Winding-direction follow-up: reference comparison and Fourier directional tests

The new reference comparison and the Fourier directional tests are complete. No statistically significant directional winding pattern is detected with either method under the conditional shuffle null, and the two methods give different labels for 12,119 of 30,910 common targets.

1. Agreement with reference labels, and coverage

Coverage. Of 49,211 attempted targets, the Fourier method assigned a sign to 49,007 (204 have no retained tensor). SpArcFiRe resolved 30,910 and left 18,301 unresolved. On the 30,910 targets with both labels the methods agree for 18,791 and disagree for 12,119. A sign assignment is not a quality guarantee.

Reference comparison. For cases with a strong Galaxy Zoo reference label, the chart shows how often each method gave the opposite direction among the cases it assigned. Fourier assigned every such case; SpArcFiRe abstained on some. The fractions are therefore conditional on different assigned sets and are not a fair whole-population accuracy ranking.

Limits. Reference labels are human classifications, not physical truth. DEV and EVAL were both exposed during development: neither is an untouched or representative validation set, and they are never pooled. The separate CzSL expert cohort is historical context only; it lies outside the current Fourier parent.

EVAL split: reference disagreement (opposite direction / assigned cases) with nominal Clopper-Pearson 95% intervals for these conditional counts. Notes give opposite/assigned; assigned/reference cases. CW ref rows count CW-to-ACW disagreement, ACW ref rows ACW-to-CW.

DEV split, shown separately and never pooled with EVAL. Intervals exclude reference error, the vote threshold, selection into the reference set and any transfer to the science sample. These are reference disagreements, not classifier error rates.

2. Directional tests: no significant pattern detected

Amplitude. The dipole amplitude is 2|b| in A = 2p - 1: a one-sided cosine coefficient, not a hemisphere contrast or a pooled imbalance. Fourier labels give 1.65% on full coverage (approximate 95% Wald envelope 0.00 to 6.03%) and 2.16% on the common set (0.00 to 7.68%). The unchanged SpArcFiRe result is 2.27% (0.00 to 7.79%).

Tests. Four new tests were declared before any draw: a dipole likelihood ratio and a maximum hemisphere contrast over 648 axes, each on full and common coverage, with 1,024 label shuffles per coverage set. The largest observed contrasts in A are 2.04 and 2.49 percentage points. All four Bonferroni-adjusted p-values are 1.000. The earlier SpArcFiRe test (p = 0.684) is separate and not part of this family.

Meaning. Under a null that fixes positions and the CW count and treats labels as exchangeable, no statistically significant directional winding pattern is detected. This is not a physical-spin or cosmic-parity result, and absence of significance does not prove exact symmetry.

Fitted dipole amplitude with approximate 95% Wald envelopes. The envelopes are asymptotic, have no guaranteed coverage and exclude classification and selection uncertainty. Best-fit axes are not established and are not shown. The SpArcFiRe row is the retained earlier result, not recomputed.

Conditional plus-one permutation p-values with the four-test Bonferroni correction. The maximum over axes is recomputed in every shuffle, which accounts for that grid search; the likelihood ratio refits a continuous direction. The correction covers these four tests only. Full and common coverage overlap and the two algorithms are correlated. Spatial selection and classification effects are not removed. Monte Carlo intervals are in the appendix. pp = percentage points of A.

3. Where the two methods disagree, and scope

Disagreement by brightness and size. The chart gives the share of common targets on which Fourier and SpArcFiRe labels differ, in bins frozen before any winding outcome was read. These are descriptive proportions: no intervals or tests were computed and no cause is inferred from any bin. Empty bins are marked; bins with fewer than 30 common targets are flagged sparse.

Takeaway. The new reference comparison and the Fourier directional tests are complete. No statistically significant directional winding pattern is detected with either method under the conditional shuffle null, and the two methods give different labels for 12,119 of 30,910 common targets.

Scope. Nothing was reclassified and the historical SpArcFiRe test was not rerun. The optional group and cluster audit was not run in this phase; group results are unchanged. Axis-dependent maps remain descriptive, not evidence of a preferred direction. No calibration-adjusted population asymmetry was calculated.

Checks. Independent numerical checks reproduced the tables (21 checks) and certified 2,050 saved optima with no new optimisation or draw. They confirm arithmetic, not the scientific assumptions.

Disagreement share (disagree / common, percent) by frozen r-magnitude and size bins; notes give disagree/common counts. Declination, score-magnitude and exposure-count bins and all denominators are in the appendix. Method context only, not validation of this pipeline: arxiv.org/html/2404.06617v2 and arxiv.org/abs/1610.07060.

Limitations and scope

• Reference labels are human classifications, not physical truth. DEV and EVAL are development-exposed and never pooled. Fractions conditional on assigned cases are not whole-population accuracy.
• A Fourier sign is not a quality guarantee. No calibration is transferred to the science sample and no calibration-adjusted population asymmetry was calculated.
• The conditional null fixes positions and the CW count and assumes labels are exchangeable over those positions. Spatial selection, classification effects and correlated labels are not removed.
• The Bonferroni correction covers the four declared new tests only. The historical SpArcFiRe test (p = 0.684) is separate. Full and common coverage overlap and the algorithms are correlated, so the tests are dependent.
• The Monte Carlo CP95 interval describes the precision of the estimated exceedance probability at 1,024 draws per coverage set. It is not an amplitude interval, is neither p-value and is not Bonferroni-adjusted.
• Wald envelopes are asymptotic approximations. Best-fit axes and axis-dependent maps are descriptive; no preferred direction is established.
• No physical-spin or cosmic-parity detection. Absence of significance is not proof of exact symmetry.
• Strata proportions are descriptive: no intervals, tests or causal reading. Group and cluster results are unchanged; the optional group audit was not run in this phase.
• The CzSL expert cohort is retained historical context outside the current Fourier parent and is not pooled.
• Numerical verification confirms arithmetic on pinned inputs, not the scientific assumptions. Poster v55 is unchanged; this report is its dated supplement.
• Later items in this list are verbatim stage caveats written before the reference comparison ran. Where one calls Fourier signs uncalibrated, it means no calibration is applied or transferred to the science sample; conditional reference counts for the exposed Galaxy Zoo subsets now exist and are shown on page 1.
• Null: unrestricted conditional exchangeability of the Fourier labels over the fixed positions of each coverage set, with the CW count fixed. Spatially correlated labels or position-dependent selection or classification efficiency violate it; a small p-value does not exclude those explanations.
• Fourier signs are uncalibrated image-winding proxies; no accuracy transfer, physical spin or parity claim.
• Fourier and SpArcFiRe are correlated algorithms on the same objects; agreement or disagreement between them is not independent replication. Full and common coverage overlap, so the four tests are dependent.
• p_plus_one is the permutation p-value; p_bonferroni4 corrects this declared four-test family only, not the project's earlier choices. exceedance_cp95 is a Clopper-Pearson Monte Carlo interval for the null exceedance probability at 1024 draws; it is neither of those p-values and is not Bonferroni-adjusted.
• Covariance and the amplitude envelope are asymptotic Wald approximations without guaranteed coverage. The fitted direction and the arg-max axis are descriptive only.
• Exploratory follow-up on exposed data; observed maps are reproduced from the frozen labels as an input check.
• Counts are conditional on the GZ1 0.8 vote-fraction reference class within one development-exposed split; they are not population error rates, not physical truth and not an untouched validation. No calibration transfer to the science sample: the reference rows are northern (none below Dec -30) and not representative. DEV and EVAL are never pooled. Clopper-Pearson intervals are nominal 95% binomial intervals for those conditional counts only. |chi| is a score magnitude, not a quality probability, and is not used as a filter. Strata were frozen before any winding outcome was read; sparse rows are descriptive; no p-values.
• sparse = common < 30; shown, flagged, descriptive only
• original GZ1 vote-fraction column ref (0.8); ref06 unused
• Independent recomputation of counts, strata and interval equations from the same pinned inputs. Numerical agreement only: not a method acceptance, not a calibration transfer, not physical truth.
• Numerical/algebra verification at saved estimates only. Inference still rests on conditional label exchangeability and the stated model; no acceptance is implied.
• Numerical verification only; inference requires exchangeability and stated model.

• https://arxiv.org/html/2404.06617v2 Patel and Desmond: explicit monopole/dipole likelihood, mock recovery and treatment of direction search. Methodological context only.
• https://arxiv.org/abs/1610.07060 Hayes, Davis and Silva: selection into a winding catalogue can bias handedness independently of classification. Context only.

Frozen strata: declination_deg

Frozen strata: fourier_abs_chi

Frozen strata: magnitude_dered_mag_r

Frozen strata: nobs_r

Frozen strata: size_shape_r

Parent coverage

Reference comparison, DEV (never pooled)

Reference comparison, EVAL (never pooled)

Retained CzSL context (historical, not pooled)

Four new tests, including Monte Carlo CP95

Fourier fits (approximate Wald; direction descriptive)

verify_calibration summary

verify_inference summary

Retained SpArcFiRe dipole content

Retained SpArcFiRe verification summary