Numerical Artifacts of the Computational Universe: Detecting Planck-Scale Truncation Errors and Anisotropy in Physical Experiments

Numerical Artifacts of the Computational Universe: Detecting Planck-Scale Truncation Errors and Anisotropy in Physical Experiments Anonymous Author November 26, 2025 Abstract If the universe is a dynamical system realized by underlying discrete computational processes, then finite memory, finite clock frequency and finite numerical precision must leave extremely weak but in-principle detectable “numerical artifacts” in macroscopic physics. In computational universe models prototyped by quantum cellular automaton (QCA), this paper provides a systematic analytical framework, uniformly expressing spatial lattice discretization near Planck scale, finite-precision rounding errors, and algorithm-level resource scheduling (“lazy computation”) as higher-dimensional operators and weak Lorentz-violating parameters in effective field theory, relating them to existing and anticipated experimental constraints. The main conclusions of this paper are: (1) Any local QCA on a cubic lattice approximating Lorentz-invariant dispersion relations necessarily includes direction-dependent O((pa)4) anisotropic corrections in highmomentum dispersion, causing weak differences in group velocities of extremely high-energy particles and photons across spatial directions; (2) If underlying hardware uses finite-precision number domains not completely eliminated by perfect reversible computation or error-correction coding, it manifests macroscopically as conservation law noise strengthening with energy, giving upper bounds on “effective truncation error” indicator ϵeff in ultra-high-energy cosmic rays and long-baseline gravitational wave propagation; (3) If there exists “computational resolution allocation” adaptive to local entropy or observer information density, slight effective constant drift may appear in cosmological void regions, such as spatial modulation of fine-structure constant αor vacuum energy density Λeff . Combining ultra-high-energy cosmic rays, energy-dependent time delays in gammaray bursts, high-precision optical resonators and atomic clock network observations, this paper provides currently obtainable quantitative constraints and proposes several “computational forensics” experimental proposals oriented toward next-generation observation facilities, hoping to find traces of computational universe in noise. Keywords: Simulation hypothesis; Quantum cellular automaton; Numerical artifacts; Lorentz invariance tests; Ultra-high-energy cosmic rays; Gamma-ray bursts; Atomic clock networks; Fine-structure constant 1 Introduction & Historical Context The simulation hypothesis at philosophical level is most representatively expressed in Bostrom’s argument for “ancestor simulation”, i.e., under the premise that highly developed civilizations can run large-scale high-fidelity simulations, the prior probability of us being in a simulation may far exceed being in “base reality”. Meanwhile, physics conceptions of computational universe—including reversible computation, cellular automaton universe, information gravity, etc.—attempt to view physical laws as emergent forms of some underlying computational rules. 1 In such conceptions, Beane et al. first proposed a clear testable idea: if the universe runs as lattice gauge field theory numerical simulation on a cubic lattice, then lattice spacing a cannot be infinitely small, and its finite value will manifest as direction-dependent cutoff and anisotropy in GZK cutoff structure in high-energy cosmic ray spectra, yielding lower bound a−1≳1011 GeV. Recently, Vazza et al. systematically quantified physical realizability of simulated universe from information energy cost and astrophysical observations perspectives, pointing out that reproducing observed universe under finite computational resources must face stringent energy and entropy budget constraints. On the other hand, numerous high-precision experiments continuously test basic postulates of special and general relativity. Including modern Michelson–Morley experiments using cooled optical resonators, giving upper limits on light speed anisotropy ∆c/c ∼10−15 level; Fermi-LAT analysis of gamma-ray burst time delays pushing scale of linear energy-dependent light speed correction close to or beyond Planck scale; and atomic clock comparisons giving tight limits on temporal variation and spatial variation of fine-structure constant. Combined with quasar absorption lines, cosmic microwave background and cosmological data, constraints on αand other fundamental constants’ variation on cosmological scales are also becoming stringent. The above work shows: even temporarily setting aside simulation hypothesis, high-precision testing of Lorentz invariance, conservation laws and constant stability itself has become a mature experimental field. However, existing discussions often view “simulated universe” and “quantum gravity corrections”, “effective field theory higher-dimensional operators” etc. as independent frameworks from each other. This paper’s goal is: on quantum cellular automaton discrete ontology, introduce explicit “computational resource limitation” assumptions, transforming “universe as numerical simulation” into a set of specific effective parameterizations, thus establishing a “computational forensics” directly interfaceable with existing high-precision experiments. Its core ideas are: 1. View spatial lattice discretization as QCA geometric structure, analyzing its impact on dispersion relations and symmetries; 2. Abstract finite numerical precision as weak random breaking of unitarity and conservation laws, providing quantitative mapping to macroscopic observations; 3. View resource scheduling and resolution adaptation as low-frequency modulation of effective constants in spacetime, using atomic clocks, cosmological observations and large-scale structure data for constraints. This work does not attempt to prove simulation hypothesis “holds”, but provides: if the universe can be approximated by such computational models, what experimental constraints any reasonable simulation must satisfy; conversely, once phenomena significantly conflicting with these constraints are discovered, it may provide non-trivial support for such computational universe models. 2 Model & Assumptions 2.1 QCA Prototype of Computational Universe We adopt the following abstract model to describe computational universe: 1. Spacetime is represented as three-dimensional cubic lattice Λ ∼ =aZ3, with time discretized in steps ∆t. 2. Each lattice site x∈Λ has finite-dimensional local Hilbert space Hx∼ =Cd, with global Hilbert space H=Nx∈ΛHx. 2 3. Time evolution is given by local unitary operator Uwith finite action range R, i.e., |ψ(t+ 1) = Uψ(t), U =Y X⊂Λ UX, where each UXacts only on finite subsets with diameter ≤R. 4. In long-wavelength limit a→0,∆t→0, effective dynamics of single-particle excitations can be approximated by Lorentz-invariant continuous field theory (such as scalar field, Dirac field or Maxwell field). In “ideal mathematical QCA”, Uis exactly unitary, with lattice spacing aand time step ∆tonly as scaling parameters, not introducing any numerical errors. Under simulation hypothesis, hardware (or formal structure realization) actually executing Uis subject to following constraints:  Numerical domain is finite-precision: such as finite-word-length integers or floating-point numbers;  Storage capacity is finite: can only accommodate finite numbers of lattice sites and internal degrees of freedom;  Computational resources are finite: number of logic gates executable per step is finite, thus requiring scheduling and approximation. In this model, we view “ideal QCA” as target dynamics, with deviations from hardware implementation all viewed as “numerical artifacts”. These artifacts are parameterized into three types: 1. Lattice anisotropy artifacts: Since Λ has only finite rotation group Ohsymmetry, Lorentz invariance in long-wavelength limit only approximately holds in window pa ≪1, with systematic anisotropic corrections appearing in high-momentum or high-frequency bands. 2. Finite-precision artifacts: Each unitary evolution step can only be approximately implemented in hardware, introducing random or biased numerical rounding errors, causing weak breaking of unitarity and conservation laws. 3. Resource scheduling artifacts: To reduce resource consumption, simulators may adopt sparse updates, coarse lattices or lower precision representations for “low-complexity regions”, causing equivalent physical constants to slowly vary across spatial regions. At model level, we introduce three dimensionless parameters to characterize their magnitudes:  Lattice scale parameter ηa=aMP, where MPis Planck energy scale;  Effective truncation error parameter ϵeff , representing relative error in energy/probability conservation per local evolution step;  Resolution modulation parameter δres(x), representing deviation of effective constants from average values in different cosmological environments. Our goal is to extract upper bounds or “detectable windows” on ηa, ϵeff, δres from various observations. 3 3 Main Results (Theorems and alignments) 3.1 Theorem 1: High-Energy Anisotropic Dispersion of Cubic Lattice QCA Consider three-dimensional QCA satisfying translation invariance and locality, approaching scalar or optical wave equations in long-wavelength limit. Its single-particle dispersion relation ω(k) can be expanded in region |k|a≪1 as ω2(k) = c2k2+m2c4+a2 β1 3 X i=1 k4 i+β2X i<j k2 ik2 j +O(a4k6), where k= (kx, ky, kz), β1, β2are dimensionless coefficients determined by QCA local structure. In massless limit m= 0, group velocity vg(k) = ∇kω(k) exhibits direction dependence at a2order. Specifically, for given magnitude |k|=k, group velocity corrections along lattice axis k= (k, 0,0) and along body diagonal k= (k/√3, k/√3, k/√3) satisfy ∆vaxis g−∆vdiag g=O(a2k3)= 0, therefore anisotropy in light speed or high-energy particle propagation speed must appear in high-energy region. This conclusion is consistent with early analysis of lattice anisotropy artifacts in lattice gauge field theory, also forming basis for using cosmic rays and high-energy photons to constrain “cosmic lattice spacing”. 3.2 Theorem 2: Conservation Law Noise Scaling from Finite-Precision Rounding Errors Assume underlying hardware introduces local non-unitary perturbations with relative scale ϵ≪1 when implementing single-step evolution U, approximable by Lindblad-type effective evolution ρ(t+ ∆t) = Uρ(t)U†+ϵD[ρ(t)] + O(ϵ2), where Dis some completely positive dissipative superoperator. If errors are approximately independent and identically distributed in time, then after Nevolution steps, standard deviation of expectation value deviation of some conserved quantity Q(such as total energy or momentum) from ideal value satisfies σQ(N)∼√N ϵ Qchar, where Qchar is characteristic scale. Taking universe age TUand Planck time tPto define effective step number N∼TU/tP∼1061, then only when ϵeff ≲10−30 can nearly strict conservation laws still be observed macroscopically. Here ϵeff is effective truncation error after possible error-correction coding and reversible computation elimination. This estimate shows: if universe is QCA simulation realized with finite precision, underlying must use extremely efficient error correction and reversible logic to be compatible with existing observations on energy and momentum conservation. 4 3.3 Proposition 3: Resource Scheduling and Spatial Modulation of Effective Constants Suppose simulator adopts “hierarchical resolution” strategy: selecting different discrete steps (a(x),∆t(x)) and numerical precision nbit(x) based on local complexity indicator function χ(x) (such as local entropy density or observer density). For low-energy effective field theory, this is equivalent to introducing spatially slowly-varying renormalization constants gi(x) in Lagrangian. For example in electromagnetic interaction term LEM =−1 4ZF(x)FµνFµν +Zψ(x)¯ ψiγµDµψ+··· , will cause effective fine-structure constant α(x)∝Z2 ψ(x) ZF(x) to exhibit spatial modulation. If ZF, Zψvary smoothly on cosmological scales, linearization approximation ∆α/α ≈κ∆χcan be used. Existing atomic clock comparisons, quasar absorption lines and cosmological data give upper limits on ∆α/α typically in 10−7–10−15 range, thus can be directly transformed into constraints on resource scheduling intensity κ∆χ. 4 Proofs 4.1 4.1 Proof of Theorem 1: Derivation of Cubic Lattice QCA Anisotropic Dispersion For brevity, take three-dimensional scalar field’s discrete wave equation as example, whose continuous form is ∂2 tϕ=c2∇2ϕ. On cubic lattice adopt standard second-order difference discretization Laplacian (∇2 aϕ)(x) = 1 a2 3 X i=1 [ϕ(x+aˆei) + ϕ(x−aˆei)−2ϕ(x)] . Let ϕ(x, t) = exp [i(k·x−ωt)] , substituting into discrete equation yields dispersion relation ω2(k) = 4c2 a2 3 X i=1 sin2kia 2. Expand under |kia| ≪ 1 sin kia 2=kia 2−(kia)3 48 +O(a5k5 i), thus sin2kia 2=k2 ia2 4−k4 ia4 48 +O(a6k6 i). Substituting gives 5 ω2(k) = c2 3 X i=1 k2 i−c2a2 12 3 X i=1 k4 i+O(a4k6). First term is isotropic c2k2, second term is component-dependent Pk4 iform. Note Pk4 iis not constant under SO(3), only remaining symmetric under finite group Oh. Introducing mass term m2c4does not change high-momentum anisotropic term structure, only slightly modifying dispersion in low-energy region. For more general QCA, as long as maintaining translation invariance and locality, long-wavelength limit can be characterized by effective Hamiltonian Heff =cα·p+βmc2+a2X i,j γijp2 ip2 j+··· where α, β, γij are various matrices or constants, pi=ℏki. Expanding energy spectrum squared can obtain Pk4 istructure isomorphic to above scalar case, thus Theorem 1 holds for broad QCA models. To show direction dependence, compare two wavevector types: 1. Axial: k= (k, 0,0). Then ω2 axis(k) = c2k2−c2a2 12 k4+O(a4k6). 2. Body diagonal: k= (k/√3, k/√3, k/√3). Then 3 X i=1 k4 i= 3 k √34 =k4 3, therefore ω2 diag(k) = c2k2−c2a2 36 k4+O(a4k6). Difference at O(a2k4) between them is ω2 axis(k)−ω2 diag(k) = −c2a2 18 k4+O(a4k6)= 0, thus group velocity vg=∂ω ∂k corrections also exhibit same-order difference, i.e., ∆vaxis g−∆vdiag g=O(a2k3). Therefore, as long as ais finite and high-energy particles can probe region ka ∼1, lattice anisotropy must become observable effect. For massless photons, this means light speed becomes direction function near Planck energy scale, which is precisely basis idea for using cosmic ray arrival direction distribution to constrain lattice spacing. 6 4.2 4.2 Proof of Theorem 2: Random Walk Scaling of Finite-Precision Errors Consider Kraus representation of single-step evolution ρ7→ X α KαρK† α, where in ideal unitary case there is only one operator K0=Uwith U†U=I. Finite-precision implementation can be viewed as adding small perturbations near U, such that K0=p1−ϵ2U, Kj=ϵLj, j = 1, . . . , M, satisfying PαK† αKα=I,ϵ≪1. Let Qbe observable conserved in ideal case, i.e., [Q, U] = 0. After single step, change in expectation value of Qis δ⟨Q⟩= Tr(Qρ′)−Tr(Qρ), where ρ′=X α KαρK† α. Expanding gives δ⟨Q⟩=ϵ2X j Tr QLjρL† j−ϵ2Tr(Qρ) + O(ϵ3). If Ljsatisfy some “zero average” condition (such as symmetric noise), single-step expectation deviation can be zero, but variance is non-zero. Define δQn=⟨Q⟩n−⟨Q⟩n−1, then under independent identically distributed approximation, its variance E[(δQn)2]∼ϵ2Q2 char, where Qchar is typical scale determined by Ljand ρ. Cumulative deviation after total steps N ∆QN= N X n=1 δQn satisfies E[∆QN] = 0,Var(∆QN) = NVar(δQ1)∼Nϵ2Q2 char, i.e., σQ(N)∼√N ϵ Qchar. Taking Qas total energy or momentum, Qchar approximately system total energy or typical particle energy. If wanting to maintain macroscopically observed conservation precision on universe age scale, such as relative deviation σQ/Qchar ≲10−n, then have ϵeff ≲10−n/√N. If N∼1061, then even n= 10 requires ϵeff ≲10−40. In reality, conservation law test precision varies by system, but this order-of-magnitude estimate already shows: if “floatingpoint number”-like direct rounding errors exist, their bare value ϵmust be greatly suppressed 7 through reversible computation and error-correction coding, with only extremely weak “residual noise” ϵeff manifesting macroscopically. In astrophysical environments, such as propagation step number for ultra-high-energy cosmic rays from source to Earth can be estimated as NUHECR ∼L λint , where Lis propagation distance, λint is typical mean free path of interaction or scattering. Even adopting macroscopic step λint ∼1 Mpc coarse-graining, L∼100 Mpc still gives NUHECR ∼102. At this time if ϵeff ∼10−20, cumulative energy relative deviation only 10−19 level, far below typical observation uncertainty. Conversely, if viewing Planck time as basic step, constraints become extremely stringent. This shows different physical processes and time scales can separately constrain different combinations of ϵeff , providing hierarchical picture for “computational universe” error budget. 4.3 4.3 Proof of Proposition 3: Effective Field Theory Expression of Resource Scheduling and α(x)Modulation In general effective field theory, renormalization factors Zi(µ) depend on energy scale µ, but assumed spatially uniform. If introducing slow dependence on spatial coordinate x, then at tree level can view it as external field background. Fine-structure constant definition is α(x) = e2 4πε0ℏc Z2 ψ(x) ZF(x), therefore ∆α α≈2∆Zψ Zψ−∆ZF ZF . Suppose resource scheduling strategy provides high resolution in high-complexity regions, i.e., Zi=Z(0) i, in low-complexity regions has slight deviation δZi. If δZi/Z(0) i∼λ∆χlinearly depends on complexity indicator function change, then ∆α α∼λα∆χ. Atomic clock comparison experiments give upper limits on αtemporal drift |˙α/α|≲10−17 yr−1 level, while quasar absorption lines and cosmic microwave background give constraints on spatial drift |∆α/α|≲10−7–10−5level, depending on redshift and sample. In “lazy computation” scenario, can view cosmic voids and dense galaxy clusters as two extremes of χ, with relative difference ∆χ∼1, thus above results directly transform to upper limits on λα. These quantitative relationships will be further concretized in later model application sections. 5 Model Apply This section maps these three types of numerical artifacts to existing or anticipated observation windows. 5.1 5.1 Lattice Anisotropy and Ultra-High-Energy Cosmic Ray Spectra Beane et al. studied impact of QCD model realized on cubic lattice on cosmic rays, pointing out if universe is numerical simulation using Wilson fermions and cubic lattice, then due to Brillouin zone structure in lattice momentum space, high-energy cosmic ray spectrum cutoff 8 will exhibit anisotropy related to lattice directions. From QCA perspective, this idea can be generalized as:  Lattice dispersion relation begins deviating from continuous Lorentz form when ka ≲1;  For ultra-high-energy protons or nuclei, their energy spectrum upper limit and propagation attenuation will depend on their direction in momentum space relative to lattice axis;  If observed GZK cutoff exhibits systematic bias in celestial sphere related to some fixed direction, which cannot be simplified as result of local astrophysical structure (such as Galactic magnetic field or source distribution), can be viewed as candidate signal for lattice anisotropy. Currently Pierre Auger Observatory and Telescope Array observations of ultra-high-energy cosmic ray arrival direction anisotropy show anisotropy related to large-scale structure, but no universal deviation pointing to some fixed cosmological axis has been found. Existing data typically transforms to upper limits on ∆c/c around 10−20–10−21 level in energy band E∼ 1019–1020 eV, thus giving lower bound a−1≳1011–1012 GeV, consistent with lattice gauge field theory analysis. 5.2 5.2 Energy-Dependent Light Speed and Gamma-Ray Burst Time Delays If rewriting lattice and finite-precision artifacts together as effective Lorentz-violating terms, high-energy photon dispersion relation can be written as E2=p2c2"1 + ξ1E EQG +ξ2E EQG 2 +···#, where EQG is equivalent quantum gravity or lattice energy scale, ξnare coefficients controlled by numerical artifacts. Effective light speed change with energy in first-order approximation is v(E)≈c1−n+ 1 2ξnE EQG n. For transient sources at redshift z(GRB, blazar outbursts, etc.), arrival time delay between different energy photons ∆t∼n+ 1 2H0 ξn ∆En En QG Zz 0 (1 + z′)n pΩm(1 + z′)3+ ΩΛ dz′. Fermi-LAT analysis of several bright GRBs shows no significant energy-dependent delay in E∼GeV range, thus giving strong limits on linear term n= 1 and quadratic term n= 2: for linear case EQG ≳EP, for quadratic case EQG ≳1010 GeV typical order. In computational universe framework, EQG can be related to lattice spacing aand effective truncation error ϵeff. For example if numerical artifacts mainly from lattice, then EQG ∼ℏc/a; if mainly from finite precision, then ξnwill be some function of ϵeff. Thus GRB time delay constraints can be rewritten as joint upper bounds on aand ϵeff . 5.3 5.3 Light Speed Anisotropy and Modern Michelson–Morley Experiments Modern Michelson–Morley type experiments adopt ultra-stable frequency optical resonators, comparing light speed differences on two mutually perpendicular interferometer arms, achieving scanning of different spatial directions through Earth’s rotation and revolution. M¨uller et al. compressed light speed anisotropy relative upper limit to ∆c/c ≲10−15 level through cooled 9 observable Qsensitivity to each macroscopic event is δQτ∼√NτϵQchar, while experimental relative precision on Qat this scale is σQ/Qchar, then have ϵeff ≲σQ/Qchar √Nτ . Different physical processes give different Nτand σQ: 1. Laboratory energy conservation tests: Such as atomic transition energy level fine structure, time scale τ∼10−8s, precision σE/E ∼10−12–10−15. 2. Astrophysical processes: Such as pulsar rotation period evolution, τ∼107–108s, precision σf/f ∼10−15–10−18. 3. Cosmological processes: Such as cosmic expansion history and CMB anisotropy, τ∼ 1013–1017 s, precision depends on observables. Considering above various constraints simultaneously, can construct “error manifold” of ϵeff at different energy and time scales, further delimiting which types of computational universe implementations are incompatible with existing observations. For example if some implementation causes extremely large Nτin high-energy processes, then its constraint on ϵeff is stronger than low-energy processes; conversely, if underlying algorithm adopts more refined reversible computation in high-energy region while allowing slightly larger errors in low-energy region, constraint structure may invert. C Appendix C: Simplified Model of “Lazy Computation” and Cosmic Voids To quantitatively examine possible α(x) drift under “lazy computation” hypothesis, construct following simplified model: 1. Divide universe into two types of regions: dense region D(galaxies and galaxy clusters) and void region V; 2. In dense region, simulator uses high-resolution parameters (aD,∆tD, nD); in void region uses coarse-resolution parameters (aV,∆tV, nV), where aV> aD, ∆tV>∆tDor nV< nD; 3. Assume electromagnetic field renormalization constants in these two types of regions satisfy Z(V) F=Z(D) F(1 + δF), Z(V) ψ=Z(D) ψ(1 + δψ), then ∆α α=αV−αD αD≈2δψ−δF. If assuming δF, δψare analytic functions of aand ∆t, for example δF∼c1(a2 V−a2 D)Λ2+c2(∆tV−∆tD)Λ + ··· , where Λ is some renormalization cutoff, then can express ∆α/α as combination of aV, aD,∆tV,∆tD. Substituting observed |∆α/α|upper limit yields constraints on allowed resolution differences. 16 For example if quasar absorption line observations show |∆α/α|≲10−6in range z∼ 2–4, then roughly can consider |2δψ−δF|≲10−6. If renormalization coefficients have O(1) coefficients for a2Λ2, then must require |a2 V−a2 D|Λ2≲10−6, in case Λ ∼MP, this means even in cosmic voids, compared to dense regions, lattice spacing can only increase by less than relative 10−6order of one Planck length. This result shows: under relatively natural renormalization assumptions, “lazy computation” has almost no adjustment room on cosmological scales, unless its impact on low-energy electromagnetic effects is precisely canceled in some way. 17