Universality of the 4D Discrete d'Alembertian Instability: Deterministic Hypergraph Evidence and a Floquet Diagnostic

Abstract

We ask whether a deterministic hypergraph rewrite rule of the Wolfram class can support a Lorentz-invariant, massless propagating mode with a calculable leading Planck-scale correction $\omega^2 = c^2 k^2 [1 + \alpha (\hbar c k / E_P)^2 + \cdots]$. Working on a single causal history of a fixed arity-3 rule (here labelled $R_{323}$ and defined explicitly below) and applying the standard four-layer Benincasa-Dowker-Sorkin (BBD) discrete d'Alembertian, we find that the rule's measured shell coefficients fail the three moment-cancellation conditions required for a massless branch. A symbolic expansion shows that $\alpha$ drops out of the orders at which it should appear, and the numerical proxy value $\alpha \approx -9.35$ produced by an isotropic-shell ansatz is an artifact (eigenvector residuals $\sim 1$, $\omega/k$ monotonically decreasing). The structural form of this obstruction matches the stochastic 4D instability reported by Aslanbeigi, Saravani, and Sorkin (ASS 2014) for Poisson-sprinkled causal sets, suggesting that the 4D instability of the discrete d'Alembertian is universal across deterministic and stochastic substrates. We additionally introduce Floquet/Bloch-wave analysis on the periodic quotient of a causally invariant rewrite graph as a diagnostic method, which to our knowledge is new. We conjecture but do not show that the ASS 2014 generalized causal-set d'Alembertian family, with its non-locality scale, sidesteps the obstruction when applied to $R_{323}$.

1. Introduction

A central question in any discrete approach to quantum gravity is whether the long-wavelength continuum limit reproduces the relativistic massless dispersion $\omega = c k$ exactly, or only approximately with a calculable Planck-scale correction. The phenomenological target is the standard Lorentz-invariance-violation (LIV) ansatz $$ \omega^2 = c^2 k^2 \left[1 + \alpha \left(\frac{\hbar c k}{E_P}\right)^2 + \mathcal{O}\!\left(\frac{k^4}{E_P^4}\right)\right], $$ with $\alpha$ a dimensionless O(1) coefficient determined by the microscopic dynamics. Current Fermi-LAT GRB timing (Vasileiou et al. 2013) and IceCube high-energy neutrino observations bound $|\alpha| \lesssim 1$ at this order. A theory of discrete spacetime is in interesting territory only if it predicts $\alpha$ as a function of its microscopic data, rather than treating $\alpha$ as a free parameter.

Two broad programs sit at this question with very different microscopic data. Causal set theory (CST; Bombelli-Lee-Meyer-Sorkin 1987, Sorkin 2007) generates spacetime as a Poisson sprinkling into a Lorentzian manifold; the discrete d'Alembertian on such sprinklings has been computed (Sorkin 2007; Benincasa-Dowker 2010; Glaser 2014; Belenchia-Benincasa-Dowker 2016, henceforth BBD 2016). Aslanbeigi, Saravani, and Sorkin (ASS 2014) showed that in 4D the resulting operator is unstable: its spectrum contains exponentially growing modes, and they introduced a one-parameter family of generalized operators with a non-locality scale that suppresses the instability and acts as "a genuinely Lorentzian perturbative regulator". The Wolfram physics project (WPP; Wolfram 2020, Gorard 2020) generates spacetime as the causal graph of a deterministic hypergraph rewrite rule; the same dispersion question on this substrate is much less developed, and no direct WPP analog of the ASS 2014 instability result has been published.

This paper computes the dispersion of a localized causal-edge perturbation (LCEP, a coherent fluctuation in coarse-grained hyperedge density) on a single causal history of one explicit Wolfram-class rule, using the BBD 2016 four-layer operator. The arithmetic is straightforward; the result is a negative one. Three moment-cancellation conditions required for a massless branch are not satisfied by the rule's measured shell coefficients, and the symbolic expansion of the dispersion relation shows that the would-be coefficient $\alpha$ drops out of the orders at which it could appear. A numerical proxy fit gives $\alpha \approx -9.35$ but with structural pathologies (eigenvector residuals near unity, phase velocity monotonically decreasing) that disqualify it as a physical prediction.

We make three claims about what this means. First, structural parallels to ASS 2014's stochastic 4D instability suggest that the 4D instability is substrate-independent: it holds for at least one deterministic graph and for the standard random sprinkling. Second, the Floquet/Bloch-wave method we used to extract the symbolic expansion is, as far as we have been able to verify, novel in this setting and is a useful diagnostic in its own right. Third, the ASS 2014 generalized operator offers a concrete and published path to a possibly stable result that has not yet been applied to a WPP causal graph. These are stated as a conjecture and a research program, not as theorems.

We try to be careful throughout to distinguish (a) what was actually computed, (b) what was structurally observed but not proven, and (c) what is conjectured. Open issues — including the unproven causal invariance of $R_{323}$ and the absence of a faithful Lorentzian embedding of its causal graph — are collected in Section 7.

2. Setup

2.1 The hypergraph and the rewrite rule

Let $H = (V, E)$ be a 3-uniform directed hypergraph: $V$ a vertex set, each edge $e \in E$ an ordered triple $(a,b,c) \in V^3$. Multi-edges and self-loops are allowed. We commit to one rewrite rule, which we label $R_{323}$: $$ \{x,y,z\},\ \{y,u,w\}\ \longrightarrow\ \{x,y,z\},\ \{y,u,w\},\ \{z,u,v\},\ \{v,w,x\}. $$ Read: whenever two existing hyperedges share their second vertex (call it $y$), introduce one fresh vertex $v$ and two new hyperedges $(z,u,v)$ and $(v,w,x)$. The two pre-existing hyperedges are retained.

The label "$R_{323}$" is local to this paper. Hypergraph rewrite rules do not yet have a community-wide Wolfram-code-style taxonomy in the peer-reviewed literature (cf. Gorard 2020), so we give the algebraic signature explicitly so the construction is reproducible.

A rewrite event is a single application of the rule. Two events $e_1, e_2$ are causally ordered ($e_1 \prec e_2$) iff some hyperedge created by $e_1$ appears in the LHS pattern matched by $e_2$. The resulting DAG is the causal graph $C$, the discrete analog of a Lorentzian manifold.

A scheduling that produces a unique final hypergraph (up to isomorphism) regardless of the total order in which independent events are fired is called causally invariant. We assume $R_{323}$ is causally invariant in the limit of large event counts; numerical confluence has been checked for similar rules (Gorard 2020) but is not proven for $R_{323}$. This assumption is load-bearing for Lorentz covariance and we return to it in Section 7.

2.2 The propagating mode

We adopt the localized causal-edge perturbation (LCEP) as our notion of a propagating massless excitation. Coarse-graining the hypergraph at length scale $L$ with $\ell \ll L \ll V^{1/3}$ yields an edge-density field $n(x,t)$. An LCEP is a deviation $\delta n(x,t)$ satisfying: 1. propagation at the causal speed $c \equiv \ell/\tau$ in the $L \to 0$ limit, 2. transverse normalization, $\int \delta n\, d^3x = 0$ over a wavelength, 3. coherent lifetime exceeding the rewrite period $\tau$.

This is closer to a graviton/sound mode on the discrete substrate than to a gauge boson with internal indices; the LCEP definition is silent on $U(1)$ polarization and charge. Trugenberger's combinatorial-quantum-gravity "graph phonon" (Trugenberger 2017) is the closest published analog. We use "photon" only as shorthand for the LCEP target and not as a claim that gauge structure has been derived.

2.3 The discrete d'Alembertian

For a finite causal set we use the standard four-layer 4D retarded operator (Benincasa-Dowker 2010; BBD 2016): $$ B\phi(x) = \frac{4}{\sqrt{6}\,\ell^2} \left[ -\phi(x) + \sum_{y \in L_0(x)} \phi(y) - 9 \sum_{y \in L_1(x)} \phi(y) + 16 \sum_{y \in L_2(x)} \phi(y) - 8 \sum_{y \in L_3(x)} \phi(y) \right], $$ where $L_k(x)$ is the inclusive past layer containing events $y \prec x$ with exactly $k$ intervening events: $N(y,x) = |\{z : y \prec z \prec x\}| = k$. The integer coefficients $(1,-9,16,-8)$ with diagonal $-1$ are the 4D-specific weights chosen so that, after ensemble averaging over Poisson sprinklings of flat Minkowski space, the operator reproduces the continuum $\Box\phi$ to leading order (BBD 2016, Eq. 3).

A clarification for non-CST readers. There are two distinct integer sequences floating around the literature, and a careful reader of Glaser 2014 and BBD 2016 must keep them separate. The Taylor-expansion weights $c_n = (-1)^{n-1}/n!$ from the continuum kernel are the dimension-general building blocks; the 4D layer coefficients $(1,-9,16,-8)$ and diagonal $-1$ are the specific integer linear combinations that yield $\Box$ in $d=4$ after averaging. We use the latter throughout.

ASS 2014 generalized this operator to a one-parameter family parameterized by a non-locality scale $\epsilon$; this family is what we conjecture is relevant for a future stable computation. We do not implement it here.

3. Computation

3.1 Generation of the causal graph

We instantiate $R_{323}$ from the seed $$ \big[(0,1,2),(1,2,3),(2,3,4),(3,4,0),(4,0,1)\big] $$ and apply the rule in a random but reproducible (seed-fixed) order for $500$ events, producing $1005$ hyperedges. Statistics (full code available on request): - mean immediate in-degree = mean out-degree = $1.884$, - maximum causal depth $= 11$, - mean transitive past size $= 12.228$, - causal-distance shell means at distances $d = 1\ldots 5$ are $(1.884,\, 2.914,\, 3.226,\, 2.276,\, 1.120)$, - inclusive layer occupancies $|L_0|,\ldots,|L_3|$ have means $(1.680,\, 2.174,\, 2.126,\, 1.750)$.

Update-order sensitivity (caveat). A breadth-first deterministic update order on the same seed produces max depth $\approx 3$ at $500$ events; a depth-biased order produces an almost one-dimensional chain. The above statistics correspond to one particular reproducible random schedule. This sensitivity is a direct symptom of the unproven causal invariance of $R_{323}$ at finite $N$ and is flagged again in Section 7.

3.2 The literal BBD bracket fails on constants

Applying the BBD coefficients to a constant field and averaging over events gives $$ -1 + \langle |L_0|\rangle - 9\langle|L_1|\rangle + 16\langle|L_2|\rangle - 8\langle|L_3|\rangle = 1.130. $$ A continuum d'Alembertian must annihilate constants. The literal finite-graph operator on $R_{323}$ does not, by an O(1) margin. This is a structural finite-size and shell-statistics mismatch, not a roundoff error.

3.3 The three moment-cancellation conditions

Under an isotropic-shell plane-wave ansatz $\psi(y) \approx \psi(x)\,\exp(-i\omega d\tau)\,J_0(k\,d\,s)$ for a $y$ at causal distance $d$, with $A_d$ the (possibly reweighted) coefficient in shell $d$, the Bloch eigenvalue is $$ \Lambda(k,\omega) = D + \sum_{d=1}^{4} A_d\,e^{-i\omega d\tau}\,J_0(k\,d\,s). $$ Setting $q = k\ell$ and writing $\omega = c q(1 + \alpha q^2/2)$, a SymPy expansion through $q^4$ yields the coefficients reproduced verbatim in the supplementary computation. A massless branch requires the following three conditions: $$ (1)\quad D + A_1 + A_2 + A_3 + A_4 = 0, $$ $$ (2)\quad A_1 + 2A_2 + 3A_3 + 4A_4 = 0, $$ $$ (3)\quad A_1 + 4A_2 + 9A_3 + 16A_4 = 0. $$ These are, respectively, the zeroth, first, and second moment-cancellation conditions familiar from CST (Sorkin 2007; BBD 2016). The measured $R_{323}$ shell weights do not satisfy any of them.

3.4 $\alpha$ drops out at the relevant orders

The crucial observation: the $q^3$ and $q^4$ coefficients of $\Lambda(k,\omega)$ have the form $$ \text{coeff}(q^3) \propto -\alpha\,(A_1 + 2A_2 + 3A_3 + 4A_4) + \text{($\alpha$-free moments)}, $$ $$ \text{coeff}(q^4) \supset -\alpha\,(A_1 + 4A_2 + 9A_3 + 16A_4)\cdot(c\tau)^2/2 + \cdots $$ The factors multiplying $\alpha$ are exactly the first and second moments — i.e., the LHS of conditions (2) and (3). If the moments cancel, the $\alpha$ contribution vanishes at these orders, postponing $\alpha$ to higher order. If the moments do not cancel — which is the case for $R_{323}$ — then there is no massless branch around $\omega = c k$ at all, and $\alpha$ is not the leading correction to a relativistic dispersion: it is the subleading correction to a non-relativistic one. This is the explicit algebraic obstruction.

3.5 The numerical proxy is an artifact

To check what the "fit anyway" answer would look like, we assign event time by topological depth, one spatial coordinate by the first non-constant eigenvector of the normalized graph Laplacian, take $\psi_e = \exp(i k x_e - i \omega t_e)$, minimize $\langle |B\psi|^2\rangle$ over $\omega$ for each $k$, and fit $\omega/k = c_{\mathrm{fit}} + \mathrm{slope}\cdot k^2$. The result is $c_{\mathrm{fit}} = 0.0689$, slope $= -0.322$, giving $\alpha_{\mathrm{proxy}} \approx -9.35 \pm 0.82$.

This number is within the $[10^{-2}, 10^{2}]$ window we set as the experimentally interesting range in the setup of §2, but it is not a physical value of $\alpha$. The diagnostics rule it out:

1. Eigenvector residuals $\|B\psi - \lambda\psi\|/\|B\psi\| \approx 0.999$ for all sampled $k$. The graph-Laplacian plane waves are almost completely not eigenvectors of $B$.
2. $\omega/k$ decreases monotonically toward zero over the sampled $k$ range $[0.1, 0.5]$ rather than approaching a finite low-$k$ speed. There is no light-cone limit being recovered.
3. The "real" matrix spectrum is a triangularity artifact. The retarded $B$ is lower-triangular in event-creation order; with a renormalized diagonal $D_{\mathrm{ren}} = 2.165$ chosen post hoc to annihilate constants, every diagonal entry is identical and all eigenvalues of $B$ collapse to $D_{\mathrm{ren}}$. The numerically observed real spectrum is a property of triangular matrices, not evidence that the continuum limit has a real plane-wave spectrum.

We therefore record $\alpha_{\mathrm{proxy}} \approx -9.35$ for documentation and flag it as an artifact. The defensible statement is that, under the BBD operator and the embedding we used, $\alpha$ is undefined for $R_{323}$ at first order in $k\ell_P$.

4. Discussion

4.1 Substrate-independence of the 4D instability

ASS 2014 showed that the standard 4D BBD operator $B$ on a Poisson-sprinkled flat causal set has an unstable spectrum — exponentially growing modes — and that the analogous 2D operator is stable. Their construction is stochastic (averaging over sprinklings) and their proof is spectral.

What we observe deterministically on $R_{323}$ is structurally parallel rather than identical. The deterministic finite-graph operator (i) fails the moment-cancellation conditions on a constant field by an O(1) margin, (ii) yields a Bloch-wave expansion in which $\alpha$ drops out at the orders required, and (iii) produces a numerical proxy with phase velocity decreasing monotonically toward zero. None of (i)-(iii) is a direct measurement of ASS 2014's exponentially growing modes, and we have not computed the full spectrum of $B$ on $R_{323}$ in the same sense ASS 2014 did. What both cases share is the structural feature: in $d=4$, the moment-cancellation conditions necessary for a stable, propagating massless branch are not generically satisfied at finite layer counts, regardless of whether the causal structure comes from random sprinkling or deterministic rewriting.

The right framing is therefore that we have structural evidence, not a proof, that the 4D instability of the BBD operator is substrate-independent. The proof in the deterministic case would consist of (a) a direct spectral analysis of $B$ on $R_{323}$ in the limit of large event count, and (b) a demonstration that the failure mode reduces to ASS 2014's in the appropriate limit (e.g., that the sprinkling-equivalent causal set of $R_{323}$, if one exists, gives the ASS 2014 result). Neither step is done here.

If the structural evidence is borne out, this matters for two reasons. First, it suggests the instability is a feature of the operator and the dimension, not of any particular substrate's randomness, simplifying the search for stable formulations. Second, it provides the first quantitative bridge from the WPP family of theories to the CST literature on operator construction: the same $(1,-9,16,-8)$ coefficients can in principle be applied on a Wolfram-class graph, and they fail there in a structurally similar way to how they fail on a sprinkled causal set.

4.2 Floquet/Bloch-wave analysis on causal graphs

The expansion in §3.3 treats the action of $B$ on plane waves indexed by a wavevector and frequency on the causal graph. This is conceptually Bloch/Floquet analysis applied to the periodic quotient of the rewrite system rather than to a crystal lattice. The method is standard in solid-state physics; its application to causal-set-style operators is, to the best of our literature review, new. CST has historically used spatial-sprinkling spectral methods that do not require periodicity (Belenchia-Benincasa-Dowker 2016); WPP has used graph-distance and Myrheim-Meyer dimensional analysis (Gorard 2020) rather than spectral momentum-space techniques.

We present this method here as a diagnostic and not as a definitive computation. Its diagnostic value is exactly what was used in §3: it produced an explicit algebraic obstruction (the moment-cancellation conditions and the dropout of $\alpha$) in a few lines of symbolic algebra, rather than requiring a full numerical eigenvalue scan. It is most likely to be useful for causally invariant, periodic-quotient rewrite systems where the unit cell of the periodic structure can be identified and the Bloch ansatz can be made rigorous. For finite, non-periodic causal histories (such as the $R_{323}$ instance we computed), Floquet analysis is heuristic; one would want a verified periodic quotient or an explicit treatment of finite-size effects before treating its output as predictive.

4.3 A path forward: the ASS 2014 generalized operator

ASS 2014's generalized causal-set d'Alembertian family is parameterized by a non-locality scale $\epsilon$ and "acts as a genuinely Lorentzian perturbative regulator" for the unstable modes. We conjecture, but do not show, that applying this family to the $R_{323}$ causal graph would yield a stable Bloch-wave branch with a calculable $\alpha = \alpha(\epsilon, R_{323})$.

This is an explicit conjecture to be tested. The minimum technical content of the test is: (i) implement the ASS 2014 generalized operator on $R_{323}$'s finite event DAG, (ii) repeat the Bloch-wave expansion of §3.3, (iii) check whether moment-cancellation can be satisfied for $\epsilon$ in some range, and (iv) extract $\alpha(\epsilon)$ and ask whether it has a finite $\epsilon \to 0^+$ limit or whether $\epsilon$ must be retained at a Planck-scale value. We do not predict the outcome.

Two cross-checks would strengthen the conjecture. First, applying the generalized operator on a Poisson sprinkling and recovering ASS 2014's stability result (a baseline). Second, applying it on a small selection of sibling Wolfram-class rules (cf. Gorard 2020's rule catalog) and asking whether the resulting $\alpha$ is rule-dependent. If $\alpha$ depends on the rule, that is the long-sought non-trivial prediction; if it doesn't, the WPP substrate is doing less work than the operator.

5. Open problems and falsifiability

5.1 Causal invariance of $R_{323}$ is unproven

We assumed throughout that $R_{323}$ is causally invariant in the large-event-count limit, but the update-order sensitivity we observed at $N = 500$ (max depth $11$ random, $\approx 3$ BFS, near-chain depth-biased) means causal invariance is not established at this size. Without it, no observer-independent ω(k) exists and the entire Bloch ansatz is suspect. Direct test: numerical confluence check on $R_{323}$ at $N \ge 10^6$.

5.2 The "$R_{323}$" label needs canonicalizing

The label is local to this paper and does not match any published Wolfram code. We provide the algebraic signature explicitly in §2.1; future work should propose a community-friendly Wolfram-code-style taxonomy for arity-3 rules so that "$R_{323}$" can be cross-referenced.

5.3 The numerical $\alpha_{\mathrm{proxy}} \approx -9.35$ is an artifact

It should not be cited as a physical prediction or compared to Fermi-LAT / IceCube LIV bounds. The eigenvector residuals near unity and the monotonic phase velocity falloff disqualify it. We report it only as a diagnostic.

5.4 No faithful Lorentzian embedding of $R_{323}$ has been constructed

The graph-Laplacian coordinate we used for §3.5 is a spectral drawing, not a Lorentzian-manifold approximation. Without a faithful embedding, the Poisson volume factor $V(d)$, shell degeneracy, and spatial isotropy are assumptions rather than measured quantities, and the "isotropic shell" Bloch ansatz is approximate at best. Constructing such an embedding (or proving none exists) is the largest single open piece of technical work.

5.5 Falsifiability of the universality claim

The natural test: apply the same BBD-operator Bloch-wave diagnostic to a small library of Wolfram-class causally invariant rules drawn from Gorard's 2020 catalog (or its successors). If they all fail moment-cancellation in structurally similar ways to $R_{323}$, the universality claim gains support. If a non-trivial fraction of rules do satisfy moment-cancellation and yield finite $\alpha$, the claim is falsified — i.e., the 4D instability is then a feature of $R_{323}$, not of the operator-dimension pair.

6. Conclusion

We attempted to derive the first Planck-scale correction to the photon dispersion on a single causal history of a deterministic Wolfram-class hypergraph rewrite rule, using the standard BBD discrete d'Alembertian. The attempt failed in an informative way: the three moment-cancellation conditions required for a massless branch are not satisfied by the measured shell coefficients of $R_{323}$, and a symbolic Bloch-wave expansion shows that the would-be coefficient $\alpha$ drops out at the orders at which it should appear. The numerical proxy $\alpha \approx -9.35$ is an artifact of the chosen embedding and renormalization. This structural pattern of failure parallels the stochastic 4D instability of ASS 2014 closely enough that we conjecture the 4D instability of the BBD operator is substrate-independent, and we have introduced Floquet/Bloch-wave analysis on causal graphs as a new diagnostic. A concrete path forward — the ASS 2014 generalized operator family applied to $R_{323}$ — remains as a falsifiable conjecture for future work.

Acknowledgements

Numerical code and supplementary materials available on request. The mathematical errors and conceptual choices remain the responsibility of the corresponding author.

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