Dependently Reconstructible Distinguishability (DRD): A Functorial Theory of Information Carriers, Graded Spectral Capacities, and Causal Integration on Compact S³ Geometry

This preprint develops Dependently Reconstructible Distinguishability (DRD) as a definition-first framework for the layer beneath standard information theories. Shannon theory, Kolmogorov complexity, and quantum information all quantify information over a presupposed alphabet; DRD asks what makes an alphabet distinguishable, carried, preserved, erased, and operationally realizable in the first place. Its guiding formula is information = dependently reconstructible distinguishability: not a self-existing substance, not something created from nothing, but a functorial package of distinctions carried by mathematical and physical structures. The mathematical core replaces a single “amount of information” by a two-level profile D♯ = (D♯top, D♯op). The topological layer records reduced homology, torsion, cohomology ring structure, homotopy data, and characteristic classes; the operational layer adds metric, spin, bundle, spectral, and resolution data. The main structural results are conservation, attributed erasure, and transport. Structure-preserving morphisms preserve D♯; if D♯ changes, a non-invertible forgetting operation must be identified — quotient, deletion, category change, or coarse graining. Suspension transports distinguishability upward: H̃₀(S⁰) ≅ H̃₁(S¹) ≅ H̃₂(S²) ≅ H̃₃(S³) ≅ ℤ, so minimal distinction is not annihilated but re-represented on richer carriers. The operational layer works on compact S³_R geometry. At resolution ε, DRD gives exact graded spectral capacities for scalar, transverse-traceless tensor, and Dirac alphabets, with leading asymptotics N₀ ∼ ⅓(R/ε)³, N_TT ∼ ⅔(R/ε)³, and N_D ∼ ⅔(R/ε)³, in agreement with Weyl scaling. Capacity alone is not enough: causal integration is bounded by light propagation on S³. The Physical Integration Ratio PIR(T) measures how much of the carrier’s capacity can be causally integrated by time T, with cubic short-time onset and the exact symmetry PIR(t_coh/2)=1/2. The physical reading is deliberately cautious. The Planck grain ℓP, tP=ℓP/c is treated as the lower operational boundary of distinguishable causal-geometric reconstruction, not as a derivation of fundamental physics from topology. The paper does not replace Shannon, Holevo, or Landauer; it supplies the pre-statistical object those theories presuppose: the physically realizable alphabet of distinguishable states. It also makes the volume–area tension central: spectral capacities scale like volume, while gravitational entropy bounds scale like area. This is not hidden but stated as a major open problem, with gravitational dressing, holographic projection, and backreaction-based selection listed as competing resolutions. In compact form, DRD v2.0 says: distinction requires a carrier; erasure requires an operator; physical information requires grain and causal order; compact S³ gives finite graded capacity; and no distinction disappears for free. The framework is not a completed theory of quantum gravity, but a checkable formal laboratory for information carriers, invariants, erasure, operational resolution, and quasi-local reconstruction.