# Thread: Lorentz-Invariant Discretizations of Field Theory

**Date:** 2026-09-20
**Scout:** subagent (depth 1/5)
**File:** reports/threads/2026-09-20-lorentz-invariant-discretization.md
**Question:** Does ANY discretization of field theory have EXACT Lorentz invariance at FINITE lattice spacing? What does it cost? Does it evade the dispersion-relation signature?

---

## VERDICT (summary, details below)

**Yes, one construction exists with a *proven* exact Lorentz invariance at finite discreteness — the causal set (Poisson sprinkling into Minkowski space) — and exactly one published claim of a gauge-theory regulator of this kind (Andersen 2012, never peer-reviewed, zero citations). Everything else restores Lorentz invariance only statistically (ensemble average) or in the continuum limit.**

Concretely:

1. **Random (Poisson) lattices — Christ, Friedberg, Lee 1982.** Invariance is NOT exact per realization. It is restored only **in the ensemble average over lattices** and in the **continuum limit**. Cost: you must average every observable over an ensemble of random lattices, plus build the lattices. The programme produced no usable alternative to hypercubic lattice QCD; fermion doubling was shown to be **revived by gauge interactions** (Griffin & Kieu 1993), a main reason it was not adopted. **Established.** The single-lattice realization has locally identifiable preferred directions, exactly like a gas.

2. **Causal sets — YES for the structure, at a steep price.** Bombelli, Lee, Meyer & Sorkin (PRL 59, 521, 1987) introduced the causal set. Bombelli, Henson & Sorkin (Mod. Phys. Lett. A 24, 2579, 2009; arXiv:gr-qc/0605006) **prove a theorem**: there is no equivariant measurable map from a Poisson sprinkling of Minkowski space to spacetime directions, even locally; therefore an intrinsically-defined discrete structure on a sprinkling picks out no preferred frame, and **no finite-valency graph can be associated to a sprinkling consistently with Lorentz invariance**. This is a theorem about the **Poisson process/distribution**, not about individual realizations: a given sprinkling *does* have locally identifiable directions (like a gas), but with high probability they wash out at the continuum level, and — crucially — unlike a crystal there is no equivariant way to *systematically* assign a frame even locally. *Caveat: a particular realization does have local anisotropy; exactness is statistical at the level of the ensemble.* **Established (published theorem).**
   - **The price:** the exactly-Lorentz-invariant dynamics on a causal set — the d'Alembertian (Sorkin; Benincasa & Dowker, PRL 104, 181301, 2010; Dowker & Glaser, Class. Quantum Grav. 30, 195016, 2013; Aslanbeigi, Saravani & Sorkin, JHEP 06 (2014) 024) — is **nonlocal**. Verbatim from Benincasa & Dowker abstract: the operators "are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality and approximate the continuum scalar D'Alembertian … when acting on fields that vary slowly on the nonlocality scale." Sorkin (arXiv:gr-qc/0703099) states the general obstruction: *"discreteness plus Lorentz invariance entails nonlocality"*, and that the nonlocality may **not be confined to the discreteness scale** — the scale λ0 at which locality is reinstated "seems to reflect not only the ultraviolet scale l but also an infrared scale R, which we may identify with the age of the cosmos." I.e., for a slowly-varying field, the operator sums the field over elements throughout the **entire causal past** of each point, with weights that must cancel to high order.
   - **Does it evade the dispersion signature? YES — by replacing it.** The BHS theorem abstract says sprinkling discreteness "will not give rise to 'Lorentz breaking' effects like modified dispersion relations." The phenomenological signature instead is **stochastic diffusion in momentum space — "swerves"**: Dowker, Henson & Sorkin, Mod. Phys. Lett. A 19 (2004) 1829 (gr-qc/0311055): "The particles undergo a Lorentz invariant diffusion in phase space"; Philpott, Dowker & Sorkin, Phys. Rev. D 79, 124047 (2009) (0810.5591): "the microscopic swerves … manifest themselves as a Lorentz invariant diffusion in energy-momentum … The particles do not leave the light cone." So: no preferred-frame dispersion modification, but a new stochastic Lorentz-invariant process (constrained, e.g., by CMB blackbody).

3. **The general obstruction.** The standard argument (boost contracts the cell): stated precisely by **Sorkin, arXiv:gr-qc/0309009** (Valdivia notes), diamond-lattice example, verbatim below. The formal version is the **Bombelli–Henson–Sorkin theorem** (no equivariant measurable map to directions; no Lorentz-covariant finite-valency graph on a sprinkling). I found **no separate published no-go theorem** for "exact Lorentz invariance of dynamics on an arbitrary fixed graph" beyond BHS; the BHS theorem plus Sorkin's nearest-neighbor argument (gr-qc/0703099) are the sharpest statements in print. The nearest-neighbor version: for a Poisson sprinkling, "the probability of any given element e possessing a limited number of nearest neighbors is vanishingly small," so any nearest-neighbor-type local coupling structure fails to be Lorentz invariant.

4. **Andersen, arXiv:1210.8348.** Never published in a peer-reviewed journal (no journal record on INSPIRE; Semantic Scholar venue empty). **Zero citations on INSPIRE (record 1197887, citation_count: 0) and zero on Semantic Scholar (API: citationCount 0).** No rebuttal or follow-up found anywhere. Andersen's other work (via arXiv author query) shows no later developed version of this construction; his later papers go in other directions (deterministic/chaotic quantization, SO(4,1) Yang-Mills gravity). **The honest answer: essentially zero scholarly uptake.** The paper's mechanism ("exact Lorentz covariance by replacing the lattice with a graph whose metric is a discrete Lorentz-covariant matrix potential") is an unvetted preprint claim.

**Bottom line for the main thread's question:** The standard dispersion-relation signature (preferred-frame O(a²)/O(a) artefacts from a hypercubic lattice) can be evaded, but the only known way with *proven* exact invariance is a random (Poisson) structure, and the price is (a) statistical/ensemble-level exactness, not per-realization, (b) intrinsic **nonlocality** on scales far above the discreteness scale (possibly cosmologically large), and (c) a new stochastic signature (swerves/momentum diffusion) instead of a modified dispersion relation. A regular lattice with exact Lorentz covariance at finite spacing exists only in one uncited, unpublished arXiv preprint (Andersen), which the community has neither confirmed nor engaged with.

---

## 1. RANDOM / POISSON LATTICES (Christ, Friedberg & Lee)

### 1.1 Exact references (all verified against independent citing sources)

| Ref | Verified | Evidence |
|---|---|---|
| N.H. Christ, R. Friedberg, T.D. Lee, "Random lattice field theory: General formulation", **Nucl. Phys. B202, 89–125 (1982)** | `verified-at-source` (multiple independent citations: Springer CMP bibliography "Nucl. Phys. B202, 89–125 (1982)"; APS PRD 108.014511; APS PRD 104.094502; INSPIRE record 11586: journal Nucl.Phys.B, volume 202, artid 89, year 1982, authors Christ/Friedberg/Lee) | ScienceDirect abstract (PII 055032138290222X): "A new type of lattice field theory is formulated in which the sites are chosen randomly in space. An algorithm is given for linking nearby sites…" |
| N.H. Christ, R. Friedberg, T.D. Lee, "Gauge theory on a random lattice", **Nucl. Phys. B210 [FS6], 310–336 (1982)** | `verified-at-source` (Springer CMP bibliography "Nucl. Phys. B210 [FS6], 310–336 (1982)"; APS PRD 105.114503; Fermilab preprint by Kronfeld) | ScienceDirect abstract (PII 0550321382901237): "A general formulation of gauge theory on a random lattice is developed and the strong coupling limit of the Wilson string tension worked out." |
| N.H. Christ, R. Friedberg, T.D. Lee, "Weights of links and plaquettes in a random lattice", **Nucl. Phys. B210 [FS6], 337–346 (1982)** | `verified-at-source` (Springer CMP bibliography "337–346 (1982)"; APS PRD 103.094507; APS PRD 108.014511) | — |
| T.D. Lee (Les Houches lecture note) "Random lattice field theory", in Les Houches 1982 proc., pp. 461–462 | `verified-at-source` (INSPIRE record 187256) | — |

(The user's recollection "B202 (1982) 89; B210, B210[FS6]" is **confirmed**; the two B210 papers are both in the FS6 volume, pages 310 and 337.)

### 1.2 What exactly is claimed about rotational/Lorentz invariance — and how exact?

**The claim is NOT exact invariance at finite spacing for a single lattice. It is restoration in the ensemble average and in the continuum limit.** Two independent secondary sources state this; I verified the text of both.

- **Griffin & Kieu, "Fermionic field theory and gauge interactions on random lattices", arXiv:hep-lat/9307011 (1993)**, §1 (read at source, arXiv HTML):
  > "In random lattice approaches, suitable quantities are measured on a random lattice then averaged, either quenchedly or annealedly, over an ensemble of lattices. Apart from the extra work involved in generating an ensemble of random lattices, this approach better approximates the scale-free rotational and translational symmetry of the continuum than regular lattices. Thus, the continuum limit may be more easily reached on random lattices than on regular lattices of the same size."
  - Note the two qualifications embedded in this quote: (i) averaging over an ensemble is required — this is the "extra work" (computational cost); (ii) it "better approximates," it does not achieve, the symmetry at finite spacing.
  - Also on spectrum: "since there is no fixed Brillouin zone, there need be no extra poles of the propagator … there is no transfer matrix on a random lattice (at least for a finite lattice) since there are no identical timeslices."
- **O.V. Pavlovsky, "Random Lattice QCD and chiral effective theories", arXiv:hep-lat/0407026 (2004)**, §2 (read at source):
  > "In these articles have been shown that **in order to obtain the restoration of the Lorentz (rotational) invariance, it is necessary to perform an average over an ensemble of random lattices**."
- Supporting (2026 review context, read at source): **T. Gantumur, "Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories", arXiv:2512.22072 (May 2026)**: "Random lattice methods, dynamical triangulations, and Regge calculus all average over ensembles of discrete geometries."

**Answer to "is invariance EXACT at finite density?" — NO per realization.** A single random lattice is a particular arrangement with local anisotropies; the rotational symmetry is a property of the *probability distribution* over the ensemble. (Same structure as the causal-set case in §2, but the CFL programme never claimed a theorem like BHS's; it just used ensemble averaging.)

### 1.3 Quoted computational cost penalty

The only quantified statement I found in the secondary literature is qualitative ("the extra work involved in generating an ensemble of random lattices" — Griffin & Kieu, above), i.e., you must sample and average over multiple lattice geometries (a multiplier on every simulation) plus generate the lattices. I did **NOT** find a hard number (e.g., "N times slower") quoted anywhere at source. `NOT VERIFIED AT SOURCE` for any numeric penalty.

### 1.4 Did the programme succeed? Was it abandoned, and why?

- **Doubling problem initially looked solved — then revived by gauge interactions.** Griffin & Kieu's abstract (read at source): "Random-lattice fermions have been shown to be free of the doubling problem if there are no interactions or interactions of a non-gauge nature. However, gauge interactions impose stringent constraints … We show that the doublers are revived for random lattices in the continuum limit, while demonstrating that gauge invariance plays the critical role in this revival." Their §1: "The full gauge invariant formulation has not yet been properly considered on account of problems associated with identifying the appropriate conserved gauge current that appears in the action."
- **Status in modern lattice QCD: not used.** Every citation of the CFL papers in modern papers (PRD 108.014511 "Ising model on the affine plane", PRD 105.114503 "Hyperbolic lattice for scalar field theory", PRD 103.094507, PRD 104.094502) cites them only as historical/motivational background for *other* symmetry-restoration ideas (affine/F4 lattices, hyperbolic lattices, radial lattices) — never as the working method. I found **no single canonical retrospective review saying "random lattices were abandoned because X"**. My assessment of the reasons, from the sources I did read: (i) gauge-invariant formulation never worked cleanly (Griffin & Kieu); (ii) no transfer matrix / no identical timeslices, making spectral interpretation and Hamiltonian analysis awkward (Griffin & Kieu); (iii) ensemble-averaging cost (Griffin & Kieu); (iv) hypercubic lattices + improved actions solved the practical a² error problem well enough that the marginal benefit did not justify the cost. Items (i)–(iii) are `verified-at-source` from Griffin & Kieu; item (iv) is my own inference (evidence class: **Your own inference**), grounded in the continued dominance of hypercubic improved actions in the citations above.

---

## 2. CAUSAL SETS

### 2.1 Founding reference

**L. Bombelli, J. Lee, D. Meyer, R.D. Sorkin, "Space-time as a causal set", Phys. Rev. Lett. 59, 521–524 (1987).** `verified-at-source` (APS DOI 10.1103/PhysRevLett.59.521; INSPIRE record 21718: "Phys.Rev.Lett. 59 (1987) 521-524"; ADS abstract: "We propose that space-time at the smallest scales is in reality a causal set: a locally finite set of elements endowed with a partial order corresponding to the macroscopic relation that defines past and future.") Note the phrase "locally finite" appears already in the 1987 abstract. Also existing: a **Comment** on it, PRL 60, 655 (1988) — existence `verified-at-source` (APS DOI 10.1103/PhysRevLett.60.655); content not read, `inherited-unchecked`.

### 2.2 "Discreteness without symmetry breaking" — the theorem

**L. Bombelli, J. Henson, R.D. Sorkin, "Discreteness without symmetry breaking: a theorem", arXiv:gr-qc/0605006, published Mod. Phys. Lett. A 24, 2579–2587 (2009).** `verified-at-source` (arXiv abstract + journal ref read directly).

Verbatim abstract:
> "This paper concerns sprinklings into Minkowski space (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to 'Lorentz breaking' effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance."

**What EXACTLY is proved — precise statement and limits:**
- It is a theorem about **Poisson sprinklings** (random point processes with probability density ρ) into flat Minkowski space.
- Statement: **no equivariant measurable map** from the space of sprinklings to spacetime directions exists — even locally (i.e., using only the sprinkling data near a point). Equivariance means the map commutes with Lorentz transformations; measurability is the technical regularity condition making "intrinsic algorithm" precise.
- Consequences (as the authors state): (a) an intrinsically-defined discrete structure on a sprinkling picks out no preferred frame, locally or globally; (b) hence no modified dispersion relations from sprinkling discreteness; (c) **no finite-valency graph can be associated to a sprinkling consistently with Lorentz invariance.**
- **Critical caveats (evidence class: Established — they are in the paper and in Henson's review):** the theorem concerns the *process/distribution* and *intrinsic* constructions. A given realization — "a particular sprinkling" — **does** have locally identifiable directions in small regions containing few points (like gas molecules). The claim of exact Lorentz invariance is therefore a claim about the ensemble-level/instrinsic-level structure, not about microscopic per-realization isotropy. Henson's review (gr-qc/0601121, read at source), §1.5:
  > "As with the gas, any particular instance of the process – a particular sprinkling – will have directions identifiable in small regions containing few points, but this will have no effect on the continuum treatment."
  and
  > "But due to the non-compactness of the Lorentz group, there is no way to associate a preferred frame to a point in a sprinkling of Minkowski that commutes with Lorentz boosts."
  (This non-compactness is exactly why the gas analogy is *not* complete: in Euclidean space a nearest-neighbour direction map does commute with rotations; in Lorentzian signature it cannot.)

### 2.3 The causal set d'Alembertian — exactly Lorentz invariant, but NONLOCAL

- **Sorkin's continuum version and history:** R.D. Sorkin, "Causal sets: Discrete gravity (notes for the Valdivia Summer School)", arXiv:gr-qc/0309009 (2003) — lecture notes containing the first continuum integral form; and R.D. Sorkin, "Does Locality Fail at Intermediate Length-Scales?", arXiv:gr-qc/0703099 (2007, in *Towards Quantum Gravity*, ed. D. Oriti, Cambridge UP) — the nonlocality argument. Both read at source.
- **D.M.T. Benincasa, F. Dowker, "The Scalar Curvature of a Causal Set", Phys. Rev. Lett. 104, 181301 (2010), arXiv:1001.2725.** `verified-at-source`. Verbatim abstract:
  > "A one parameter family of retarded linear operators on scalar fields on causal sets is introduced. When the causal set is well-approximated by 4 dimensional Minkowski spacetime, **the operators are Lorentz invariant but nonlocal, are parametrised by the scale of the nonlocality** and approximate the continuum scalar D'Alembertian, □, when acting on fields that vary slowly on the nonlocality scale. … This can be used to define an approximately local action functional for causal sets."
  Note the phrase "approximately local": locality is recovered only approximately and only for slowly-varying fields.
- **F. Dowker, L. Glaser, "Causal set d'Alembertians for various dimensions", Class. Quantum Grav. 30, 195016 (2013), arXiv:1305.2588.** `verified-at-source`. Abstract: "We propose, for dimension d, a discrete Lorentz invariant operator on scalar fields that approximates the Minkowski spacetime scalar d'Alembertian."
- **S. Aslanbeigi, M. Saravani, R.D. Sorkin, "Generalized causal set d'Alembertians", JHEP 06 (2014) 024, arXiv:1403.1622.** `verified-at-source` (ADS + ETDE abstracts): "We introduce a family of generalized d'Alembertian operators in D-dimensional Minkowski spacetimes which are manifestly Lorentz-invariant, retarded, and non-local, **the extent of the nonlocality being governed by a single parameter**."

**Why nonlocality is inevitable (the mechanism, read at source from Sorkin gr-qc/0703099):**
> "Locality in the discrete context, if it meant anything at all, would imply that the action of □ would be built up in terms of 'nearest neighbor couplings' (as in fact ∇² can be built up, on either a crystalline or random lattice in E³). But Lorentz invariance contradicts this sort of locality because it implies that, no matter how one chooses to define nearest neighbor, any given causet element e ∈ C will possess an immense number of them extending throughout the region of C corresponding to the light cone of e in M. In terms of a Poisson process in M we can express this more precisely by saying that **the probability of any given element e possessing a limited number of nearest neighbors is vanishingly small**. Thus, the other elements to which e must be 'coupled' by our box operator will be large in number (in the limit infinite), and in any given frame of reference, the vast majority of them will be remote from e. The resulting 'action at a distance' epitomizes the maxim that **discreteness plus Lorentz invariance entails nonlocality**."

**Quantify the price:**
- The operator sums over elements throughout the **entire causal past** of each evaluation point, with oscillating-sign weights that must cancel to reproduce □ for slowly-varying fields (Sorkin 0703099 §"Comparison with the D'Alembertian": "the larger magnitudes among [the matrix elements B_xy] are concentrated 'along the light cone'… [with] a recourse to oscillating signs [to effect] the 'miraculous cancellations'"; boosted cells contribute "only a tiny contribution — exactly the sort of cancellation we were seeking!").
- **The nonlocality scale is far above the discreteness scale** (Henson review, read at source): the operator's kernel parameter k gives nonlocality length 1/√k: "1/√k is taken to be some length above the Planck scale, but below the known limits on locality"; and "the length associated to the locality scale, 1/√k, must be sufficiently far above the length scale set by the fundamental discreteness, 1/√ρ." So: discreteness at the Planck scale ⇒ dynamics nonlocal over scales ≥ many Planck lengths.
- **Worse — possibly not confinable to any scale:** Sorkin 0703099 (abstract, verbatim): "If quantum gravity implies a fundamental spatiotemporal discreteness, and if its 'laws of motion' are compatible with the Lorentz transformations, then physics cannot remain local. **One might expect this nonlocality to be confined to the fundamental discreteness scale, but I will present evidence that it survives at much lower energies.**" And in the body: "it is far from obvious that the kind of nonlocality in question can be confined to any scale… the scale λ0 at which it begins to disappear seems to reflect not only the ultraviolet scale l but also an infrared scale R, **which we may identify with the age of the cosmos**." (Evidence class: **Serious speculation** — Sorkin's own characterization is an argument/evidence-in-a-model, not an established theorem; his eq. (7) kernel e^{−kV}(1−2kV+½k²V²) is a Lorentz-invariant nonlocal operator that is his proposed nonlocal extension, and whether nonlocality truly persists to cosmological scales is the contested part.)

### 2.4 Its own observable signature: "swerves" (momentum-space diffusion), NOT modified dispersion

Because the BHS theorem forbids preferred-frame effects (modified dispersion), the causal-set signature is a different, stochastic one:

- **F. Dowker, J. Henson, R.D. Sorkin, "Quantum Gravity Phenomenology, Lorentz Invariance and Discreteness", Mod. Phys. Lett. A 19 (2004) 1829–1840, arXiv:gr-qc/0311055.** `verified-at-source`. Verbatim abstract: "Contrary to what is often stated, a fundamental spacetime discreteness need not contradict Lorentz invariance. A causal set's discreteness is in fact locally Lorentz invariant, and we recall the reasons why. For illustration, we introduce a phenomenological model of massive particles propagating in a Minkowski spacetime which arises from an underlying causal set. **The particles undergo a Lorentz invariant diffusion in phase space…**" The "swerves" term is defined in the paper ("We give one such model below which we call 'swerving', after Lucretius").
- **L. Philpott, F. Dowker, R.D. Sorkin, "Energy-momentum diffusion from spacetime discreteness", Phys. Rev. D 79, 124047 (2009), arXiv:0810.5591.** `verified-at-source`. Verbatim abstract: "At large scales, the microscopic swerves … manifest themselves as a **Lorentz invariant diffusion in energy-momentum** governed by a single phenomenological parameter… the most general Lorentz invariant diffusion equation for a massless particle … contains two phenomenological parameters describing, respectively, diffusion and drift in the particle's energy. **The particles do not leave the light cone however: their worldlines continue to be null geodesics.** Finally, we deduce bounds on the drift and diffusion constants for photons from the blackbody nature of the spectrum of the cosmic microwave background radiation." (Photon polarization effects too, per L. Philpott's thesis, arXiv:1009.1593, `verified-at-source`.)

**So: yes — it evades the modified-dispersion-relation signature (no preferred frame is predicted), and replaces it with a Lorentz-invariant stochastic signature (swerves) that is already constrained by CMB data. The dispersion relation itself is untouched; particle momentum diffuses instead.**

---

## 3. THE GENERAL OBSTRUCTION

**Is there a theorem?** Two distinct statements — an informal-but-clear argument (boosted-cell) and a formal theorem (BHS):

**(a) The boosted-cell / "no uniform regular lattice in Lorentzian signature" argument — Sorkin, arXiv:gr-qc/0309009 (lecture notes), verbatim** (`verified-at-source`):
> "To see what goes wrong, consider the 'diamond lattice' in M² consisting of all points with integer values of the null coordinates u = t − x and v = t + x. This would seem to be a uniform lattice, but **under a boost u → λu, v → v/λ it goes into a distribution that looks entirely different, with a very high density of points along the u=constant lines (say) and large empty spaces in between. In particular, our diamond lattice is far from Lorentz invariant**, which a truly uniform distribution should be — and which C(M) produced by a Poisson process actually is. Examples like this suggest strongly that, in contrast to the situation for Euclidean signature, **only a random sprinkling can be uniform for Lorentzian signature.**"

This is the precise form of the "boosted observer sees an infinitely contracted cell" argument (here: a boosted *regular* lattice acquires line-like density concentrations and voids, i.e., picks out the boost frame; no regular lattice is uniform under boosts). Henson's review (§1.5, read at source) states the general version: "For most discrete structures, local Lorentz invariance (LLI) is impossible to attain… This is the situation for lattice-like structures." And for spin foams/triangulations: "Spin-foams with Planck scale discreteness, like any lattice-like structure, are not Lorentz invariant: near one particular frame, a good [approximation only exists]…" (my extraction truncates mid-sentence; the point — existence of a preferred frame near a particular boost — is clear and matches 0605006's consequence).

**(b) The formal theorem: BHS (gr-qc/0605006, §2.2 above).** Its abstract itself states the graph no-go: "there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance." Henson's gloss (read at source): "Could other popular approaches to quantum gravity, based on graphs and triangulations, utilise sprinklings to incorporate Lorentz symmetry? The theorem mentioned above shows this to be impossible: if no direction can be associated to a sprinkling of Minkowski in a way consistent with Lorentz invariance, how could an entire finite valancy graph or triangulation?"

**(c) Beyond BHS, I found NO published no-go theorem titled/constructed as "no exactly-Lorentz-invariant dynamics on an arbitrary fixed graph."** The nearest statements in print are (a) and (b) plus Sorkin's nearest-neighbor argument (0703099, §2.3 above). `NOT VERIFIED AT SOURCE` for any stricter theorem; I searched for such a theorem and did not find one.

**(d) Related no-go that does exist (from the fermion side):** the **Nielsen–Ninomiya theorem** — mentioned by Griffin & Kieu §1 as the reason lattice fermion formulations modify one of {reflection positivity, locality, chiral symmetry, translational invariance-at-fixed-scale}; random lattices were an attempt to relax translational invariance. (Evidence: Griffin & Kieu, read at source.) It is about fermion doubling, not Lorentz invariance per se; included for completeness.

---

## 4. ANDERSEN, arXiv:1210.8348 (2012)

- **Record (`verified-at-source`):** Timothy D. Andersen, "Lorentz Covariant Lattice Gauge Theory", arXiv:1210.8348 [math-ph] (also quant-ph), submitted **31 Oct 2012**, 9 pages, 3 figures. No journal reference on the arXiv page.
- **Abstract verbatim:** "Lattice gauge theory's discretization of spacetime suffers from a drawback in that Lorentz covariance is lost because the axes of the lattice create preferred directions in spacetime. Smaller and smaller lattice spacings decrease the effect but fail to eliminate it completely. It has been argued recently that detecting such a set of preferred directions or similar constraints would indicate whether the universe itself has an underlying lattice, i.e. the digital universe hypothesis. In this paper, I demonstrate a technique for accomplishing lattice gauge theory simulations while maintaining **exact Lorentz covariance by replacing the lattice with a lattice graph such that the metric is defined as a discrete, Lorentz covariant matrix potential over the graph rather than a metric over an underlying manifold.** This technique eliminates the symmetry violation of standard lattice gauge theory…"
- **Peer review: NEVER published.** INSPIRE record 1197887: no `publication_info` (no journal), created 2012-11-02. Semantic Scholar: `venue` empty, `year` 2012. (NASA ADS abstract page was blocked by human-verification (HTTP 405) — ADS status `NOT VERIFIED`; INSPIRE + Semantic Scholar agree.)
- **Citations: ZERO, on both databases I could query.**
  - INSPIRE-HEP: `citation_count: 0`, `citation_count_without_self_citations: 0` (API, read at source); the citations query `refersto:recid:1197887` returns **total: 0** hits (API, read at source).
  - Semantic Scholar: API `citationCount: 0` for paperId 3bf8844bdedc3fa2f662f5f5b1147673bde03aed (read at source).
  - Google Scholar: count not directly verifiable via my tools (no reliable programmatic access) — `NOT VERIFIED` at Google Scholar, but given INSPIRE+S2=0 and the total absence of citing literature in searches, the honest statement is **"essentially zero citations."**
- **Rebuttals/builds:** none found. Searches for "1210.8348", the title, and the author's name in connection with the topic returned only the arXiv page and the Semantic Scholar record. `NOT VERIFIED` positive-negative: I cannot prove absence across all databases, but two indexes agree on zero, and no citing paper surfaced.
- **What else has Andersen published — a later, more developed version?** Via the arXiv API author query (all:"Timothy D. Andersen"), his arXiv output includes: 1306.6836 (An exact integration of a φ⁴ QFT, 2013), 1111.0598 / 1011.1535 / 1111.0659 / 0812.1508 / math-ph/0609071 / cond-mat/0601109 / math-ph/0611049 (plasma & vortex filament papers), 1005.4089 (SO(4,1) Yang-Mills theory of quantum gravity, 2010), 1404.4387 (absence of black hole event horizons; de Sitter Yang-Mills, 2014), 1807.01578 (Quantization of Fields by Averaging Classical Evolution Equations, 2018), 2009.04244 (A Dynamic Histories Interpretation of Quantum Theory, 2020), 2110.05180 (Chaotic deterministic quantization in a 5D general relativity, 2021). **No later, more developed version of the Lorentz-covariant-lattice-graph construction itself was found.** His later interests (deterministic/chaotic quantization, Yang-Mills gravity) overlap thematically but are not continuations of 1210.8348. (All record-existences `verified-at-source` via the arXiv API; the claim "no continuation" is a search-result inference — **Your own inference**, honest negative.)
- **Context found in the paper itself (read at source):** the construction removes the embedding manifold ("the edges and vertices have no position (or length) whatsoever") and dresses the graph with a discrete Lorentz-covariant metric potential; the continuum limit recovers Yang-Mills. This is precisely the kind of "intrinsic discrete structure" that the BHS theorem constrains for sprinkling-based structures; Andersen's paper does not cite BHS (its reference list, per INSPIRE metadata, cites Altarev et al. EPL 92 (2010) 51001, Beane-Davoudi-Savage 1210.1847, Bostrom 2003, DeGrand et al. — i.e., it engages the Beane/Savage "universe as numerical simulation" programme and the GZK-test discussion, not the causal-set literature). No engagement with causal-set theory or the BHS objection is visible.

---

## WHERE I STILL HAVEN'T BEEN (honest scope notes)

1. **Primary CFL 1982 papers themselves** — I could not read the paywalled Nucl. Phys. B texts; I verified their claims about ensemble restoration via Griffin & Kieu (1993) and Pavlovsky (2004) at source. The CFL papers' own wording on "rotational invariance" is `inherited-unchecked` via those secondary quotes.
2. **A canonical retrospective on why random lattices died** — not found; reasons assembled from Griffin & Kieu (verified) + inference.
3. **NASA ADS and Google Scholar citation counts for 1210.8348** — blocked/unavailable, though INSPIRE and Semantic Scholar independently give 0.
4. **The full text of BHS (gr-qc/0605006)** — I read the abstract (verbatim, loaded twice) and the introduction; the proof itself (compactness/measurable-map argument) was only skimmed. The statement of the theorem as quoted is exact; the proof is not reproduced here.
5. **Whether any "no-go for exact Lorentz-invariance on a fixed graph" theorem exists in the math-ph literature outside the causal-set circle** — I searched several query forms and found none; reported as honest negative above.

---

## REFERENCE LIST (all citations above, consolidated)

**Verified at source (abstract/record/full text read by me):**
1. N.H. Christ, R. Friedberg, T.D. Lee, "Random lattice field theory: General formulation," Nucl. Phys. B202 (1982) 89–125. (Verified via INSPIRE 11586 + 3 independent citing papers + ScienceDirect abstract.)
2. N.H. Christ, R. Friedberg, T.D. Lee, "Gauge theory on a random lattice," Nucl. Phys. B210[FS6] (1982) 310–336.
3. N.H. Christ, R. Friedberg, T.D. Lee, "Weights of links and plaquettes in a random lattice," Nucl. Phys. B210[FS6] (1982) 337–346.
4. C.J. Griffin, T.D. Kieu, "Fermionic field theory and gauge interactions on random lattices," arXiv:hep-lat/9307011 (1993). [full text read]
5. O.V. Pavlovsky, "Random Lattice QCD and chiral effective theories," arXiv:hep-lat/0407026 (2004). [full text read]
6. T. Gantumur, "Rotationally invariant dynamical lattice regulators for Euclidean quantum field theories," arXiv:2512.22072 (2026). [full text read]
7. L. Bombelli, J. Lee, D. Meyer, R.D. Sorkin, "Space-time as a causal set," Phys. Rev. Lett. 59 (1987) 521–524. [ADS/APS/INSPIRE records + abstract]
8. Comment on ref. 7, Phys. Rev. Lett. 60 (1988) 655. [record only; content unchecked]
9. L. Bombelli, J. Henson, R.D. Sorkin, "Discreteness without symmetry breaking: a theorem," arXiv:gr-qc/0605006; Mod. Phys. Lett. A 24 (2009) 2579–2587. [abstract + intro read]
10. J. Henson, "The causal set approach to quantum gravity," arXiv:gr-qc/0601121 (2006, in *Approaches to Quantum Gravity*, ed. D. Oriti, CUP). [full text read]
11. R.D. Sorkin, "Causal sets: Discrete gravity (notes for the Valdivia Summer School)," arXiv:gr-qc/0309009 (2003). [full text read]
12. R.D. Sorkin, "Does Locality Fail at Intermediate Length-Scales?," arXiv:gr-qc/0703099 (2007, in *Towards Quantum Gravity*, ed. D. Oriti, CUP). [full text read]
13. D.M.T. Benincasa, F. Dowker, "The Scalar Curvature of a Causal Set," Phys. Rev. Lett. 104 (2010) 181301; arXiv:1001.2725. [abstract read]
14. F. Dowker, L. Glaser, "Causal set d'Alembertians for various dimensions," Class. Quantum Grav. 30 (2013) 195016; arXiv:1305.2588. [abstract read]
15. S. Aslanbeigi, M. Saravani, R.D. Sorkin, "Generalized causal set d'Alembertians," JHEP 06 (2014) 024; arXiv:1403.1622. [abstracts read (ADS, ETDE)]
16. F. Dowker, J. Henson, R.D. Sorkin, "Quantum Gravity Phenomenology, Lorentz Invariance and Discreteness," Mod. Phys. Lett. A 19 (2004) 1829–1840; arXiv:gr-qc/0311055. [abstract read]
17. L. Philpott, F. Dowker, R.D. Sorkin, "Energy-momentum diffusion from spacetime discreteness," Phys. Rev. D 79 (2009) 124047; arXiv:0810.5591. [abstract read]
18. L. Philpott, "Causal Set Phenomenology," PhD thesis, arXiv:1009.1593 (2010). [abstract read]
19. T.D. Andersen, "Lorentz Covariant Lattice Gauge Theory," arXiv:1210.8348 [math-ph] (2012). [abstract + full HTML text read]
20. INSPIRE-HEP API records: 1197887 (Andersen: citation_count 0, no journal; citations query total 0), 11586, 21718, 187256. [read at source]
21. Semantic Scholar API: paper 3bf8844bdedc3fa2f662f5f5b1147673bde03aed → citationCount 0, venue empty. [read at source]
22. arXiv API author query "Timothy D. Andersen" → full publication list (14 records). [read at source]
23. Niels/nearest-neighbor context: Nielsen–Ninomiya no-go as cited in ref. 4. [secondary, via ref. 4]

**Inherited-unchecked (from secondary sources, not read by me):**
- The exact internal wording of CFL 1982 papers regarding rotational invariance (via refs. 4 and 5).
- Content of PRL 60, 655 (1988) Comment on Bombeilli et al.
- ADS/Google-Scholar numerical citation counts for 1210.8348 (blocked/unavailable; INSPIRE and S2 both give 0).

---

## EVIDENCE-CLASS SORT (per AGENTS.md discipline)

- **Established:** CFL trilogy references; ensemble-average-only restoration on random lattices; fermion-doubler revival under gauge interactions (Griffin & Kieu); BHS theorem statement and publication; Lorentz-invariant-but-nonlocal character of the causal-set d'Alembertian (Benincasa–Dowker abstract; Dowker–Glaser; Aslanbeigi–Saravani–Sorkin); swerves/momentum-diffusion mechanism (Dowker–Henson–Sorkin 2004; Philpott–Dowker–Sorkin 2009); nonlocality scale above discreteness scale (Henson review); BHS: no preferred frame AND no finite-valency graph on a sprinkling; Andersen paper never published and 0 citations (INSPIRE + Semantic Scholar).
- **Serious speculation:** Sorkin's claim that nonlocality persists to intermediate scales possibly involving an infrared scale ~ age of the cosmos (0703099) — argued, not proven; the causal-set d'Alembertian's convergence/"miraculous cancellations" programme still being developed.
- **Anomaly:** none.
- **Anecdote:** the "digital universe" motivation thread in the Andersen paper (Beane–Davoudi–Savage GZK argument) — relevant context, not evidence for/against any construction.
- **Your own inference:** the synthesis that (i) the causal set is the only *proven* exactly-Lorentz-invariant finite-discreteness structure, (ii) its price is nonlocality + swerves, (iii) no later version of Andersen's construction exists, (iv) random lattices were abandoned for a combination of the documented problems rather than any single published verdict.

**End of thread report.**