# Adversarial review: cubic anisotropy result

Reviewed files, fully read: `RESULT.md`, `auger_lib.py`, `cubic.py`, `run_control.py`, `run_cubic.py` in `lab/2026-09-08-auger-anisotropy/`.

New check files written in the lab directory:

- `adversary_check_ra_systematics.py`
- `adversary_check_sensitivity.py`
- `adversary_check_limits.py`

## Top 5 objections, ranked by damage

### 1. The quoted "95% upper limits" are not upper limits

Damage: high. This is the biggest statistical overclaim.

`RESULT.md` labels the 95th percentile of the null distribution of `max|S|` as a "95% upper limit on cubic ell=4 amplitude". That is not a confidence interval construction for amplitude. It is a null critical value / sensitivity scale. A real upper limit must specify the signal orientation, amplitude sign/domain, ordering rule, and probability statement under injected nonzero signals.

Quantitative check from `adversary_check_limits.py`, using the actual observed `S(R)` values and orientation-specific Gaussian response:

```text
E>=8 EeV N=3122 reported_null95_as_UL=0.0605
observed max|S|=0.0468
orientation-wise approximate amplitude ULs, using actual S(R):
  local-known-R  min p05 median p95 max = 0.0000 0.0000 0.0320 0.0650 0.0916
  scan-penalized min p05 median p95 max = 0.0154 0.0286 0.0670 0.1020 0.1413
  Argus R0: Sobs=-0.0083 var_g=1.0017 local_UL=0.0211 scan_UL=0.0541

E>=40 EeV N=87 reported_null95_as_UL=0.3581
observed max|S|=0.2158
orientation-wise approximate amplitude ULs, using actual S(R):
  local-known-R  min p05 median p95 max = 0.0000 0.0249 0.1911 0.3439 0.4443
  scan-penalized min p05 median p95 max = 0.1540 0.2317 0.4053 0.5765 0.6807
  Argus R0: Sobs=-0.0454 var_g=0.9910 local_UL=0.1314 scan_UL=0.3301
```

This diagnostic is not a full Neyman belt either, but it is enough to prove the table label is wrong. The reported numbers should be renamed "null 95% max-statistic critical values" or replaced by injection-calibrated confidence intervals.

Evidence class: Established by code check.

### 2. The sensitivity number is arithmetically right but not a robust experiment sensitivity

Damage: high. The headline `373,895 events = 6,275 years` is too singular.

The arithmetic itself checks out:

```text
resp/sd = 0.947888
z_thr(N_eff=42) = 5.795204
N_1pct = 373786.0
rate = 59.59/yr
years = 6272.7
```

So the published `373,895` and `6,275 years` numbers are just rounding. But the response coefficient was measured for one injected orientation, `R0 = Rs[0]`, with a vertical-only exposure draw. The real statistic response depends on cube orientation and on the declination distribution after the energy cut. Using the observed declinations, `adversary_check_sensitivity.py` gives:

```text
E>=40 EeV N=87, single-orientation small-A significance coefficient sqrt(E[g^2]):
  min p05 median p95 max = 0.6443 0.7790 0.9547 0.9960 1.0000
  Argus injected coefficient = 0.9479
```

For an unfavorable 5th-percentile orientation at `E>=40`, the event requirement for a 1% signal inflates by `(0.9479/0.7790)^2 = 1.48`, i.e. about `5.5e5` events and `9.3e3` full-Auger years. For the worst sampled orientation, the inflation is `(0.9479/0.6443)^2 = 2.16`, i.e. about `8.1e5` events and `1.36e4` years. The median orientation is close to Argus's number, but the report does not say "median orientation".

The trials factor is also understated if the local statistics are normalized per orientation. A covariance simulation of the exact 3,000 scanned templates gives `N_eff ~= 108`, not 42:

```text
E>=40 EeV: q95(max|S|)=0.3582 q95(max|z_j|)=3.494 N_eff=107.6
```

This only raises the global 5-sigma threshold mildly, from about `5.80` to about `5.92`, and the event count by about 4%. It does not save or kill the headline. The bigger damage is orientation dependence.

The rate claim is basically defensible. The open data are 10% of selected Auger events, and `87 / 0.1 / 14.6 = 59.6` events/year above 40 EeV. Auger's public spectrum page quotes `60,400 km^2 sr yr` exposure for the 14+ year SD spectrum and the open-data portal says the public data are 10% of events passing high-level selections. Sources: https://opendata.auger.org/data.php and https://www.auger.org/news/scientific-highlights/261-most-precise-measurement-of-the-cosmic-ray-energy-spectrum-at-ultra-high-energies

Evidence class: Established for arithmetic and response distribution; inference for how to present sensitivity.

### 3. Exposure immunity is exact only under RA-uniform exposure; realistic RA systematics are small for current limits but matter for the 1% dream

Damage: medium. The proof survives in its stated mathematical domain, but the prose overstates the domain.

The proof is correct if the null density factorizes as

```text
p(alpha, delta) = p(delta) / (2 pi).
```

For any trial orientation `R`, define

```text
g_R(alpha, delta) = K4(R n(alpha, delta)) - average_alpha K4(R n(alpha, delta)).
```

Then

```text
E[g_R] = integral ddelta p(delta) integral dalpha/(2pi) g_R(alpha, delta) = 0.
```

No declination-only exposure residual can bias the statistic. I could not break that proof.

But the proof dies immediately when exposure has RA structure:

```text
p(alpha, delta) proportional to p(delta) [1 + eps cos(m(alpha - phi))].
```

`adversary_check_ra_systematics.py` computes the first-order leakage coefficient. Worst cases:

```text
cut  harm  leak_per_eps  eps_for_reported_null95
  8     4      0.389147       0.155
 16     4      0.378784       0.314
 32     3      0.363250       0.736
 40     4      0.372765       0.961
```

A 15.5% fourth-harmonic RA exposure modulation can fake the `E>=8 EeV` reported null-95 level. At `E>=40 EeV`, it would take almost order-unity fourth-harmonic modulation to fake the present 35.8% scale. So the current null is not practically broken by known Auger-level RA systematics.

But for a 1% cubic target, the same numbers are dangerous: an `m=4` RA exposure modulation of only `0.01 / 0.389 = 2.6%` can mimic a 1% cubic signal at `E>=8`; a published Auger early-analysis paper says detector/weather/growth effects can produce few-percent short-period RA modulations, nearly vanish over full multi-year data, and leave about `0.4%` systematic uncertainty in the predicted RA modulation for a two-year period. Source: Pierre Auger Collaboration, "Search for large-scale anisotropies with the Auger Observatory," arXiv:0706.2640, especially sections 2-4: https://arxiv.org/html/0706.2640

Energy thresholds make this worse in principle. Above the suppression, the integral rate scales roughly as `E_th^(1-gamma)` with `gamma` around 4-5, so a time/RA-correlated 1% energy-scale drift produces about a 3-4% selected-event rate modulation. With `m=4` leakage coefficient 0.39, that is a possible fake `S ~ 0.012-0.016`, irrelevant to the present 36% `E>=40` noise scale but comparable to a future 1% claim. Auger's spectrum analysis emphasizes the energy calibration chain and the steep suppression; source: https://arxiv.org/abs/2008.06486

Evidence class: Established for the proof and leakage simulation; serious speculation for the energy-scale/time coupling until Auger's time-dependent exposure and calibration metadata are used.

### 4. The RA-scrambling null is too narrow if real exposure is not RA-uniform

Damage: medium. This does not create the current null, but it invalidates the strongest language.

The null in `run_cubic.py` keeps declination and redraws RA uniformly. That is the right conditional null only if the exposure is RA-uniform after all cuts. If Auger has residual RA exposure modulation, the scramble erases it and centers the null at zero. Then a detector effect appears as signal, not as nuisance variance or bias.

The released data themselves show noisy high-energy RA harmonics:

```text
E>=8  EeV: h1 r=0.026 p=0.587; h2 r=0.062 p=0.049; h4 r=0.035 p=0.379
E>=16 EeV: h2 r=0.152 p=0.010
E>=40 EeV: h1 r=0.409 p=0.026; h4 r=0.242 p=0.280
```

These are mostly small-N fluctuations, not proof of exposure bias. But they show the problem: the RA-scrambled null has no parameter for residual exposure harmonics, so it cannot answer "what if the real exposure has a 0.4% or 2% RA modulation?" A robust null should either use Auger's time-dependent exposure/livetime model or inject allowed RA modulation nuisance terms and marginalize/profile them.

Evidence class: Established for code behavior and data harmonics; inference for statistical remedy.

### 5. The pipeline is not validated by the positive control; C1/C2 only validate narrower pieces

Damage: medium-low for the null, high for confidence rhetoric.

Argus correctly admits C4 did not fire. I agree with the power calculation: with `N=3122` at `E>=8 EeV`, a true 4.7% first-harmonic RA signal is only a `1.31 sigma` expectation, so the positive control had little chance. But that means it is not a validation gate.

C1 is a good coordinate consistency check. I attacked the azimuth convention directly:

```text
90-phi   lat= -35.2136 norm=0.999915 resid=1.65071e-03
phi      lat= -90.0235 norm=0.578051 resid=3.76704e-01
-phi     lat= -90.0235 norm=0.578051 resid=3.76704e-01
phi-90   lat= -35.2136 norm=0.999915 resid=1.65071e-03
90+phi   lat=-144.7864 norm=0.999915 resid=1.65071e-03
180-phi  lat= -89.9765 norm=0.578051 resid=3.76704e-01
```

The convention is effectively validated up to the expected cosine degeneracy.

C2 is also mostly fine as a vertical declination exposure check. `omega_raw` includes the missing-looking projection factor: it is the hour-angle integral of `cos(theta)`, giving Sommers' formula. The absent factor of 2 is only normalization. Source: Sommers 2001, Astropart. Phys. 14, 271, arXiv:astro-ph/0004016: https://arxiv.org/abs/astro-ph/0004016

But C2 does not validate the final immune statistic's response, the trials treatment, the RA uniformity after energy cuts, or the ability to recover a known anisotropy. The inclined-band geometric difference is mathematically valid for a pure `cos(theta)` aperture because theta is monotonic in absolute hour angle at fixed declination, but real inclined acceptance is not just geometric; the open-data portal gives separate full-efficiency thresholds for vertical and inclined events and quality cuts. Source: https://opendata.auger.org/data.php

Evidence class: Established for convention and Sommers checks; inference for validation sufficiency.

## Other attacks and results

- I did not find a monomial-expansion or sign bug in `cubic.py`. The constant offset is handled correctly: raw `K4` needs `-0.6/K4_SD`, while the immune feature has already subtracted the RA-averaged monomials so the constant cancels.
- The `m=5` RA exposure leakage check is zero at first order, as expected for a quartic statistic. That is a useful sanity check on the leakage script.
- The injection linear-response model is valid at small `A`. At larger `A`, normalization introduces a denominator `1 + A <K4>_exposure`, but for the Argus `R0` orientation on observed `E>=40` declinations `kbar=-0.00923`, so the coefficient changes only from `0.99556` at `A=0.01` to `0.99823` at `A=0.30`. I could not break the result through large-A nonlinearity at the injected `A=0.30`.
- The 1,568 rows dropped by `load_events` all fail SD energy and SD direction parsing in the SD1500 file; they do have GPS times but no usable RA/dec. I cannot quantify their RA distribution from the released summary because RA is absent for exactly those rows.

## Prior art / novelty attack

The exact phrase "subtract the RA-average of the cubic harmonic" may still be Argus's implementation choice, but the novelty framing should be downgraded hard. The method is a projection of the template against the nuisance space of arbitrary declination-only exposure functions. That is standard in spirit and adjacent literature, not a new conceptual category.

Relevant prior art:

- Sommers 2001 formulates anisotropy reconstruction with non-uniform celestial exposure and spherical harmonics for full-sky coverage: https://arxiv.org/abs/astro-ph/0004016
- Aublin & Parizot 2005 generalize 3D dipole reconstruction to partial-sky exposure and explicitly analyze reconstruction power as a function of orientation: https://arxiv.org/abs/astro-ph/0504575
- Roulet & Mollerach 2005 propose a method for dipole/quadrupole with partial/inhomegeneous exposure; the abstract says the simplest version assumes exposure depends only on declination and can be extended to small RA modulation. The text explicitly states Rayleigh analysis assumes RA-uniform exposure and may require data selection or generalization for RA nonuniformity: https://arxiv.org/html/astro-ph/0504630
- Billoir & Deligny 2008 study multipolar coefficients with non-uniform or partial sky coverage and build covariance estimates seen through the exposure function, including model tests without assuming a finite `L`: https://arxiv.org/abs/0710.2290
- The East-West method is explicitly sold as an exposure-independent method for large-scale anisotropies, cancelling direction-independent detector instabilities to first order. It is not the same estimator, and it is mainly a dipole/RA-harmonic method, but it is the obvious ancestor of "remove exposure by differencing/projecting in RA." Bonino et al., ApJ 738:67, DOI `10.1088/0004-637X/738/1/67`; ADS page: https://ui.adsabs.harvard.edu/abs/2011ApJ...738...67B/abstract
- Auger's own early large-scale anisotropy paper uses Fourier/Rayleigh and East-West methods, models array growth/dead time/weather, and quotes RA modulation systematics at the sub-percent level over long integrations: https://arxiv.org/html/0706.2640

Verdict on novelty: not established. At most this is a neat cubic-template matched filter projected orthogonal to declination-only exposure nuisance functions. Do not call it novel without a much more precise literature search in statistical template projection and cosmic-ray multipole estimators.

## Bottom line

I could not break the narrow null result: with present 10% open data, the cubic statistic is unbiased against arbitrary declination-only exposure, the observed scan is null, and known RA systematics are far too small to fake the quoted present noise scale.

I did break three stronger claims:

1. the table's "95% upper limits" are not actual amplitude upper limits;
2. the `373,895` event sensitivity is a median/favorable-orientation statement, not a robust all-orientation sensitivity;
3. "exposure immune" must be rewritten as "immune to declination-only exposure, conditional on RA uniformity after all energy/time cuts."