A Closure-Cost Route to MOND-like Galactic Dynamics

A Closure-Cost Route to MOND-like Galactic Dynamics: The Reconstructive Halo This exploratory note investigates whether the closure-cost structure of the reconstruction programme can reproduce MOND-like galactic dynamics without introducing a new particle halo. It is explicitly classified as non-constitutive: it is not part of the published core nor of the main working spine of the programme. Purpose The note asks what radial closure-cost profile is imposed when a local reconstruction, such as a galaxy, must match the global cosmological reconstruction. The central question is whether this matching naturally produces a logarithmic potential, and therefore flat galactic rotation curves. Main Assumption The key load-bearing hypothesis is that local–global matching is effectively two-dimensional or scale-invariant. In other words, the matching is carried by the dilation coordinate ρ = ln(scale) rather than by the ordinary three-dimensional Laplacian. Without this assumption, the harmonic solution gives a Newtonian 1/r profile and no halo-like behaviour. Main Result Under the scale-invariant matching assumption, the closure cost becomes logarithmic: ΔS(r) ∝ ln r. This induces a logarithmic potential and therefore flat rotation curves, reproducing the characteristic MOND-like galactic phenomenology. Acceleration Scale The note shows that the natural acceleration scale is of order cH₀, linking the galactic transition scale to the cosmological closure scale. However, the exact MOND value is not forced. The bare relation a₀ = cH₀ overshoots the empirical MOND acceleration by a factor of about 5.5. A previous attempt to insert a factor 1/2π is rejected: the modular temperature is universal and cancels in the local–global crossover. Interpretation The construction can reproduce the form of MOND and the order of its acceleration scale, but it does not derive the precise MOND constant. The remaining numerical factor is an unfixed order-one normalisation ratio between local and global closure costs. What It Explains flat galactic rotation curves; the logarithmic halo-like potential; the qualitative radial acceleration relation; the order-of-magnitude coincidence between the MOND scale and cH₀. What It Does Not Explain The Bullet Cluster: because the reconstructive halo tracks baryons, it cannot reproduce the observed lensing offset. The CMB acoustic peaks: the construction does not provide a non-baryonic gravitating component before recombination. The exact MOND acceleration scale: the numerical factor remains unfixed. Status The note is best read as a controlled exploratory result. It shows that a MOND-like profile can emerge from the closure-cost framework under a clear scale-invariance assumption, but it also states the limitations explicitly. It is therefore a possible derivation of MOND-like behaviour, not a replacement for dark matter.