Dark Matter as Superimposed Cosmic Webs: A Methodology for Testing Independent Structure Formation Across ψ-Branches

Within the phase space dimensionality framework, dark matter may consist of ordinary baryonic matter on gravitationally coupled but electromagnetically isolated ψ-branches. Papers I and II in this series found no evidence for quantized dark matter fraction signatures (f_DM = (N-1)/N) in SPARC galaxy rotation curves — neither in the global distribution nor in radial profiles. This paper argues that those null results are expected if each branch undergoes independent structure formation, producing its own cosmic web of galaxies, filaments, and voids. Under this independent structure formation (ISF) hypothesis, the dark matter distribution is a superposition of N-1 independent cosmic webs, naturally yielding smooth, continuous f_DM values rather than discrete quantized signatures. A feasibility study using 84 SPARC galaxies detects apparent spatial autocorrelation of f_DM residuals (Moran's I = 0.67, p = 0.006), but confound analysis reveals this signal is entirely driven by galaxy group membership — particularly the Ursa Major cluster — and vanishes after deconfounding (I = 0.025, p = 0.53). SPARC lacks the sample size and spatial coverage for a definitive test. We propose three follow-up tests with sufficient discriminating power: (i) residual spatial autocorrelation of f_DM in volume-limited spectroscopic surveys (~360,000 BOSS galaxies), where the ISF hypothesis predicts positive correlation at scales below ~50 Mpc while ΛCDM predicts none after controlling for mass and environment; (ii) dark matter power spectrum suppression from weak lensing, where ISF predicts suppression by a factor (N-1)/N² with distinctive scale dependence; and (iii) galaxy-mass dark substructure from strong lensing, where ISF uniquely predicts dark matter concentrations at 10¹⁰–10¹² M☉ — masses where ΛCDM predicts visible galaxies should be present. A convergence test requiring four independent N estimates to agree prevents the model from being unfalsifiable despite its increased flexibility. Third paper in a series exploring observational tests of the phase space dimensionality framework. All analysis code and data are available by request. Version 2 notes: Updated following external review (ChatGPT 5.2, OpenAI). The key revision addresses an internal consistency requirement in the original formulation: the power spectrum suppression formula (Eq. 2 in v1) assumed zero cross-branch correlation (ε = 0), which is inconsistent with the moderate gravitational coupling invoked to explain the observed mass–f_DM correlation. Version 2 introduces the generalized power spectrum (Eqs. 2–3) parameterized by a scale-dependent cross-branch correlation coefficient ε(k), which interpolates between fully independent branches (ε = 0, maximum suppression) and fully locked branches (ε = 1, no suppression, recovers ΛCDM). The correlation function ε(k) is physically constrained to be monotonically increasing with scale, bounded by ε → 0 at sub-galactic scales and ε → 1 at super-cosmic-web scales. The convergence test (Section 5) is updated to reflect that the model has three effective parameters (N, amplitude, and transition scale of ε) constrained by four independent probes. ChatGPT 5.2 (OpenAI) added as computational collaborator in acknowledgements. No changes to the SPARC feasibility analysis, the confound results, or the proposed follow-up tests (I–III). Keywords: dark matter, large-scale structure, gravitational lensing, cosmic web, multi-brane cosmology, independent structure formation, galaxy rotation curves, SPARC, weak lensing, strong lensing substructure