TESTING SPIN PARITY IN GALAXIES, GROUPS AND CLUSTERS Research notes accompanying poster v74 | Duho Kim | KAS 2026 Fall 12 October 2026 These notes summarize retained results. No scientific calculations, fits, classification, catalogue matching or random draws were repeated for this HTML revision. The selected print PDF is preserved unchanged. Current public companion: https://nebulamind.net/studies/spin-parity-2026.html MAIN FINDINGS Galaxies: neither method establishes a preferred sky direction under the reported conditional tests. Agreement between winding labels is not accuracy. Group/cluster H depends on selection and centre choice; conditional sign-null tests show no excess. These are observable proxies and conditional analyses. They do not establish physical rotation or cosmic parity violation. GALAXIES: KEEP THE DENOMINATORS AND QUANTITIES SEPARATE Attempted target inventory: 49,211. SpArcFiRe: 30,910 resolved; 15,479 CW, 15,431 ACW; 18,301 unresolved. Fourier: 49,007 assigned; 24,644 CW, 24,363 ACW; 204 lack a retained image tensor. Shared sample: 30,910 targets, including every disagreement. Labels agree on 18,791. Coverage and agreement do not establish accuracy. Observed pooled imbalance is +0.155% for SpArcFiRe and +0.573% for Fourier. It is not the dipole amplitude or a map maximum. Saved directional fits: SpArcFiRe resolved sample: dipole amplitude 2.27%; approximate 95% Wald envelope 0.00-7.79%; global conditional label-shuffle p=0.684. Fourier full sample: 1.65%; approximate envelope 0.00-6.03%. Fourier shared sample: 2.16%; approximate envelope 0.00-7.68%. The Fourier full/shared dipoles and maximum hemisphere contrasts form one four-test family; all Bonferroni-adjusted p-values are 1. Classification, selection and dependence uncertainty remain. Historical SpArcFiRe calibration: Galaxy Zoo only: conditional 95% range -17.93% to +10.03%. CzSL human-reference only: -4.37% to +4.56%. Shared transferable directional errors: point -0.06%, range -4.44% to +4.32%. Cohort-specific error bracketing: -19.07% to +11.10%, no unique point estimate. The first three scenarios use M3 simultaneous binomial-margin control; the last uses M5 and a different model. All assume equal CW/ACW resolution. These are not current Fourier calibrations or validated full-population corrections. CURRENT GROUP / CLUSTER SAMPLE Both methods use the same 1,713 groups and 9 clusters, N=1,722. The fixed 3 Mpc / 3,000 km/s joint proximity screen removed 216 groups and one cluster from 1,929 fitted groups plus ten clusters. Of 1,934 eligible groups, one failed source validation and four lack a primary-aperture fit. Regression fits a constant plus sine and cosine to member line-of-sight velocity versus angle. Half-plane extracts the first harmonic of rotating mean-velocity contrasts. Both are projected-motion estimators, not confirmed three-dimensional spins. Each system receives equal weight in H. H = -(sum_i q_i) dot (sum_i g_i) / [N(N-1)/2] q_i = n_i cross g_i; n_i is the unit sky direction; g_i is the unit projected velocity-gradient vector. Zero fitted vectors remain zero and count in N. H is dimensionless, not a velocity or a percentage of rotating systems. All numbers in the following table are 10^4 H: Method Primary pooled value Leave-one-system sensitivity range Regression +0.673972 +0.023 to +1.319 Half-plane +0.210983 -0.416 to +0.839 Leave-one-system ranges are not confidence intervals. The 12 literature-only systems in the earlier 102-system pilot are outside this current matched sample. READING THE MAPS Each cell is a candidate dividing-axis direction, not an object position. It compares a toward hemisphere with its opposite on the same 648-axis, 10-degree grid. Opposite axes swap the two halves and reverse the contrast by construction. EP/MW are fixed ecliptic/Galactic reference poles. Galaxy colour: A(toward)-A(opposite), A=(CW-ACW)/N, in percentage points. Shared display scale saturates at +/-5 percentage points; observed map ranges are +/-2.1 (SpArcFiRe) and +/-2.0 (Fourier). These ranges are not uncertainties. Halves with fewer than 30 galaxies or boundary ties are masked. Group/cluster colour: 10^4 [H(toward)-H(opposite)], a dimensionless scaled contrast. A displayed 100 means Delta H=0.01, not 100%. The linear scale saturates at +/-100; both maps use identical masks and at least three systems per half. The archived mean of two correlated maps is not an independent third measurement. CONDITIONAL SIGN-NULL TESTS Whole-system benchmark: 2,048 stored fair sign patterns; one sign per system shared by both methods. Positions, unoriented axes, magnitudes and selection are fixed. Four raw p-values, in order absolute pooled H regression/half-plane and maximum absolute map contrast regression/half-plane: 0.854075, 0.953636, 0.958516, 0.594436. Bonferroni correction within this four-test family gives 1 for every test. Spatial blocks: four fixed Cartesian partitions using 50/100 Mpc cubes with zero/half-cell origins. Each partition has 2,048 patterns, one sign shared by all systems in a block and both methods. The first block is fixed positive. Number of blocks: 1,166; 1,152; 556; 558, respectively. The four statistics across four partitions form a separate 16-test family. Raw p=(b+1)/2049, adjusted p=min(1,16*p). Exceedance includes ties within 1e-12+1e-10*abs(observed). Partition Statistic b/2048 Raw p Adjusted p Wilson95 MC-exceedance interval L50O0 absH-reg 1743 0.851147 1 [0.834999,0.865835] L50O0 absH-hp 1946 0.950220 1 [0.939902,0.958803] L50O0 maxmap-reg 1949 0.951684 1 [0.941496,0.960133] L50O0 maxmap-hp 1177 0.574915 1 [0.553175,0.595959] L50OH absH-reg 1766 0.862372 1 [0.846701,0.876551] L50OH absH-hp 1961 0.957540 1 [0.947894,0.965431] L50OH maxmap-reg 1960 0.957052 1 [0.947360,0.964992] L50OH maxmap-hp 1167 0.570034 1 [0.548271,0.591116] L100O0 absH-reg 1813 0.885310 1 [0.870723,0.898342] L100O0 absH-hp 1986 0.969741 1 [0.961382,0.976313] L100O0 maxmap-reg 1994 0.973646 1 [0.965757,0.979735] L100O0 maxmap-hp 1301 0.635432 1 [0.614171,0.655830] L100OH absH-reg 1768 0.863348 1 [0.847721,0.877482] L100OH absH-hp 1985 0.969253 1 [0.960837,0.975883] L100OH maxmap-reg 1955 0.954612 1 [0.944690,0.962788] L100OH maxmap-hp 1206 0.589068 1 [0.567410,0.609991] The intervals describe finite Monte Carlo exceedance-probability uncertainty, conditional on these models. They are not uncertainty in H or in physical parity. An adjusted p=1 is a capped multiplicity correction, not a probability of randomness. The maps were already seen before these follow-ups; partitions and methods are correlated, not independent replications. The cubes do not calibrate a physical correlation length or a cosmological null. Within-block pair contributions remain fixed. Consequently the signed-H null mean need not be zero. A decomposition of within-block contribution divided by observed numerator is a signed algebraic ratio, not a percentage explained or a causal attribution. Moment comparisons are descriptive only. ERROR AND CHOICE SENSITIVITY The later cluster formal-error completion recovered identity-matched positive redshift errors for all 1,322 retained cluster spectra. The same 1,713 groups and 9 clusters remain selected. Formal errors use an independent Gaussian model with fixed reference redshifts; this model is not calibrated here. Members and centres are fixed. It excludes membership, centring, finite velocity-sample inference, reference-redshift and correlated/systematic error uncertainty. Updated conditional quantiles across 1,024 perturbations (16th,50th,84th), in 10^4 H: Regression: [+0.225361,+0.770468,+1.256272]. Half-plane: [-0.244285,+0.279904,+0.821382]. These are not confidence intervals or detection significance. Stored group normal rows and covariance roots were reused, with new cluster-only draws; reconstruction differences mean comparisons with old outputs need not isolate cluster noise alone. Centre-angle stress: four deterministic 100 kpc east/west/north/south shifts change only the member-angle origin. Members, velocities, sky centres, tangent bases, selection and nine cluster vectors remain fixed. Thirty-two groups fail angle coverage in at least one shift. Matched comparisons therefore use the same 1,681 groups plus nine clusters (N=1,690): Condition Regression Half-plane (10^4 H) Matched baseline +0.53 +0.09 East +2.83 +4.31 West -2.50 -3.65 North -4.89 -4.03 South +0.39 +1.53 No full-N=1,722 shifted map is implied. The subset may favour richer or extended groups; 100 kpc is not a calibrated centre error. Saved member-cut sensitivity uses the same 1,040 groups plus nine fixed clusters (N=1,049). Of the original groups, 664 fail the smaller aperture's member-count requirement and nine its azimuthal coverage requirement. P15 is the 1.5 Mpc baseline, S075 the 0.75 Mpc aperture, V15C the 1.5 Mpc aperture with 1,500 km/s velocity cut: Variant Regression Half-plane (10^4 H) P15 -4.367723 -3.951468 S075 -3.955682 -3.654016 V15C -4.316008 -3.911802 Even baseline P15 is negative on this subset, unlike the positive primary sample. Compare cuts within the matched subset; cross-sample differences also change composition. V15C changes members for only one of 1,040 groups; apparent stability is not independent evidence of robustness. MEMBERSHIP AND CENTRES There are 23,623 retained member occurrences: 22,301 in groups and 1,322 in clusters. A bounded external redMaPPer retrieval obtained 7,025 member rows for 133 candidate hosts. Local association found 326 unique positional candidates: 158 across 30 groups and 168 across two clusters. Of these, 118 lack external spectroscopic redshift. Occurrences and unique counterparts are different units; their ratio is not a direct coverage fraction. PMem belongs to the external redMaPPer host model, not to our native group/cluster definitions. Pfree, luminosity and radial terms are separate. Unknown coverage is not non-membership. No probability weighting or refitting was applied. External PCen0..PCen4 and five candidate-centre positions are retained, with their acceptance=false status. They are not transfer-calibrated native centre probabilities or a Gaussian covariance. The 26,111-row host table versus 26,311 in the abstract remains unresolved. DESI Gfinder's documented RANK is categorical, not a membership posterior. Further work must establish host identity and coverage, validate membership and centring calibration on appropriate truth/mocks, and assess a joint model before changing the scientific weights. CONTROLLED GROUP-ESTIMATOR VALIDATION (SUPPORTING METHODS ONLY) The completed experiment used eight fixed ring geometries (N=12/24/48/96; uniform/skewed angles), noise-only cases and sine/split fields. Sine mean=A*cos(phi-psi); split mean=A/sqrt(2)*sign(cos(phi-psi)); sigma=1 iid Gaussian scatter. Amplitudes .25/.5/1 and phases 0/45/90/135 degrees form 200 total case rows, including eight noise-only rows. Each geometry/method has a separate threshold: the 974th of 1,024 ordered calibration amplitudes. Evaluation uses 256 trials per case; a strict amplitude>threshold is a toy flag, not physical-rotation classification. Evaluation noise is shared across cases and methods. The 59,392 total randomized production-core evaluations consist of 8,192 calibration datasets and 51,200 case datasets based on 2,048 unique evaluation noise draws. The preselected N=24 uniform, psi=45 degree, A=1 example gives noise/sine/split flags: Regression 11/256,230/256,208/256; half-plane 8/256,229/256,208/256. It illustrates estimator ambiguity between the prescribed sine and split patterns. It is not a model of two physical bound subclusters or a cosmological prediction. Fixed rings do not represent realistic radial distributions, interlopers or selection. Cluster-specific estimators were not tested; no population H/parity power was measured. All 200 stored aggregate rows are provided in spin-parity-toy-validation-v74.json. Its pointwise Wilson95 intervals condition on estimated thresholds and toy models, omitting calibration-quantile uncertainty. They are not simultaneous claims over the grid. Separate numerical implementation checked every retained dataset and row without new RNG. Such checking verifies operations, not the realism of the assumptions. READING THE RECORD The current HTML leads with the v74 real-data findings. Earlier v69 HTML and the 102-system pilot remain linked as archives; missing-data claims in those older documents can be superseded by later retrieval. The evidence index identifies retained source products by hash without exposing local paths or private rows. Physical print proof, printer-specific colour and PDF/X certification remain outside the digital checks. Existing illustrative cutouts retain their native detail. Model-assisted reviews are not conventional journal peer review or independent observations. PRIMARY REFERENCES AND SOURCE DEFINITIONS Lee & Pen (2001): https://arxiv.org/abs/astro-ph/0008135 Shim et al. (2025 revision): https://arxiv.org/abs/2406.06080v2 Patel & Desmond (2024): https://arxiv.org/abs/2404.06617 DESI Gfinder: https://data.desi.lbl.gov/doc/releases/dr1/vac/gfinder/ redMaPPer schema: https://cdsarc.cds.unistra.fr/ftp/J/ApJS/224/1/ReadMe Castellani et al. (2025), section2.1: https://arxiv.org/html/2508.18354v1 Its literal selection c*abs(delta_z)<3*sigma_fit, with sigma_fit=c*sigma_z/(1+z_cl), corresponds to a rest-frame cut 3*sigma_fit/(1+z_cl). It is not multiplied by (1+z_cl). This convention is documented; no membership correction or refit was applied here.