This preprint opens an information-theoretic branch of the compact S³ reconstruction programme. Its starting point is simple: many physical theories use the word “information” before defining the primitive it quantifies. Shannon theory measures statistics over an already given alphabet; this work asks what makes an alphabet physically distinguishable in the first place. The proposed primitive is distinguishability, formalized on carriers by the profile Dₖ (X) =rank H̃ₖ (X). Information is therefore not treated as a self-existing substance, nor as something emerging from nothing, but as dependently reconstructible distinguishability. The mathematical core contains three structural results. First, invariance: the distinguishability profile Dₖ (X) is preserved under structure-preserving re-description, such as homotopy equivalence. Second, attributed erasure: distinctions are not erased “by magic”; any change of profile must exhibit an operator — deletion, identification, quotient, or change of category. In the minimal case S⁰=−1, +1, the class H̃₀ (S⁰) ≅ℤ records a two-component distinction. Collapsing S⁰ → * is a quotient, not a homotopy equivalence; it erases the invariant rather than continuously deforming it. Third, carrier transport: the suspension isomorphism carries the distinction upward, H̃₀ (S⁰) ≅H̃₁ (S¹) ≅H̃₂ (S²) ≅H̃₃ (S³) ≅ℤ. Thus minimal distinction is not annihilated by suspension; it is re-represented on richer carriers. The physical layer is deliberately cautious. The Planck scale is not derived from topology and is not treated as the beginning of the universe. Instead, ℓP and tP=ℓP/c are used as the lower metric-causal grain at which distinguishability can be operationally reconstructed in effective physics. The compact carrier S³ then supplies a finite spectral alphabet: scalar modes on S³R below resolution ε give N (M) =⅙ M (M+1) (2M+1), with M=⌊√ (1+R²/ε²) ⌋ and asymptotic growth N∼ (1/3) (R/ε) ³. This is a capacity theorem for reconstructible distinguishability on a compact geometry, not a holographic entropy law and not a replacement for Shannon information. The guiding formula of the paper is: information = dependently reconstructible distinguishability. The work does not claim that information is a substance, that distinctions are absolutely created, or that topology alone produces physics. It gives a formal laboratory in which distinctions, carriers, invariance, erasure, operational resolution, and compact S³ capacity can be stated exactly. In this sense the paper supplies the lower information-theoretic layer of the broader programme: S⁰ gives minimal distinction, suspension transports it through carriers, S³ provides compact reconstructible geometry, and the Planck grain marks the lower operational boundary of physical distinguishability.
Batenin et al. (Thu,) studied this question.