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arxiv: 2606.04910 · v1 · pith:JXL26MCRnew · submitted 2026-06-03 · 🧮 math.DG

Ollivier-Ricci Curvature for Causal Sets

Joe Barton , Samu\"el Borza , Jona R\"ohrig This is my paper

Pith reviewed 2026-06-28 04:20 UTC · model grok-4.3

classification 🧮 math.DG
keywords Ollivier-Ricci curvaturecausal setsLorentzian optimal transporttimelike Ricci curvaturecausal diamondsdiscrete curvature
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The pith

A novel Ollivier-Ricci curvature for causal sets is defined using Lorentzian optimal transport and recovers timelike Ricci curvature from discrete order data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A mesoscopic notion of Ollivier-Ricci curvature is defined for causal sets by using Lorentzian optimal transport between probability measures on causal diamonds along maximal chains. An independent asymptotic formula shows that this transport distance recovers timelike Ricci curvature up to higher order terms in the continuum limit. The resulting discrete curvature satisfies local-to-global propagation, obeys timelike Bonnet-Myers theorems, and reproduces the constant curvature of Minkowski, de Sitter and anti-de Sitter space in high-density sprinkling experiments.

Core claim

Using Lorentzian optimal transport on measures supported in causal diamonds, the authors construct a discrete Ollivier-Ricci curvature along maximal chains in a causal set that approximates the timelike Ricci curvature of the continuum spacetime approximated by the causal set.

What carries the argument

Lorentzian optimal transport distance between probability measures on causal diamonds, used to define curvature along maximal chains via idle versus transport cost differences.

If this is right

  • Local curvature bounds propagate to global statements.
  • Timelike Bonnet-Myers theorems hold for the discrete curvature.
  • Numerical experiments on Poisson sprinklings recover constant curvature signatures for known spacetimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This approach may enable curvature-based diagnostics for emergent spacetime in causal set models of quantum gravity.
  • Similar transport-based curvatures could be explored in other discrete Lorentzian geometries beyond causal sets.

Load-bearing premise

The new Lorentzian asymptotic formula recovers timelike Ricci curvature up to higher-order terms from transport distances on nearby causal diamonds.

What would settle it

A failure of the high-density sprinkling experiments to recover the expected constant-curvature values for Minkowski, de Sitter or anti-de Sitter space would falsify the claim that the discrete curvature captures the continuum timelike Ricci curvature.

Figures

Figures reproduced from arXiv: 2606.04910 by Joe Barton, Jona R\"ohrig, Samu\"el Borza.

Figure 1
Figure 1. Figure 1: Riemannian and Lorentzian version of Ollivier Ricci–curvature [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The two causal diamonds used in the asymptotic expansion of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical experiments for the Ollivier–Ricci curvature of causal sets gener [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Spaces of constant positive Ollivier Ricci curvature [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Empirical distributions of the renormalised curvature from 50,000 trials in [PITH_FULL_IMAGE:figures/full_fig_p030_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Mean and one standard deviation bands of the renormalised curvature [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Discretisation error in 2D Minkowski space. Log–log plot of the empirical [PITH_FULL_IMAGE:figures/full_fig_p032_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: A commutative diagram between diamonds. makes the diagram commute. The map (22) is well defined for ε > 0 sufficiently small because expx is injective on D˜ ε(x, v), so each x˜ ∈ expx (D˜ ε(x, v)) can be written uniquely as x˜ = expx (u) with u ∈ D˜ ε(x, v), and for such u we have ˜t ε exit(x, v, u) > 0. Moreover, Dε(x, v) and D˜ ε(x, v) are star-shaped along future timelike rays from x, so scaling u by t … view at source ↗
read the original abstract

We introduce a novel notion of Ollivier--Ricci curvature for causal sets using Lorentzian optimal transport. The construction is motivated by a new Lorentzian asymptotic formula of independent interest, which recovers timelike Ricci curvature, up to higher-order terms, from the transport distance between probability measures on nearby causal diamonds. Passing to the discrete setting, this leads to a mesoscopic notion of Ricci curvature defined along maximal chains and built from probability measures on causal diamonds. We study several variants, including idle and Lin--Lu--Yau type curvatures, prove local-to-global propagation results and timelike Bonnet--Myers theorems, and compute the curvature for a range of explicit causal sets. We design high-density Poisson sprinkling numerical experiments recovering the expected constant-curvature signatures of Minkowski, de Sitter, and anti-de Sitter space. These results provide evidence that the construction captures timelike Ricci curvature from order-theoretic data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Ollivier-Ricci curvature for causal sets via Lorentzian optimal transport between probability measures supported on causal diamonds along maximal chains. The definition is motivated by a new continuum asymptotic formula claimed to recover timelike Ricci curvature (up to higher-order terms) from transport distances; the authors prove local-to-global propagation results and timelike Bonnet-Myers theorems, and report numerical experiments on high-density Poisson sprinklings that recover the constant-curvature signatures of Minkowski, de Sitter, and anti-de Sitter space.

Significance. If the asymptotic formula holds with controlled error terms, the construction supplies an intrinsic, order-theoretic notion of timelike Ricci curvature that bridges discrete causal sets to continuum Lorentzian geometry. The numerical recovery of expected continuum signatures and the statements of local-to-global and Bonnet-Myers results constitute concrete strengths that would be of interest to both discrete geometry and quantum-gravity communities.

major comments (2)
  1. Abstract and opening paragraphs: the new Lorentzian asymptotic formula is presented as the independent-interest motivation that licenses the discrete definition, yet no explicit statement of the formula, the choice of measures on the diamonds, or the precise error term appears in the provided text. Because the mesoscopic curvature along maximal chains is defined by direct analogy with this continuum identity, the absence of a rigorous derivation or reference establishing the claimed recovery of timelike Ricci curvature is load-bearing for the geometric justification of the entire construction.
  2. Numerical experiments section (as described in the abstract): the Poisson-sprinkling tests are said to recover the expected constant-curvature signatures, but the abstract supplies no information on the concrete choice of probability measures, the mesoscopic scale of the diamonds relative to the sprinkling density, or the statistical controls used to compare discrete curvature values against the continuum Ricci tensor. Without these details the experiments cannot be assessed as evidence that the discrete definition reproduces the asymptotic formula rather than merely reproducing qualitative features.
minor comments (2)
  1. Notation for the idle and Lin-Lu-Yau variants should be introduced with a short comparative table or explicit formulas to avoid ambiguity when the variants are later used in the theorems.
  2. The manuscript should include a brief remark clarifying whether the probability measures on the diamonds are required to satisfy any regularity (e.g., compact support, absolute continuity) that might be needed for the asymptotic expansion to hold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments highlight areas where the manuscript can be made more self-contained and reproducible; we have revised accordingly while preserving the core contributions.

read point-by-point responses

  1. Referee: Abstract and opening paragraphs: the new Lorentzian asymptotic formula is presented as the independent-interest motivation that licenses the discrete definition, yet no explicit statement of the formula, the choice of measures on the diamonds, or the precise error term appears in the provided text. Because the mesoscopic curvature along maximal chains is defined by direct analogy with this continuum identity, the absence of a rigorous derivation or reference establishing the claimed recovery of timelike Ricci curvature is load-bearing for the geometric justification of the entire construction.

    Authors: We agree that an explicit statement of the asymptotic formula belongs in the introduction for clarity. The revised manuscript adds a new paragraph (now labeled as Equation (1.1)) that states the formula: for suitable probability measures μ, ν supported on nearby causal diamonds separated by timelike distance ε, the Lorentzian Wasserstein distance satisfies W(μ,ν) = ε(1 - (1/6) Ric(v,v) ε² + O(ε³)), where v is the timelike vector between the diamonds. The measures are taken to be normalized volume measures on the diamonds (with a brief discussion of alternatives). A self-contained derivation sketch, relying on the Lorentzian version of the Kantorovich duality and Taylor expansion of the volume form, appears in Section 2, together with references to related continuum results. This addresses the load-bearing justification without altering the subsequent discrete construction. revision: yes

  2. Referee: Numerical experiments section (as described in the abstract): the Poisson-sprinkling tests are said to recover the expected constant-curvature signatures, but the abstract supplies no information on the concrete choice of probability measures, the mesoscopic scale of the diamonds relative to the sprinkling density, or the statistical controls used to compare discrete curvature values against the continuum Ricci tensor. Without these details the experiments cannot be assessed as evidence that the discrete definition reproduces the asymptotic formula rather than merely reproducing qualitative features.

    Authors: We accept that the numerical section required additional technical detail. The revised version now specifies: (i) the probability measures are the normalized counting measures on the vertices inside each causal diamond; (ii) the mesoscopic scale is chosen so that each diamond contains on average 20–30 points (approximately 5–8 times the mean sprinkling length); (iii) results are averaged over 200 independent Poisson sprinklings per spacetime, with error bars given by the standard deviation across realizations, and a direct quantitative comparison (via L² distance) to the continuum Ricci values is included in a new table. These controls allow the reader to assess agreement with the asymptotic formula within the predicted higher-order error. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained with independent continuous formula and external numerical validation

full rationale

The paper introduces a new Lorentzian asymptotic formula as a result of independent interest, then defines the discrete curvature from it along maximal chains. Numerical sprinkling experiments recover known continuum curvature signatures (Minkowski, de Sitter, anti-de Sitter) without fitting free parameters to the curvature definition itself. No self-citations are load-bearing for the central claim, no fitted inputs are relabeled as predictions, and no ansatz or uniqueness theorem reduces the result to its inputs by construction. The derivation chain remains non-circular under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on extending optimal transport to the Lorentzian causal-set setting and on the validity of the new asymptotic formula; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Lorentzian optimal transport is well-defined between probability measures supported on causal diamonds
    Invoked to define the discrete curvature from the continuous asymptotic formula.

pith-pipeline@v0.9.1-grok · 5688 in / 1113 out tokens · 24516 ms · 2026-06-28T04:20:31.790947+00:00 · methodology

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