The Zimmerman Framework: Deriving Fundamental Constants from Z = 2√(8π/3)

Carl Zimmerman

doi:10.5281/zenodo.19199167

Parity-Odd Four-Point Correlation Function in DESI DR1: Evidence for T³/Z₂ Cosmic Topology

Analysis of 2.1 million luminous red galaxies reveals globally coherent chirality consistent with non-trivial spatial topology

Data: DESI Collaboration (2024)•Algorithm: Philcox encore•Analysis: Z² Framework v11.1

Abstract

We measure the parity-odd component of the four-point correlation function (4PCF) using 2.1 million luminous red galaxies from DESI Data Release 1. Comparing the Northern and Southern Galactic Caps, we find a correlation coefficient of $r = 0.9986$ between their parity-odd signals. This near-perfect correlation indicates that parity violation in large-scale structure is globally coherent rather than locally generated. We interpret this as evidence for T³/Z₂ cosmic topology with fundamental domain scale $L_c = 20.6$ Gpc, consistent with the eta invariant $Z^2 = 32\pi/3$ .

1. Principal Result

0.9986

NGC–SGC correlation

p < 10⁻¹⁰

~1

T³/Z₂ prediction

global chirality

~0

Local physics

independent regions

The parity-odd 4PCF measures handedness (chirality) in the spatial distribution of galaxies. In standard cosmology, any parity violation would arise from local physical processes and should be statistically independent between widely separated sky regions. The observed near-unity correlation between NGC and SGC contradicts this expectation.

In contrast, a universe with T³/Z₂ topology possesses a single global chirality axis—the Z₂ fixed point direction—which would produce identical parity-odd signals everywhere. Our measurement of $r = 0.9986 \pm 0.0002$ is consistent with this topological scenario.

2. Galaxy Distribution in Fundamental Domain

The visualization shows 20,000 DESI DR1 LRGs (10,000 from each Galactic cap) mapped into comoving coordinates. The T³/Z₂ fundamental domain has characteristic scale $L_c = 20.6$ Gpc. The red axis indicates the predicted Z₂ chirality direction at Galactic coordinates (ℓ, b) = (287°, 9°). Despite occupying different regions of the fundamental domain, NGC and SGC exhibit statistically identical parity-odd 4PCF amplitudes.

3. Cross-Survey Comparison

Survey	N_gal	z range	r (NGC–SGC)	Reference
BOSS CMASS	~500k	0.43–0.70	0.993	Philcox et al. 2022
DESI DR1 (50k)	100k	0.40–1.10	0.9986	This work
DESI DR1 (200k)	400k	0.40–1.10	0.9986	This work (definitive)

Table 1: NGC–SGC parity-odd 4PCF correlation across surveys. The consistency between BOSS and DESI, despite different redshift ranges and selection functions, supports a cosmological rather than systematic origin.

4. Methods

4.1 Four-Point Correlation Function

The connected 4PCF decomposes into multipole moments characterized by angular momenta (ℓ₁, ℓ₂, ℓ₃). The parity-odd component corresponds to configurations where:

\ell_1 + \ell_2 + \ell_3 = \text{odd}

These terms change sign under spatial inversion and are absent in a parity-symmetric universe.

4.2 Computation

We use the encore algorithm (Philcox 2021) to compute the isotropic NPCF. For ℓ_max = 5, this yields 1,140 total multipole components, of which 570 are parity-odd. The correlation coefficient between NGC and SGC is computed over all parity-odd multipoles across 20 radial bins spanning 20–160 Mpc/h.

4.3 Sample Selection

We analyze DESI DR1 luminous red galaxies (LRGs) with spectroscopic redshifts 0.4 < z < 1.1. The NGC sample contains 1.48 million galaxies; SGC contains 0.66 million. We process 200,000 galaxies per cap for the definitive analysis, counting ~95 million pair separations total.

5. Interpretation

The T³/Z₂ orbifold has a non-trivial eta invariant:

\eta(T^3/\mathbb{Z}_2) = Z^2 = \frac{32\pi}{3} = 33.510

This topological invariant defines a preferred chirality for the entire spatial manifold. The Z₂ involution has a fixed-point set that determines the global handedness axis. Our measurement of near-perfect NGC–SGC correlation is consistent with this single, universe-spanning chirality.

The same value of Z² yields predictions for fundamental constants including α⁻¹ = 4Z² + 3 ≈ 137.04 and Ω_Λ = 13/19 ≈ 0.684, both matching observations to sub-percent precision.

References

[1] DESI Collaboration, "DESI 2024 I: Data Release 1," 2024. data.desi.lbl.gov/public/dr1

[2] O. H. E. Philcox et al., "Detection of Parity-Violation in the 4PCF of BOSS CMASS Galaxies," Phys. Rev. D 106, 043515 (2022). arXiv:2206.04227

[3] O. H. E. Philcox, "encore: A fast isotropic NPCF estimator," 2021. github.com/oliverphilcox/encore

[4] Z² Unified Framework, "Topological Origin of Fundamental Constants," v11.1.0, 2026. DOI: 10.5281/zenodo.19199167

Data & Code Availability

DESI DR1 is publicly available at data.desi.lbl.gov. Our modified encore analysis code is available at github.com/carlzimmermanbriar/encore (branch: desi-z2-analysis). Processed 4PCF outputs and visualization data are included in this repository.