A Fail-Closed Audit of Scale-to-Bulk Spatiality in a Two-Dimensional Multiscale Toy Framework

This technical note reports a fail-closed investigation of whether a genuinely two-dimensional microscopic system can acquire an effective third spatial direction through multiscale organization alone. The study separates several notions that are often conflated in discussions of emergent geometry: scale labels, information geometry, multiscale network geometry, operator locality, cross-scale dynamics, redundant bulk representations, and genuine local spatial dimensionality. Starting from two-dimensional free-field and lattice models, the analysis constructs and audits hierarchical coarse-graining maps, wavelet and multiresolution representations, relational subspace geometry, exact retarded responses, boundary-selected blocking schemes, adaptive maps, and redundant constrained bulk descriptions. Several positive intermediate results are obtained. Two-dimensional dynamics can support nontrivial multiscale information geometry; boundary-local structure can select coarse-graining features; gauge-invariant relational geometry can survive basis ambiguities; exact physical response functions can recognize multiscale and ancestor-descendant structure; and a redundant bulk representation with boundary-derived presymplectic gauge directions can be constructed without adding physical degrees of freedom. These properties, however, are not sufficient to establish an independent local third spatial dimension. For the periodic massless two-dimensional nearest-neighbour Laplacian and the explicitly tested separable dyadic db1-db4 wavelet hierarchy, an exact nearest-scale construction encounters a bounded deep-scale obstruction: after sufficient coarse-graining, the dynamically invariant coarse sector collapses to the constant mode, whose massless Laplacian stiffness vanishes. The mechanism is derived through the Fourier alias structure of the hierarchy and independently reproduced over an extended numerical grid. A second bounded result concerns the redundant multiscale embedding. When the exact boundary pullback and the exact gauge fibres are fixed, the induced presymplectic form is unique for this embedding and retains non-nearest-scale couplings. The resulting gauge redundancy is therefore exact and boundary-derived, but does not by itself generate a local metric direction along scale. Additional audits show that mass terms, larger scale bandwidth, interacting two-dimensional dynamics, and boundary-selected adaptive maps can relax particular diagnostics, but none of the tested mechanisms establishes the required combination of local metric propagation, independent spatial distance, and genuinely new physical dimensionality. The result is therefore deliberately limited. It is not a no-go theorem for emergent dimensions, holography, tensor networks, or dimensional emergence in general. It shows instead that, within the explicitly tested constructions, multiscale organization, operator locality, cross-scale dynamics, and gauge-redundant bulk representations do not by themselves turn scale into an independent local spatial dimension. The main scientific outcome is a bounded negative and methodological result: any successful continuation of a two-dimensional-to-three-dimensional emergence programme must supply an additional physical principle capable of generating genuine local metric structure along the emergent direction, rather than merely a hierarchical, informational, representational, or gauge-redundant scale coordinate. The principal bounded propositions were independently reproduced and subjected to a separate referee-style mathematical audit before release.