Homotopical Apophasis: Bilattice Symmetry, the Progenitor as Eilenberg–MacLane Space, and the Shadow of Classical HoTT

We give a homotopy-type-theoretic treatment of the apophatic progenitor grounded throughout in the paraconsistent bilattice framework. Three foundational problems are addressed jointly. First, the Universal Apophatic Progenitor A₀ is characterized by a universal property: it is the paraconsistent Eilenberg-MacLane space K (Aut (L), 1), where L = 2 × 2 is the four-valued Belnap-Dunn bilattice (over the Boolean classical sub-universe with Ω = ⊤, ⊥) and Aut (L) ≅ ℤ/2ℤ is its De Morgan symmetry group. The Higher Inductive Type (HIT) presentation of A₀ is derived from the group presentation of Aut (L), not postulated. Uniqueness up to homotopy equivalence is established via the Whitehead theorem for 1-truncated types. A complete encode-decode proof establishes π₁ (A₀) ≅ ℤ/2ℤ. Second, we verify framework compatibility rigorously via the Labeled-Path Universe Model: an explicit Hofmann-Streicher-style universe in which types are classical types equipped with bilattice valuations carried as metadata on their path spaces. We prove the No-Glutty-J Theorem: the metadata design is the unique J-compatible placement of bilattice labels, because any alternative incorporating a non-⊤L label into type membership forces the reflexivity element to satisfy a label condition contradicted by Axiom (i). We further prove that the J-computation rule holds definitionally, that transport along glutty paths is non-explosive, and illustrate the metadata principle with a worked example tracing transport along the generator ω ∈ A₀. Third, we develop the shadow Homotopy Type Theory: the unique finite-group determinization satisfying the bilattice alignment condition π₁ᵈet (α) ≅ Aut (L). In the shadow, Tight-Apartness Empty (TAE) types are free ℤ/2ℤ-sets, and their sub-TAE filtration is the Boolean orbit-power-set lattice Bₙ. The Duality Theorem collapses all TAE types to 1 under the full projection, with universal generation of classical types as a corollary. The Monad-Equivariance Theorem establishes that Det_αˢh acts on TAE (S) as a poset isomorphism Bₙ ≃ Fix (Det_αˢh) ^≤n. Finally, the Gödelian undecidability result is proved via a bilattice Lindenbaum algebra and a decidable clopen split whose classifying geometric morphism identifies the independence of G with a non-degenerate element of the shadow determinization.