# IceTracks-DR2 IC86_XI: reconstructed energy and estimated directional uncertainty

Higher reconstructed muon energy is associated with smaller estimated angular uncertainty overall, but this is not a universal rule within sky directions. The relationship reverses over much of the southern sky and turns upward in the highest energy quantiles. These data do not establish that higher true-energy neutrinos have smaller measured directional errors.

## Data and definitions

Source: supplied IceTracks-DR2 version 3.1 JSON, containing DR2_readme.txt, IC86_XI_events.csv and IC86_XI_uptime.csv. [Official dataset, version 3.1](https://dataverse.harvard.edu/dataset.xhtml?persistentId=doi:10.7910/DVN/MMIIZA&version=3.1), DOI 10.7910/DVN/MMIIZA. Analysis uses the attachment, not a fresh download. All three embedded text contents match their supplied SHA-256 hashes.

There are 127,695 events, spanning 2021-05-03 21:18:21.817 to 2022-05-23 19:46:15.813 UTC. There are 77,844 events in 2021 and 49,851 in 2022; these are partial calendar years within one detector season. MJD was interpreted as UTC as specified in the README. No missing or nonfinite numeric fields and no duplicated (run,event,subevent) keys were found. Every event lies inside a supplied uptime interval. The 1,218 intervals sum to 356.555219 days; the event table contains 1,190 distinct runs. No events were discarded in the primary analysis.

The two variables are:
- log10(E/GeV): reconstructed energy of the muon passing through the detector, not the incident neutrino's true energy.
- AngErr[deg]: estimated directional uncertainty, not the measured angular separation from a known true direction. The README describes a symmetric angular uncertainty calibrated in simulations for an E^-2 flux and explicitly specifies a 0.2° lower limit. Its numeric value should not be relabeled as a 68% or 90% containment radius without a documented conversion.

No true event energies or true directions are provided. No independent per-event quality flag is present. Run/event/subevent are identifiers; uptime is a good-data-taking list, not an independent event-quality score. Counterexamples below are descriptive, not newly invented quality classes.

## Methods

Spearman rho uses average ranks for tied values. The primary 95% intervals are percentile paired-event bootstrap intervals from 1,000 resamples, recomputing ranks each time. For efficiency, observed (energy,error) pairs are compressed into counts and multinomially resampled; this is equivalent to sampling paired events with replacement. Each point estimate was checked against scipy.stats.spearmanr.

An additional 1,000-replicate run-cluster bootstrap resamples the 1,190 runs with replacement, retaining their events and recomputing weighted midranks. This probes within-run dependence. Neither bootstrap includes detector calibration systematics or uncertainty about true neutrino properties. Intervals are pointwise, not simultaneous multiple-comparison intervals. Fixed random seed: 20261008.

Each quantile panel divides its own population into approximately equal-count reconstructed-energy deciles using pandas.qcut. Ties remain together, so counts differ. Diamonds show the median angular uncertainty; dark and light bands show the 25th–75th and 10th–90th percentiles. These are event-distribution bands, not confidence intervals. The x coordinate is the bin's median log-energy. Lines connect summaries for visibility and are not a fitted response model. Bins are recomputed within each panel; comparisons therefore concern conditional patterns, not identical energy-bin populations. Full bin bounds and counts are in quantiles.csv.

Sky strata use reconstructed J2000 declination bands [-90,-30), [-30,0), [0,30), [30,90) degrees and four 90° right-ascension sectors. These axes are assessed separately; right-ascension sectors still mix declinations. All four declination bands were also assessed separately within each year. A finer 10° declination check assesses dependence on the broad band boundaries.

## Results

| Population | Events | Spearman rho | 95% event-bootstrap interval |
|---|---:|---:|---|
| All | 127,695 | -0.5303 | [-0.5345, -0.5259] |
| 2021 | 77,844 | -0.5348 | [-0.5399, -0.5292] |
| 2022 | 49,851 | -0.5233 | [-0.5300, -0.5167] |
| Declination -90° to -30° | 23,236 | +0.2244 | [+0.2108, +0.2376] |
| Declination -30° to 0° | 21,987 | -0.3677 | [-0.3807, -0.3559] |
| Declination 0° to 30° | 50,372 | -0.2031 | [-0.2116, -0.1941] |
| Declination 30° to 90° | 32,100 | -0.1637 | [-0.1744, -0.1525] |

The overall run-cluster interval is [-0.5346,-0.5256], closely agreeing with the event bootstrap.

The median estimated uncertainty falls from 1.10° in the lowest energy bin (log-energy 1.38–2.79) to 0.20° in the ninth bin (4.68–5.05), then rises to 0.26° in the highest bin (5.06–7.20). The 90th percentile similarly rises from 0.49° in the ninth bin to 1.16° in the highest. Thus even the pooled relationship is not monotonically decreasing throughout its range.

All four right-ascension sectors have similar negative correlations: -0.5299, -0.5262, -0.5316 and -0.5334 in ascending sector order. Their individual intervals are in correlations.csv.

The southernmost broad band's positive association persists in both years: +0.2247 in 2021 and +0.2237 in 2022, with intervals entirely positive. All three other broad declination bands retain negative correlations in each year. This does not imply identical year-specific distributions or formally establish year equivalence.

The finer check is especially informative: every 10° declination band from -90° through -10° has positive rho; from -10° northward, rho is negative. All corresponding pointwise intervals exclude zero. The broad [-30°,0°) band pools two positive sub-bands with the negative [-10°,0°) band and has a negative aggregate correlation. Directional mixing can therefore obscure or reverse within-direction patterns. The overall rho should not be interpreted as a direction-independent energy effect. The released sample and detector response are direction dependent; these descriptive data alone do not identify a causal mechanism.

## Floor and rounding sensitivity

Both energy indicator and angular uncertainty lie on 0.01 grids (within floating-point precision). Average ranks handle these reported ties directly.

There are 18,818 events at exactly 0.20°, or 14.74%. Their fraction varies sharply by declination: 43.47%, 31.50%, 2.18%, and 2.16% in ascending broad-band order. It also varies by energy, reaching 50.06% in the ninth pooled energy bin.

Removing floor-valued events leaves 108,877 events and rho=-0.3768, 95% CI [-0.3821,-0.3713]. Removing values at 0.21° as well leaves 106,935 events and rho=-0.3582, CI [-0.3640,-0.3528]. This weakens but does not remove the overall negative association. These are selected-subset sensitivities, not unbiased estimates of an unobserved uncapped relationship.

Within the southernmost band, removing floor events still gives rho=+0.2129, CI [+0.1970,+0.2293]. The other broad bands remain negative. The directional reversal is therefore not solely a consequence of floor ties.

For 200 independent perturbations, add uniform jitter within ±0.005 to the log-energy indicator and to non-floor angular errors, retaining all 0.20° errors as tied. Overall rho ranges only from -0.530290 to -0.530139. This is a sensitivity experiment under a rounding-cell assumption, not a reconstruction of unrounded values and not another confidence interval. The floor is not ordinary rounding: values below it cannot be recovered. We do not assign invented sub-floor uncertainties or claim the latent unfloored correlation is known.

## Counterexamples

These examples were chosen transparently: the three largest angular uncertainties among events at or above the empirical 90th energy percentile (log-energy >=5.05), and the three smallest uncertainties among events at or below the 10th percentile (<=2.79). Inclusive thresholds retain ties and are not exactly the same as qcut bin membership. Tied minimum-error examples follow catalogue order.

| Run / event / subevent | log10(reconstructed muon E/GeV) | Estimated uncertainty | Declination |
|---|---:|---:|---:|
| 136407 / 18743252 / 0 | 5.87 | 5.13° | -58.723° |
| 135365 / 24677728 / 0 | 6.46 | 5.10° | -56.204° |
| 136517 / 38306686 / 0 | 6.21 | 5.04° | -26.909° |
| 135349 / 12162625 / 0 | 2.79 | 0.20° | +8.085° |
| 135352 / 47825884 / 2 | 2.70 | 0.20° | +77.213° |
| 135363 / 22325111 / 0 | 2.78 | 0.20° | +31.790° |

For scale, log-energy 5.87 corresponds to approximately 741 TeV reconstructed muon energy, whereas 2.70 corresponds to approximately 501 GeV. These are not true neutrino energies. Among events meeting the high-energy threshold, 12.05% have estimated uncertainty greater than 1°. Counterexamples are therefore not restricted to the three displayed extreme cases. A reported 0.20° value indicates the floor; it does not prove an actual directional error of 0.20° or less.

## Delivered files and reproducibility

- energy_year_quantiles.png: pooled and calendar-year quantiles.
- sky_quantiles.png: declination and right-ascension quantiles.
- icetracks_results.zip: correlations.csv (including year-by-declination results), quantiles.csv, floor_sensitivity.csv, fine_declination.csv, counterexamples.csv, and analysis_metadata.json.
- icetracks_analysis.py: standalone analysis using the attachment as its single input. Run `python icetracks_analysis.py path/to/attached.json` in a writable output directory. Requires NumPy, pandas, SciPy and Matplotlib. It reproduces the calculations and plots, using standard Matplotlib styling in place of the workbench figure-style helper.

Input JSON SHA-256: e2da4674800865b8408e919c9e0f1291302efa3b5578903a7b1faad4da93a259. Computation used Python 3.12.14, NumPy 2.5.3, pandas 2.3.3, SciPy 1.18.0 and Matplotlib 3.11.1. The script combines executed notebook calculations and passed a syntax check; it was not separately rerun end-to-end. Published plots passed geometric label checks and visual inspection; the results archive passed integrity checking.
