµν is conformally related to the quantum Fisher information metric on holographic screens: gµν = Φ(Ω) Gµν (4) where Φ(Ω) is a conformal factor determined by local capacity density. Distance between eective points corresponds to statistical distinguishability of their local quantum states. This construction is compatible with AdS/CFT entanglement-geometry correspondences [6, 8], where minimal surfaces computing entanglement entropy correspond to geodesics in the emergent bulk geometry. 2.3 Time as Committed History Postulate 2.3 (Time as Complexity Growth) . There is no fundamental global time parameter. Let H denote the global historythe set of all events and their committed outcomes. Dene proper time locally as proportional to the growth of description complexity: dτ ∝dK(Hlocal) (5) where K(·) denotes Kolmogorov complexity of the local history. Past is the set of bits committed to the ledger; future consists of potential events subject to causal consistency. Time does not ow; the historical record extends. This connects to complexityaction dualities [14] relating dK/dt to bulk gravitational action. 2.4 Operational Complexity: Minimum Description Length Since Kolmogorov complexity K(·) is uncomputable, we require an operational proxy for physical applications. Denition 2.2 (Minimum Description Length Proxy) . For a subsystem S with boundary Σ , dene the minimum description length as: mDL(Σ; S) = min q∈Q [L(q) + L( data |q)] (6) where Q is the class of predictive models consistent with boundary data, L(q) is the description length of model q , and L( data |q) is the conditional description length of subsystem data given the model. 4 Lemma 2.1 (MDL as Complexity Proxy) . Under coarse-graining over microscopic degrees of freedom: mDL →K+O(log K) (7) where the correction is sublinear in the complexity. Furthermore, ∇mDL induces an entropic force: Fentropic =T∇S≈T∇mDL (8) reproducing Newtonian scaling to leading order when T is identied with local horizon temperature. The proof follows from the correspondence between thermodynamic entropy and algorithmic complexity established by Zurek [17] and the entropic force derivation of Verlinde [11]. Postulate 2.4 (Mass as Description Length) . The eective gravitational mass m of a subsystem S is proportional to its minimum description length: m∝mDL(Σ; S) (9) consistent with Bekenstein bounds relating mass, entropy, and horizon area [1]. 2.5 The Law of Entropic Capacity We now state the central result unifying holographic bounds with computational speed limits. Postulate 2.5 (Law of Entropic Capacity) . For any bounded region with boundary area A , the maximum information-processing rate Ω is: Ω = A 4ℓ2 P ·1 τP =A 4ℓ2 PτP (10) where ℓP=pℏG/c3≈1.616×10−35 m is the Planck length and τP=pℏG/c5≈5.391×10−44 s is the Planck time. Proposition 2.1 (Unication of Bounds) . The Law of Entropic Capacity unies three independent results: 1. Holographic bound : The boundary encodes at most N=A/4ℓ2 P bits of information [4]. 2. MargolusLevitin bound : A quantum system with energy E requires time ∆t≥πℏ/2E to transition to an orthogonal state [13]. 3. Lloyd bound : Maximum computational rate is νmax = 2E/πℏ≈6×1033 operations/second per Joule [12]. The capacity Ω counts orthogonalizations per unit time state transitions to distinguishable congurations. In natural units with kB= 1 , this is measured in nats/second; to convert to bits/second, multiply by ln 2 . Hypothesis 2.1 (Capacity Response) . When local processing demand D=mDL ·νlocal approaches the entropic capacity Ω , the system responds via: 1. Throttling : Local update rate νlocal decreases, manifesting as gravitational time dilation. 2. Expansion : Available boundary area increases, manifesting as cosmic expansion. 5 2.6 Speed of Light as Causal Propagation Bound Postulate 2.6 (Causal Update Rate) . The speed of light c is the maximum causal propagation rate on the discrete network: c≡χ= max  causal edges updated unit proper time  (11) Lemma 2.2 (Observer Invariance of χ ) . In the continuum limit of Poisson-sprinkled causal sets [15], the maximum causal propagation depth per proper time increment is observer-invariant, reproducing the Lorentz-invariant speed c . The mass-energy relation becomes: E=mDL ·χ2 (12) where E is the computational potential the capacity to inuence other addresses per unit time. 3 Deriving Gravity from Capacity Constraints 3.1 Einstein Equations as Capacity-Response Following Jacobson [10], we derive the Einstein eld equations from thermodynamic equilibrium on local Rindler horizons, reinterpreted through the capacity framework. Proposition 3.1 (Thermodynamic Derivation) . Impose the rst law of thermodynamics δQ = T δS on local Rindler horizons with:  Entropy S=A/4ℓ2 P (holographic)  Temperature T=ℏa/2πkBc (Unruh)  Heat ux δQ =TµνkµkνdA dλ (stress-energy through horizon) Equilibrium δS = 0 under Raychaudhuri focusing yields: Rµν −1 2R gµν =8πG c4Tµν (13) In the capacity framework, δS represents capacity variation on the holographic screen, and δQ is incoming computation (bit-ips) bounded by Ω . The Einstein tensor emerges as a capacityresponse tensor : spacetime curvature is the system's adjustment to maintain processing within bounds. 3.2 Capacity-to-Metric Derivation: 1+1D Toy Model To make the connection between capacity constraints and metric structure explicit, we present a simplied derivation in 1+1 dimensions. Lemma 3.1 (1+1D Capacity-Metric Correspondence) . Consider a timelike strip with screen density σ(x) and local complexity demand d(x) = mDL(x)·ν(x) . Dene the capacity action: A=Z[Λ −λ(x) (ν(x)σ(x)−Ωmax(x))] dx dt (14) where Λ is a constant and λ(x) is a Lagrange multiplier enforcing the capacity constraint νσ ≤ Ωmax . 6 Proof. Extremizing A with respect to λ enforces: ν(x) = Ωmax(x) σ(x)=σ(x)/τP σ(x)=1 τP (15) when unconstrained. In the presence of mass (complexity concentration), mDL(x)>0 increases demand. To maintain d≤Ωmax , the local update rate must throttle: ν(x)≤Ωmax(x) mDL(x) (16) Identifying local proper time as dτ =ν dt and using Ωmax =A/4ℓ2 PτP with A∝r2 for a spherical screen at radius r enclosing mass M with mDL ∝M : dτ dt =ν ν0 =Ωmax/mDL Ωmax,0/mDL,0 ≈1−2GM rc2+OGM rc22 (17) which matches the Schwarzschild time dilation to rst order: dτ dt =r1−rs r≈1−rs 2r, rs=2GM c2 (18) The Schwarzschild radius rs emerges as the scale where local demand exactly saturates capacitythe buer overow boundary. 3.3 Geodesic Motion from Capacity Gradients Proposition 3.2 (Geodesics as Capacity Optimization) . Objects follow geodesics because they drift toward regions of higher processing eciency. Dene the capacity eciency: η(x) = Ω(x) D(x)=Ω(x) mDL ·ν(x) (19) The gradient ∇η denes the downhill direction in the capacity landscape. Variational Argument. A free particle minimizes integrated proper time, which in the capacity framework corresponds to maximizing total processing accomplished. The action: S=Zdτ =Zν(x(λ)) dλ (20) is extremized when δS = 0 , yielding the geodesic equation in the emergent metric induced by capacity constraints. This reproduces Verlinde's entropic force [11]: F=T∆S ∆x≈T∇mDL (21) interpreted as the ledger's resistance to moving high-complexity structures through address space. 7 3.4 Computational Speed Limits and Black Holes Proposition 3.3 (Black Holes Saturate Computational Bounds) . For a black hole of mass M , the MargolusLevitin bound yields: νmax =2Mc2 πℏ (22) The horizon area A= 16πG2M2/c4 implies the capacity bound: Ω = A 4ℓ2 PτP =4πGM2 ℏ (23) Comparing with νmax , black holes operate at O(1) of their theoretical maximum computational rate [14], supporting the interpretation that they are the fastest information processors in nature. 4 Emergent Phenomena: Force, Motion, and Measurement 4.1 Motion as Discrete Address Rewriting In the entropic ledger, there is no continuous motion. Standard physics describes motion as coordinate translation x→x+δx ; in our framework, the state at address x is cleared and rewritten at x+δx . The speed of an object is the rate at which these address updates propagate through the causal network, bounded by χ=c . Denition 4.1 (Inertia as Rewrite Cost) . Inertia is the computational cost of re-encoding a complex data structure at a new address: Inertia ∝mDL ·( address change rate ) (24) Newton's second law F=ma emerges as the macroscopic t to this cost curve. More complex objects (higher mDL ) require greater computational eort to relocate. 4.2 Measurement as Decoherence and Commitment Rather than invoking observer-dependent collapse, we frame measurement through decoherence and quantum Darwinism [18]. Hypothesis 4.1 (Measurement as Commitment) . A measurement is an environmentallyinduced decoherence event that selects robust pointer states. In the ledger picture:  Superposition : The ledger maintains a low-description, high-symmetry stateanalogous to lazy evaluation in computation.  Decoherence : Environmental interaction forces high-resolution specication of outcomes, increasing mDL .  Commitment : The specied outcome is written to the ledger as part of the irreversible historical record H . This avoids consciousness-dependent interpretations while preserving the operational distinction between unspecied and specied degrees of freedom. 8 4.3 Interpretive Analogies (Heuristic) The following computational analogies are oered as heuristics for resource allocation under capacity constraints, not claims about literal implementation: Interpretive Analogies  Lazy evaluation ↔ Superposition : Unobserved degrees of freedom remain in low-description states, conserving computational resources.  Frustum culling ↔ Cosmological sparsity : Distant regions may be represented by compressed summaries until causal contact forces resolution.  Instance sharding ↔ Causal isolation : High-complexity regions may be eectively isolated to maintain nite global update rates. These analogies suggest how a nite-capacity system might optimize resource allocation, but do not constitute claims about external simulation. 5 Cosmological Implications 5.1 Dark Energy as Capacity Expansion Conjecture 5.1 (Dark Energy from Storage Scaling) . As global history H accumulates, more bits must be stored. The capacity law requires: dA universe dt ∝dS history dt (25) The universe expands to provide boundary area sustaining Ω . Accelerated expansion corresponds to entropy production exceeding linear growth. Proposition 5.1 (Consistency with Observed Λ ) . The observed cosmological constant Λ≈ 10−52 m−2 implies a Hubble rate H0≈70 km/s/Mpc . The entropy production rate of the observable universe (dominated by black hole formation and CMB photon production) is approximately dS/dt ∼10104 kB/Gyr [20]. The required area growth rate: dA dt ∼4ℓ2 P·dS dt ∼1034 m2/s (26) is consistent with Hubble expansion of a horizon-scale surface, supporting the conjecture. 5.2 Dark Matter as Structural Information Conjecture 5.2 (Dark Matter from Non-Luminous Complexity) . Observed gravitational eects without luminous matter sources represent non-rendered structural information : graph connectivity, constraints, and boundary conditions in the causal network. This metadata has processing cost (manifesting as gravitational eects via capacity throttling) but no associated eld excitations (photons). This is a testable hypothesis . Discriminating predictions include:  Complexity-weighted lensing residuals in low-surface-brightness galaxies vs. baryon-only ts  Dierent scaling of dark matter halos with information complexity vs. total mass  Potential detection of non-thermal correlations in dark matter distribution 9 7.9 Neurological Evidence: Criticality and Consciousness Recent neuroscience research provides complementarythough not conrmatoryevidence for the role of criticality in maintaining structured existence. Irrmischer et al. [26] investigated the eects of DMT (N,N-dimethyltryptamine) on brain criticality using EEG, nding that:  DMT shifts brain oscillations away from criticality (DFA exponent: 0.79 →0.63 in alpha band)  The shift correlates with subjective self-dissolution ( r=−0.61 , p < 0.001 )  Entropy increases while complexity decreases under DMT  The direction is toward subcritical (inhibition-dominated) regimes Important distinction : Our CSD results show AR(1) increasing as systems approach capacity limits (the warning signal). The DMT data show DFA decreasing when neural systems depart from criticality (the dissolution signature). These are complementary observations of the same phenomenon from opposite directions: Phenomenon Direction Signature Approaching criticality Toward limit AR(1) ↑ , variance ↑ Departing criticality Away from limit DFA ↓ , complexity ↓ The DMT ndings support the general principle that proximity to criticality has measurable consequences for structured existence. The brain's self requires near-critical dynamics (DFA ≈0.8 ) to maintain coherenceconsistent with the Entropic Ledger principle that existence requires operation within specic capacity regimes. Proposed test : If CSD precedes dissolution, subjects transitioning into the DMT state should show transient AR(1) elevation (the system ghting to maintain structure) immediately before the DFA collapse. This prediction could be tested with existing EEG datasets. 8 Quantitative Experimental Predictions We present falsiable predictions with specic numerical targets. 8.1 Holographic Noise Searches Hypothesis 8.1 (Holographic Noise Signature) . Discrete Planck-scale updates produce correlated position uctuations detectable via cross-correlated interferometry. The predicted crossspectrum amplitude: S1/2 h(f)∼rℓP L≈10−22 Hz−1/2 (32) for arm length L∼40 m at frequencies f∼1  10 MHz . The Fermilab Holometer [23] has achieved sensitivity ∼10−21 Hz−1/2 ; null results to date constrain but do not exclude discretization at scales below current resolution. The decorrelation length is tied to τP -paced updates: signals should correlate over distances ≲cτP∼10−35 m . 16 8.2 Complexity-Sensitive Gravitation Hypothesis 8.2 (Complexity-Dependent Gravitational Coupling) . Two objects with identical rest mass but dierent internal complexity (e.g., entangled ion chain vs. crystalline mass) may exhibit dierent gravitational inuence: ∆g g≈α·∆ ˙mDL Ω (33) where α is an O(1) coupling constant and ˙mDL is the complexity update rate. Table 7: Estimated Complexity-Gravity Eect Sizes System ˙mDL (bits/s) ∆g/g Detectability Entangled ions (100) ∼106≲10−18 Future Superconducting qubits ∼109≲10−15 Challenging BEC ( 106 atoms) ∼1012 ≲10−12 Possible Current gravitational measurements achieve ∆g/g ∼10−9 ; the predicted eects require 39 orders of magnitude improvement but may be accessible to future quantum gravimeters [25]. 8.3 Gamma-Ray Burst Dispersion Hypothesis 8.3 (Energy-Dependent Photon Speed) . Discrete spacetime structure produces energy-dependent time-of-ight: ∆t∼E EP ·D c (34) where EP=pℏc5/G ≈1.2×1019 GeV is the Planck energy and D is the source distance. For GRB photons at E∼10 TeV from redshift z∼1 ( D∼1026 m ): ∆t∼10−8s (35) Current bounds from Fermi-LAT [24] constrain |∆v|/c < 10−15 at TeV energies, approaching sensitivity to Planck-scale eects. The capacity framework predicts a linear energy dependence distinguishable from source-intrinsic spectral lags (which show dierent energy scaling). 8.4 Critical Slowing Down Universality Hypothesis 8.4 (Universal Precursors) . All complex systems approaching capacity limits exhibit critical slowing down with rising variance and AR(1) →1 , regardless of substrate. Testable domains:  Financial markets : Pre-crash volatility clustering  Power grids : Pre-blackout frequency deviations  Biological networks : Pre-extinction population uctuations  Climate systems : Pre-transition paleoclimate proxies Detection of universal scaling exponents across these domains would support the framework's claim that capacity-limit mathematics is substrate-independent. 17 9 Addressing Classical Objections 9.1 Lorentz Invariance Objection: A discrete tick introduces a preferred reference frame. Response: The tick represents a partial causal order (Lamport-like logical time), not global simultaneity. Lorentz invariance emerges statistically from random (Poisson) causal set distributions [15]. The propagation bound χ is an upper limit on causal depth per local proper time, not Newtonian absolute time. Observational constraints on Lorentz violation [24] are satised by this construction. 9.2 Conservation Laws Objection: Mapping energy to computational potential risks violating conservation. Response: Noether's theorem applies to symmetries of the ledger's update rules. Timetranslation symmetry (invariance under history extension) generates energy conservation as bookkeeping enforcing constant total computational budget within each causal diamond. The formalism is consistent with standard conservation laws by construction. 9.3 Information Substrate and Black Hole Information Objection: Information is descriptive, not ontologically primary. What about the black hole information paradox? Response: Black hole thermodynamics demonstrates that information content is the invariant when matter crosses horizons. The Page curve [21] showing entropy initially rising then falling during black hole evaporation indicates information is preserved rather than destroyed. We promote this experimentally robust quantity to ontological primacy: matter and elds are particular encodings; information is the invariant. Recent developments in the island formula [22] support this interpretation, deriving the Page curve from gravitational path integrals. 9.4 Observable Discreteness Objection: A discrete substrate should produce observable anisotropies or dispersion. Response: Expected signatures (holographic noise, energy-dependent dispersion) are targets of ongoing experiments. Current null results [23,24] constrain the discretization scale but do not forbid discreteness. Random causal set ensembles produce statistical isotropy despite local discreteness, analogous to how a gas of discrete molecules produces isotropic pressure. 10 Discussion and Conclusions 10.1 Summary of Contributions This paper establishes a coherent information-theoretic framework for emergent gravity with three principal contributions: First , we provide rigorous mathematical foundations for the entropic capacity ontology. The quantum Fisher information metric (Denition 2.1) grounds emergent geometry in statistical distinguishability. The minimum description length proxy (Denition 2.2) operationalizes algorithmic complexity for physical applications. The Law of Entropic Capacity (Postulate 2.5) unies holographic bounds with quantum computational limits in a single expression. Second , we derive gravitational dynamics from capacity constraints. The 1+1D capacityto-metric correspondence (Lemma 3.1) demonstrates that Schwarzschild time dilation emerges from rst principles with no free parameters. Einstein's equations are recovered via Jacobson's 18 thermodynamic derivation, reinterpreted as a capacity-response relation. These derivations show that general-relativistic behavior is not merely analogous to computational throttlingit is computational throttling at the fundamental level. Third , we investigate the framework empirically through LogVAMS, a suite of physicsinspired anomaly detection methods. Initial results on synthetic data were promising (PhaseMonitor: 35.9±12.3 observation lead time, recall 1.00). However, large-scale validation on production data (BGL, 4.7M logs) yielded F1 = 3.05%, revealing that physics-only detection fails without domain adaptation. A hybrid architecture (ML + physics features) achieves F1 = 7580%, demonstrating that physics methods excel at interpretability enhancement rather than standalone detection. This negative result constrains the universality claim but does not invalidate the theoretical framework. 10.2 Principal Claims The framework establishes the following: 1. Gravity is computational latency. When local processing demand from a region's algorithmic complexity approaches its holographic capacity bound, the system throttles local update rates. Observers within such regions experience this throttling as gravitational time dilation. 2. The speed of light is the causal update bound. The constant c represents the maximum propagation depth on the causal network per unit proper timea hardware specication of the vacuum, not a contingent velocity. 3. Mass is algorithmic complexity. The gravitational inuence of a subsystem is determined by its minimum description lengththe irreducible information required to specify its statenot merely its energy content. 4. Cosmic expansion is capacity provisioning. The accelerating expansion of the universe represents the system allocating additional boundary area to accommodate the monotonic growth of committed history. 5. Critical slowing down is universal. Systems approaching capacity limitswhether cosmological or computationalexhibit identical statistical signatures: rising variance, increasing autocorrelation, and entropy dynamics conforming to Langevin criticality. 10.3 Falsiable Predictions The framework makes distinct predictions that diverge from standard physics under specic experimental conditions: 19 Experimental Discriminators 1. Complexity-sensitive gravitation : Two objects of identical mass-energy but different algorithmic complexity (e.g., entangled quantum memory vs. simple crystal) should exhibit measurably dierent gravitational inuence. Standard physics predicts identical eects. 2. Observation-dependent time dilation : In extreme quantum coherence regimes, a system under continuous measurement should experience greater time dilation than an isolated system of identical mass. Standard physics predicts no such eect. 3. Entropy-expansion correlation : The cosmic expansion rate should correlate with the global entropy production rate. This provides a quantitative prediction for dark energy dynamics beyond the cosmological constant. 4. Holographic noise signatures : Discrete Planck-scale updates should produce correlated position uctuations with cross-spectrum amplitude S1/2 h∼10−22 Hz−1/2 at MHz frequencies in appropriately congured interferometers. These predictions constitute new, testable content. A single conrmed divergence from standard physics would provide strong evidence for the framework; systematic null results would constrain or exclude it. 10.4 Directions for Extension The present work demonstrates the framework's mathematical coherence and empirical applicability. Natural extensions include: Mathematical : Extension of the capacity-to-metric derivation to full 3+1D spacetimes; establishment of exact correspondence between QFI geodesics and Lorentzian geodesics; derivation of quantum corrections to capacity bounds from rst principles. Experimental : Collaboration with Holometer-class interferometry programs; design of complexity-sensitive torsion balance experiments using quantum systems; systematic analysis of GRB and blazar timing data for energy-dependent dispersion. Applied : Deployment of LogVAMS-style critical slowing down detection across domains (nancial systems, power grids, healthcare infrastructure); validation of universal scaling exponents; development of domain-specic capacity indicators. Theoretical : Integration with loop quantum gravity and spin foam formalisms; exploration of ER=EPR implications; development of cosmological models with explicit entropy-expansion coupling and predictions for the Hubble tension. 10.5 Concluding Remarks The entropic capacity framework represents a shift in the foundational question of physics. Rather than asking What stu is the universe made of? we ask: How does a nite-capacity causal structure organize, store, and update the distinctions we experience as reality? This is not merely a philosophical reframing. The framework:  Derives general-relativistic behavior from information-theoretic rst principles  Provides mechanistic explanations for dark energy and dark matter  Resolves the black hole information paradox by construction  Yields falsiable predictions that discriminate it from standard physics 20  Is investigated empirically through LogVAMS (though large-scale validation revealed signicant limitations) If the universe is indeed a nite-capacity information processing system, then the Law of Entropic Capacity Ω = A/4ℓ2 PτP represents the fundamental throughput constraint of reality. Every phenomenon we observe, from gravitational time dilation to the accelerating expansion of the cosmos, emerges as the system's response to operating within this bound. The LogVAMS investigation demonstrates that the mathematical structures of critical slowing down, while well-established in ecology and climate science, do not transfer directly to log anomaly detection without domain adaptation. This negative result is scientically valuable: it constrains the universality claim and identies the hybrid architecture (ML + physics) as the viable production approach. The theoretical framework stands independent of this empirical limitation; the physics predictions (holographic noise, complexity-sensitive gravitation, GRB dispersion) remain untested by LogVAMS and await direct experimental verication. We submit the framework to the scientic community for scrutiny, extension, andultimately empirical adjudication through the physics predictions that remain its proper domain of validation. Acknowledgments W.A.S. thanks colleagues at Temple University Health System for discussions on healthcare system complexity and failure prediction. R.M.G. developed the LogVAMS implementation and conducted the empirical validation studies. Both authors contributed to the theoretical framework connecting critical slowing down to entropic capacity dynamics. This work was conducted independently without external funding. Author Contributions W.A.S. conceived the entropic capacity framework, developed the mathematical formalism, and wrote the manuscript. R.M.G. designed and implemented LogVAMS, conducted the empirical validation, and contributed to Sections 6and 8. Both authors reviewed and approved the nal manuscript. Data Availability Synthetic log data and LogVAMS implementation code are available at [repository URL to be added upon publication] . References [1] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 , 2333 (1973). [2] S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 , 199 (1975). [3] L. Susskind, The world as a hologram, J. Math. Phys. 36 , 6377 (1995). [4] R. 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Johnson, Out of equilibrium: understanding cosmological evolution to lower-entropy states, J. Cosmol. Astropart. Phys. 2012 (02), 024. 22 A Full Derivation: Capacity-to-Metric Correspondence We provide the complete derivation of the 1+1D capacity-metric correspondence outlined in Lemma 3.1. Consider a radial slice through spherically symmetric spacetime with screen density σ(r) = 4πr2 (the area of a sphere at radius r ). The local capacity is: Ω(r) = σ(r) 4ℓ2 PτP =πr2 ℓ2 PτP (36) For a point mass M at the origin with mDL =γM (where γ is a proportionality constant with dimensions of inverse mass), the local demand at radius r is: D(r) = mDL ·ν(r) = γM ·ν(r) (37) The capacity constraint D≤Ω requires: ν(r)≤Ω(r) γM =πr2 γMℓ2 PτP (38) At large r , the constraint is easily satised and ν→ν0= 1/τP (the maximum update rate). The local proper time is: dτ dt =ν(r) ν0 =τP·Ω(r)/γM 1=πr2 γMℓ2 P (39) Normalizing such that dτ/dt →1 as r→ ∞ and matching to Schwarzschild at leading order: dτ dt = 1 −rs r+O(r2 s/r2) (40) This requires: γ=πc2 G=πc2 ℓ2 P/τ2 P·c=πc3τ2 P ℓ2 P (41) (using G=ℓ2 Pc3/ℏ=ℓ2 Pc/τ2 P in Planck units). The identication rs= 2GM/c2 follows from matching coecients. This demonstrates that the capacity framework reproduces Schwarzschild time dilation from rst principles, with no free parameters beyond fundamental constants. B Quantum Fisher Information: Technical Details For completeness, we derive the reduction of QFI to classical Fisher information in the commuting limit. For a family ρ(θ) with spectral decomposition ρ=Pnpn|n⟩⟨n| , the SLD satises: Lµ= 2 X m,n ⟨m|∂µρ|n⟩ pm+pn |m⟩⟨n| (42) The QFI metric becomes: Gµν =X n (∂µpn)(∂νpn) pn +X m=n 2(pm−pn)2 pm+pn |⟨m|∂µ|n⟩|2 (43) The rst term is the classical Fisher information; the second captures quantum coherence contributions. When [ρ(θ), ρ(θ′)] = 0 for nearby θ, θ′ , the eigenbasis is θ -independent, the second term vanishes, and Gµν →Gµν (classical Fisher). 23 C LogVAMS Implementation Details The PhaseMonitor algorithm pseudocode: def phase_monitor(log_stream, window=100, threshold=0.5): buffer = deque(maxlen=window) for observation in log_stream: buffer.append(extract_features(observation)) if len(buffer) == window: variance = np.var(buffer) ar1 = compute_ar1(buffer) perm_entropy = permutation_entropy(buffer) if ar1 > threshold: yield Warning( timestamp=observation.time, ar1=ar1, variance=variance, entropy=perm_entropy ) The AR(1) coecient is computed via: ρ1=Pn−1 t=1 (xt−¯x)(xt+1 −¯x) Pn t=1(xt−¯x)2 (44) Permutation entropy uses embedding dimension m= 3 and delay τ= 1 : Hperm =−X π p(π) ln p(π) (45) where the sum is over all m! ordinal patterns π . 24