Emergence and Continuum-Limit Tests

An emergent continuum claim requires more than large complexes or smooth average profiles. Causal dynamical triangulations provide a concrete setting in which an extended large-scale geometry is measured Ambjørn, Jurkiewicz, and Loll 2005, but a continuum claim additionally requires tuning toward a critical limit where the microscopic scale disappears relative to physical correlations. Renormalized observables, finite-size scaling, universality, restored target symmetries and causality, and low-energy gravitational dynamics must all be demonstrated with quantified errors.

Required background. Causal Dynamical Triangulations: Phases and Continuum Evidence, Causal-Set Kinematics and Dynamics, and Tensor Models and Random Geometry supply the three test cases.

Helpful background. Lines of Constant Physics and Continuum Extrapolation and Continuum Extrapolation, Bias, and Uncertainty supply inference methods.

Regulator removal

Let microscopic spacing be $a$ and lattice correlation length $\xi_{\mathrm{lat}}$. A continuum limit holds physical $\xi_{\mathrm{phys}}=a\xi_{\mathrm{lat}}$ fixed while

$$\xi_{\mathrm{lat}}\to\infty,\qquad a\to0.$$

Near a critical coupling $g_c$,

$$\xi_{\mathrm{lat}}\sim|g-g_c|^{-\nu}.$$

A line of constant physics fixes dimensionless ratios such as $m_1/m_2$ while $a$ is reduced. Gravity also needs an operational scale setter—for example a diffusion scale, curvature radius, or matter correlation length.

Application: one common checklist

Test	CDT	Causal set	Tensor ensemble
Microscopic scale	Edge length $a$	Density scale $\rho^{-1/d}$	Graph/cell scale and $N$
Critical control	Phase-boundary tuning	Dynamical coupling or nonlocality scaling	Coupling criticality and large $N$
Long-distance observable	Volume covariance, transfer matrix	Interval statistics, retarded propagation	Graph correlators, distance distributions
Missing common definition	Fully relational local observables	Established manifoldlike dynamical phase	Lorentzian causal structure

For each, measure at several sizes, fit correction-to-scaling terms, and carry autocorrelation or sampling covariance. Mark absent observables rather than replacing them with spectral dimension.

Symmetry and dynamics tests

Recover dimension and approximate homogeneity, then test diffeomorphism-compatible Ward identities or regulator independence, causal support, graviton or curvature correlators, matter coupling, and an effective Einstein equation. Universality requires different microscopic actions or measures to converge to the same renormalized data.

Adversarial control

Construct a visually extended ensemble with finite $\xi_{\mathrm{lat}}$. It fails regulator removal despite looking smooth. Change boundary conditions and microscopic measure while retuning; universal observables should agree. Fit with and without the coarsest sizes and expose drift. A single scaling exponent cannot establish the full target theory.

Scope

No surveyed discrete program has yet passed every row for four-dimensional gravity with realistic matter. Their partial results are still informative when the missing tests are explicit. Positivity and causal tests are developed on Unitarity, Reflection Positivity, and Causality Checks.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Ambjørn, J., J. Jurkiewicz, and R. Loll. “Reconstructing the Universe.” Physical Review D 72 (2005): 064014. DOI.
• Wilson, K. G., and J. Kogut. “The Renormalization Group and the $\epsilon$ Expansion.” Physics Reports 12 (1974): 75–199. DOI.