Fractal-to-Classical Spacetime Transition: IFS, DSI, and a Hausdorff Sigmoid toward FLRW Part One, General Information

Reading map 1. Pre-metric fractal substrate. The study models the Planck-scale ultra-microstructure as a compact attractor of an iterated function system (IFS). Classical geometry is therefore not assumed at the fundamental level, but treated as an effective large-scale description. 2. Minimal control of the micro-geometry. To avoid discrete pathologies such as over-connectivity, shortcuts, or branched-polymer behavior, the construction imposes operational geometric constraints (PCF-type assumptions, bounded incidence/overlap conditions) and works on a finite-resolution causal patch linking micro-geometry to observables. 3. Effective analysis without a smooth manifold. The framework is formulated through a Dirichlet form, a resistance metric, and an effective Laplacian, allowing one to discuss diffusion, spectral behavior, and propagation without assuming microscopic differentiability. 4. Dimensional transition and possible DSI residue. The passage toward the metric regime is encoded by an effective dimension HDH_DHD, used as an order parameter and modeled by a minimal sigmoid of tanh / Allen–Cahn type. The observational window controls which microstructures are effectively resolved and whether a possible DSI/log-periodic residue can remain observable. 5. Bridge to phenomenology. In Part II, the effective dimensional dynamics is mapped into a cosmological expansion history through a minimal effective closure, then confronted with data using SN+CC calibration, BAO prediction, and robustness checks. Status of the paper Points 1–3: mathematically framedPoint 4: effective / phenomenologicalPoint 5: developed and tested in Part II