The χ-Field as Dark Matter: Comprehensive Validation Across 10 Independent Tests with No Per-Galaxy Fit Parameters

This work examines whether a subset of astrophysical phenomena commonly attributed to particle dark matter can instead emerge from spatial structure in a scalar χ field within the Lattice Field Medium (LFM) framework. The dynamics are governed by a single canonical wave equation,

∂²E/∂t² = c²∇²E − χ²(x,t)E

where spatial variations of χ modify local wave propagation and effective inertial response without introducing new particle species.

Theoretical Foundation (New in This Version): We now provide the derivation chain explaining why χ-gradients gravitate. Integrating out short-wavelength E-modes via heat-kernel expansion of the one-loop effective action produces induced operators:

S_ind ⊃ ∫d⁴x√−g { ½M²_ind R + αχ²R + β(∇χ)² + γχ⁴ }

The gradient coupling β ≈ +10⁻³ is positive, meaning χ-gradients carry positive energy density ρ_χ = ½β(∇χ)². This energy gravitates through the modified Poisson equation ∇²Φ = 4πG(ρ_b + ρ_χ), providing the theoretical foundation for the phenomenological enhancement formula. The rotation curve relation v_obs = v_bar × (1 + a₀/a)^0.25 is now derived, not assumed.

The paper focuses on the low-acceleration regime, where gradients and large-scale redistribution of the χ field lead to an effective velocity enhancement relative to Newtonian expectations. An analytic treatment identifies the functional form of this enhancement and shows that, in the deep low-acceleration limit, a baryonic scaling consistent with the observed Tully–Fisher relation arises as an effective description. The characteristic acceleration scale is fixed by cosmological input, a₀ = cH₀/(2π), with no per-galaxy or per-dataset parameter tuning.

Phenomenological χ-dynamics are introduced explicitly as a closure to explore astrophysical consequences; their status and limitations are clearly delineated. The framework is tested against multiple independent observational probes across galactic and extragalactic scales, including galaxy rotation curves (SPARC), baryonic scaling relations, strong gravitational lensing, galaxy cluster mass profiles, the Bullet Cluster, dwarf spheroidals, ultra-diffuse galaxies, wide binaries, and early massive galaxies observed by JWST. Where relevant, qualitative and quantitative comparisons with standard MOND phenomenology are presented under consistent assumptions.

All analyses use published observational datasets and fixed theoretical inputs. Agreements, partial successes, and known tensions are reported explicitly. The results indicate that a collisionless, non-radiating χ-field substrate can reproduce several empirical regularities usually associated with dark matter, while remaining falsifiable in low-acceleration and small-system regimes.

Results: 9 PASS, 1 MARGINAL, 0 FAIL across 10 independent tests.

All scripts, source data, and reproduction instructions are included. This version includes a working Python implementation (lfm_induced_gravity_solver.py) that validates the complete derivation chain from GOV-01 to rotation curves.