Truncation Levels and Spacetime C: Time Layers and Spatial Layers—A Unified Language of Quotient Maps, Inverse Systems, and Projective Limits

This paper takes the kernel construction of quotient maps as its starting point andprovides a unified mathematical definition of “layers”: given a resolution parameterand an observation map, the equivalence relation induced on the microscopic configuration space by equality of observation readings yields the quotient space of layers;layers at different resolutions and the projections between them form an inverse system, whose projective limit encodes all compatible information visible to the towerbut does not constitute candidate positions for geometry. Geometry is located as thestable content of the layer tower under truncation at the coarse end—the structureconsistently induced on all layers coarser than the macroscopic scale. In this language,layers in the time direction arise from the conditional supply of causal order to observation maps: within the range of applicability of classical reconstruction theorems,causal structure determines conformal geometry; the quotient construction of time layers depends on the joint choice of reference events and a causally compatible familyof coarse grainings, rather than being an automatic product of causal order. There isno innate source of order in the spatial direction; spatial layers must be actively produced by at least one of symmetry breaking, scale selection, finite propagation speed,or topological charge. The two classes of layers share the formal skeleton of the manyto-one non-invertibility of quotient maps; their information-theoretic cost is the loss ofconditional entropy within fibers, rather than a universal increase or decrease of differential entropy. Branches divide into two classes: soft branches are driven by statisticalfluctuations and are unprotected by conserved charges; hard branches are protected byhomotopy invariants, and their discrete readings equal continuous charges only underthe condition that the sampling theorem is satisfied. The entropy bound does notprovide a proved criterion for layer splitting but a conditional organizing principle:the combinatorial capacity of layers and the metrological ambiguity of the system’sthermodynamic entropy, the absence of an intra-layer carrying-capacity upper bound,the non-uniqueness of saturation-resolution modes, and the freedom of the area-lawcoefficient jointly delimit the heuristic status of this principle. In holographic duality,the correlation between entanglement truncation and the weakening of bulk geometricconnectivity has a quantitative origin in settings with an exact factorization structure;reading it as the loss of the fiber structure of layers is a qualitative analogy rather thana microscopic mechanism. Asymptotic safety, holographic duality, and causal sets canobtain statable translations in the language of layers, but commensurability belongsto the level of language; a deep structural isomorphism has not been constructed. Thepaper distinguishes mathematical facts, conditional physical results, finite-model evidence, and programmatic proposals, and does not claim theorem status for any claimwhose conditions are not satisfied.