Topology as a Test of ΛCDM: Evidence for a Near-One-Dimensional Cosmic Web from Join Distributions and Percolation

Standard analyses of large-scale cosmic structure rely primarily on two-point correlation functions ξ( r ) and power spectra P( k ) . Distinct networks can share identical two-point statistics while differing fundamentally in graph-theoretic properties. Here, we analyze the topology of the cosmic web using join distributions P( k ) , which count independent connections (joins of order k ) within the percolating filament graph and provide a direct, non-metric probe of effective dimensional organization D eff . We apply join analysis to observational galaxy surveys spanning a broad range of redshifts 0.1≤z≤5.0 . Across five statistically independent sky regions (each 5003 Mpc3, ~106 galaxies), the observed P( k ) are statistically uniform ( χ 2 /ν 1.1 ) and quantitatively fit by a Galton-Watson branching process with offspring probability generating function f( s )=( 1−p )+p s m , mean offspring m=1.054 , yielding D eff = log 2 m≈1.054 . In contrast, the LCDM model predicts and N -body simulations exhibit a broad P( k )∝ k −α ( 1.5≤α≤2.0 ) with substantial high-order tails ( k=5 - 8 ) from hierarchical halo mergers. These high-order joins are strongly suppressed ( >25σ discrepancy ( χ 2 >1700 ) that cannot be resolved by parameter tuning ( Ω m , σ 8 ) or baryonic feedback. Three further independent empirical probes reinforce this conclusion: i) Box-counting analyses across redshift slices spanning ~12 billion years yield effective dimension D eff =− lim ϵ→0 logN( ϵ )/ logϵ and density ρ converging gradually toward 1.0 from above, D eff ( z )≈1+0.02 ( 1+z ) 0.1 . LCDM simulations do not exhibit comparable convergence. ii) Observational data show matter and voids percolate at the same threshold ρ c ≈0.20 ρ ¯ (within 2% - 3%), whereas LCDM generically exhibits separated thresholds ρ c matter ≈0.22 ρ ¯ , ρ c void ≈0.15 ρ ¯ . iii) Finally, matter and voids display identical scale-free power-law behavior N( >R )∝ R −γ with γ≈2.89 ( D=3−γ≈0.11 for void filling factor) in the scaling regime 1 R h−1, indicating phase symmetry consistent with critical phenomena.