No cubic pattern in public Auger cosmic-ray directions, at sensitivity that sees only large ones

If space were a fine cubic grid, the highest-energy cosmic rays could arrive in a pattern aligned with its axes, a signature Beane, Davoudi and Savage proposed in 2012. We searched the public Pierre Auger Observatory data for such a pattern with a method fixed in advance. The data are the 25,086 main-array events in the public 10 percent sample and the catalogue of the highest-energy events, cut at 8, 16, 32, 40 and 78 EeV. For each cut we fitted a cube-symmetric pattern over 400 orientations and compared the best fit with skies whose right ascensions were scrambled, which keeps the detector's exposure. No cut shows a cubic pattern: raw p values run from 0.19 to 0.85, and all five read 0.85 after the Benjamini-Hochberg correction.c1 The search could only have seen a strong pattern: at 8 EeV it reaches power 0.9 at a planted strength of 0.2, and at 32 EeV it reaches power 0.97 only at strength 1.0, where the sky along the cube axes is twice the average.c2 The highest-energy event, 166 EeV, bounds the lattice spacing below 3.7e-27 m, an inverse spacing of 5.3e10 GeV, which is 0.53 of the 1e11 GeV bound of Beane and colleagues.c3 This is a null whose value lies in stating how weak it is.

Lattice calculations in particle physics put space and time on a grid. Beane, Davoudi and Savage [2] asked what an observer inside such a calculation could detect if the universe itself were computed on a grid with some lattice spacingDefinition: lattice spacing lattice field theory The distance between neighbouring points of a grid on which space is modelled. On a grid, waves shorter than about twice the spacing cannot exist and the speed of a wave depends on its direction. Excludes the Planck length, which is a scale built from constants and not a measured grid. full definition. Their abstract reports that the most stringent bound on the inverse spacing, about 1e11 GeV, comes from the high-energy end of the cosmic-ray spectrum, and their published account predicts that the highest-energy cosmic rays would show the cube symmetry of the grid in their arrival directions.

The particles that could carry that signature are the ultra-high-energy cosmic rayDefinition: ultra-high-energy cosmic ray astroparticle physics A charged particle from space, a proton or heavier nucleus, arriving with an energy above about 1e18 electronvolts. It is recorded through the shower of secondary particles it makes in the atmosphere. Excludes photons and neutrinos, and the far more common cosmic rays of lower energy. full definitions, with energies of many EeVDefinition: EeV physics (units) An exaelectronvolt, 1e18 electronvolts, or 1e9 GeV, carried by a single particle. Excludes the total energy of the shower in a detector, which is spread over many secondary particles. full definition. The Pierre Auger Observatory records them with an array of ground detectors. The Auger Collaboration reported in 2017 a large-scale anisotropy above 8 EeV, a dipole of about 6.5 percent amplitude in 3e4 events, which it reads as evidence that these particles come from outside our galaxy [3]. Since 2021 the collaboration has published 10 percent of its cosmic-ray events, with a catalogue of the highest-energy events added in 2023 [1].

We asked whether the public arrival directions carry a cubic pattern, and what the most energetic event says about the spacing. The decision it informed was how to describe public cosmic-ray data in any work of ours on discretised space: as evidence, or as a weak test. Our searches found the proposal above and the Auger anisotropy results and no published cubic search on the open data; that is what the searches returned, and it does not show that none exists.

The preregistrationDefinition: preregistration research method (metascience) A dated written plan for a study, fixed and hash-locked before any result is seen: the question, the data, the statistic, the controls, the forecasts and the kill conditions. Excludes a plan revised after looking at the outcome, and a protocol written only for the record after the run. full definition and the forecast file were written and hash-locked on 31 October 2026, before the run. They name the baseline (an isotropic sky at the data's declinations), the planted answer (a cubic pattern at random orientation), the controlDefinition: control statistics and experimental design A version of the data with the structure under test removed (for example shuffled or pooled), scored the same way as the real data, so the real score has something to beat. Excludes a comparison with a different dataset or with a published number. full definition (right-ascension scrambles), the outlier checks (the top 3 events and the region around Centaurus A, with and without), the assumption changed (the energy cut and the Centaurus A exclusion), and the kill condition.

The kill conditionDefinition: kill condition research method (our usage, after Popper's falsification) A rule written in the preregistration that, when met, decides against the claim with no room for judgement after the run. Excludes reasons found after the run for setting a result aside. full definition, fixed in advance: if the cubic search gives a corrected p below 0.01, the lattice reading must still survive the Centaurus A exclusion and a second energy cut, or it dies. Each forecast and condition became a rule set whose pass or fail is a verdictDefinition: verdict research method (our usage) The pass or fail of one preregistered rule set against the run's numbers, read off by code. Excludes an interpretation written after reading the numbers. full definition read off by code.

The data are the Pierre Auger Observatory Open Data release of 2024 [1], Zenodo record 10488964, licensed CC BY-SA 4.0, fetched on 31 October 2026. We read the event summaries of the main surface array, vertical and inclined showers together with duplicates removed, which gave 25,086 events with direction and energy. This public sample is 10 percent of the recorded events. We also read the catalogue of the highest-energy surface-array events, 109 events with a top energy of 166 EeV. We report statistics only.

The cuts at 8, 16 and 32 EeV use the event summaries, and the cuts at 40 and 78 EeV use the catalogue. Table 1 gives the count at each cut.

For an arrival direction written as a unit vector with components $x$, $y$, $z$, the cubic harmonicDefinition: cubic harmonic mathematics (spherical harmonics) and crystallography The simplest pattern on a sphere with the symmetry of a cube: here x^4 + y^4 + z^4 minus its average, largest along the six cube axes and smallest along the eight diagonals. Excludes a dipole, a single hot spot, or any pattern that a rotation of the cube by a right angle would change. full definition is

The constant 0.6 is its average over the sphere, so $C$ averages to zero on an isotropic sky. It is largest, 0.4, along the six axes of the cube and smallest along the eight diagonals. A grid in space need not line up with our coordinates, so the pattern was rotated through 400 random orientations $R$.

A ground array does not watch the whole sky evenly: its exposureDefinition: exposure astroparticle physics How much of each part of the sky a detector watched over time. A ground array sees some declinations far more than others, so an isotropic sky still looks uneven in its data. Excludes the number of events, which is exposure times the true flux. full definition depends strongly on declination. The baseline sky was therefore built from 60,000 directions with uniform right ascension and declinations drawn from the data. For orientation $R$, let ${\mu }_R$ and $s_R$ be the mean and standard deviation of $C$ on this baseline, and $m_R$ its mean over the $N$ real events. The standardised excess is

and the statistic $T$ is the largest $Z_R$ over the 400 orientations. For cuts with 500 events or fewer, the baseline declinations were drawn from the 8 EeV events.

Each p value comes from a right-ascension scrambleDefinition: right-ascension scramble astroparticle physics (anisotropy searches) A control sky made by keeping each event's declination and giving it a random right ascension. The Earth's rotation makes a ground array's exposure nearly even in right ascension, so the scramble keeps the exposure and removes any pattern. Excludes a fully random sky, which would ignore the detector's exposure. full definition: each event keeps its declination and gets a random right ascension, which keeps the exposure and removes any pattern. The statistic was recomputed on 300 scrambled skies, and

with $k$ the number of scrambles at or above the real $T$ and $n$ the number of scrambles. The five p values were corrected with the Benjamini-Hochberg correctionDefinition: Benjamini-Hochberg correction statistics (multiple testing) A rule that raises a set of p values to control the expected share of false discoveries among the tests called significant. The smallest raw p is multiplied by the number of tests and larger ones by less. Excludes an uncorrected p value read on its own when several tests were run. full definition [4]: with the raw values sorted so that $p_i$ is the $i$th smallest of $M$ = 5,

and each corrected value is the smallest $q_j$ at its own rank or above.

A planted sky keeps the real events' declinations and draws right ascensions from a density proportional to

at one random orientation, so that the sky along the cube axes is $1+ϵ$ times the average. The detection powerDefinition: detection power statistics The share of planted skies, each with a pattern of known strength, in which the search finds the pattern at its preregistered threshold. Excludes the chance that a pattern exists in the real sky. full definition at strength $ϵ$ is the share of planted skies whose statistic exceeds the 95th percentile of scrambled skies. At 32 EeV the threshold came from 200 scrambles and each strength had 30 planted skies; at 8 EeV, 60 scrambles and 10 planted skies.

On a grid of spacing $a$, a particle cannot have a wavelength much shorter than the grid. In the simple picture used here, a particle of energy $E_{\mathrm{top}}$ could not propagate unless

where $Q$ = 1.973e-16 GeV m is the reduced Planck constant times the speed of light. The inverse spacing in energy units is $E_{\mathrm{top}}/\pi$.

The right-ascension scramble is the null for every p value. As a separate control at 32 EeV, 200 scrambled skies gave a mean statistic of 2.32 and a 95th percentile of 3.27; the real statistic, 1.69, sat well inside them (p value from drawsDefinition: p value (from draws) statistics The share of control draws scoring at or above the real score, with one added to the count and to the number of draws, so it is never zero. Excludes a p value from a textbook formula that assumes a distribution. full definition 0.87).

The planted checkDefinition: planted check research method; called a positive control in experimental science Data built so that the right answer is known, run through the same analysis before the real data, to show that the analysis can see the effect when it is there. Excludes any check made on the real data, which cannot show what the analysis misses. full definition measured the search's power at each strength. Figure 3 and Table 2 give the result. The planted powers are coarse: each is a share of 30 or of 10 trials.

The outlier checks removed the 28 events at 32 EeV within 27 degrees of Centaurus A, and separately the 3 highest-energy events, and recomputed the p value with 200 scrambles each.

No cut shows a cubic pattern: raw p values run from 0.19 to 0.85, and all five read 0.85 after the Benjamini-Hochberg correction.c1 Table 1 and Figure 2 give the values.

Removing the 28 events near Centaurus A or the 3 highest-energy events leaves no pattern at 32 EeV: p 0.38 and 0.71.c4

The search could only have seen a strong pattern: at 8 EeV it reaches power 0.9 at a planted strength of 0.2, and at 32 EeV it reaches power 0.97 only at strength 1.0, where the sky along the cube axes is twice the average.c2

The highest-energy event, 166 EeV, bounds the lattice spacing below 3.7e-27 m, an inverse spacing of 5.3e10 GeV, which is 0.53 of the 1e11 GeV bound of Beane and colleagues.c3 The 1e11 GeV figure was entered from memory before the run; the abstract of Beane and colleagues [2], read for this paper, gives the same value.

The study also ran a toy model of waves on a cubic grid, to check the dispersion a lattice would cause. Along an axis, the phase speed falls short of the continuum value by $(\mathrm{ka}{)}^2/24$, with $k$ the wavenumber, within 0.8 percent for $\mathrm{ka}$ up to 0.8. The group speed falls short by $(\mathrm{ka}{)}^2/8$, within 1.3 percent; our forecast had given it the phase speed's coefficient, and that forecast missed. Along a diagonal, waves run faster than along an axis by $(\mathrm{ka}{)}^2/48$, within 1.2 percent, and the model sets the highest energy along a diagonal at $\sqrt{3}$, 1.73, times that along an axis, from the grid's geometry. This direction dependence is what would imprint a cube on arrival directions.

Table 3 sets the forecasts written before the run against the outcomes.

The event forecast was made from memory of the full dataset; the public release holds a tenth of it.

The search found nothing, and the planted skies say how little that means. At 32 EeV, the highest cut with event-level data, only 155 events are public, and the search needs a pattern that doubles the flux along the cube axes before it sees one reliably. At 8 EeV the sample is twenty times larger and the threshold falls to a 20 percent modulation, but at that energy the particles are bent strongly by galactic magnetic fields, which would smear a pattern made at the source.

For scale, the dipole the Auger Collaboration measured above 8 EeV with its full data has an amplitude of about 6.5 percent [3]. A cubic pattern of that size would be far below what our search can see in the public share. The kinematic reading lands within a factor of 2 of the published bound because both rest on the same top energy and the same simple argument; it adds a check of arithmetic and no new information.

The result therefore excludes a large cubic imprint on the public arrival directions and nothing smaller. The same method on the full Auger data would set a stronger limit.