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

This Version develops a definition-first theory of information carriers on compact S³ geometry. Its central claim is precise: standard quantitative theories of information measure alphabets already assumed to exist, while DRD asks what a distinction is, what carries it, which transformations preserve it, which transformations erase it, and how much distinguishability a physical domain can causally integrate. In this framework, information is not identified with entropy, statistics, or a quantum state. It is defined as dependently reconstructible distinguishability: relational structure preserved through admissible carriers, morphisms, resolution scales, and causal order. The mathematical core is the fibrewise distinguishability profile D♯, not a single flat invariant. Homology, cohomology, characteristic classes, spin data, spectra, and operational capacities live in different structural fibres and are preserved only by morphisms that preserve the relevant structure. The theory therefore separates D♯top, D♯op, spectral data, bundle data, and physical admissibility. Its minimal distinction begins with S⁰ = −1, +1, whose reduced homology is transported through suspension as H̃₀ (S⁰) ≅ H̃₁ (S¹) ≅ H̃₂ (S²) ≅ H̃₃ (S³) ≅ ℤ. In this sense, S³ is not asserted as an arbitrary cosmological dogma, but as the first compact carrier in the suspension chain where distinction, phase, enclosure, spin, Hopf structure, and compact volume can coexist. Version 3. 0 gives special care to erasure. A distinction is not erased by vague disappearance, but by an exhibited non-invertible operation: quotient, deletion, category change, coefficient change, or spectral coarse graining. The loss is measured by the mapping-cone defect Cf = Y ∪f CX, so erasure becomes attributable rather than rhetorical. This is the topological counterpart of the thermodynamic Landauer discipline: nothing erases distinguishability for free. The paper therefore replaces the loose slogan “information is conserved” with the sharper statement: within a fixed category of structure-preserving morphisms, the invariant package is conserved; outside it, loss is attributed to a concrete forgetting morphism. The operational layer computes graded spectral capacities on round S³R. Scalar, transverse-traceless tensor, and Dirac alphabets are counted separately, with leading behaviour N₀ ∼ (R/ε) ³/3, NTT ∼ 2 (R/ε) ³/3, and ND ∼ 2 (R/ε) ³/3. These are mode counts, not yet bit counts. Version 3. 0 makes this distinction explicit through the closure chain D♯ₐ, ⏚ → nₖin → ncausal → Bcausal → Bₚhys. Causal integration is controlled by the Physical Integration Ratio PIR (T) on compact S³, with coherence time tcoh = πR/c and exact half-time symmetry PIR (tcoh/2) = 1/2 in the homogeneous shell-complete model. A central improvement of Version 3. 0 is the selector-safe treatment of the volume–area problem. The raw spectral alphabet scales like volume, while gravitational entropy bounds scale like area. DRD does not hide this tension and does not replace the carrier by a boundary screen. Instead, boundary and area constraints are treated as holonomy-compatible admissibility conditions. The gravitational selector Σgrav = Bₚhys / Bcausal chooses the gauge-invariant, gravitationally dressed, backreaction-stable, entropy-admissible code subspace from the kinematic alphabet. Thus the boundary is not an ontology of prewritten information; it is a physical selection rule on what can be realized. The interpretive principle is compact: information is neither self-being nor nothing, but dependently reconstructible distinguishability passing through a chain of carriers. The paper does not replace Shannon, Holevo, Landauer, quantum mechanics, or gravitational entropy bounds. It supplies the pre-statistical object those theories presuppose: the physically realizable alphabet of distinguishable states, constrained by carrier, grain, causal accessibility, erasure cost, and gravitational admissibility. Version 3. 0 is therefore the canonical DRD statement: selector-safe, stability-safe, holonomy-compatible, and claim-status disciplined.