Universal and Classifying Models for Identity-Eliminator-Conserved Paraconsistent HoTT

This is Paper 13 in the 20 Paper PHHT Series This paper constructs the relative classifying model for identity-eliminator-conserved paraconsistent homotopy type theory under a fixed algebraic higher-signature package. The construction separates the exposed positive HoTT base from attempted data, retained negative obstruction data, protected certificates, aggregate refinements, and certified guarded identity-elimination gates. Starting from the free exposed positive syntactic base 𝕊Id⁺, its conservative marked attempted completion ŜId, and the raw generated pointed obstruction fibration over ŜId, the paper localizes raw bridge models at the no-bypass maps forcing every positive use of retained identity obstruction to factor through a retained family certificate. The resulting free no-bypass obstruction fibration is πId : 𝒫Id → ŜId, and the universal bilateral obstruction category is 𝔹Id^univ = 𝔹Id(𝕊Id⁺,𝒫Id). The main theorem states that 𝔹Id^univ corepresents the admissible-target model-space functor. For every admissible generated no-bypass target 𝔹Id(𝒞,𝒫), there is a natural equivalence ModId(PHTT_{Π,Σ,Id,𝒰};𝔹Id(𝒞,𝒫)) ≃ FunId^Str(𝔹Id^univ,𝔹Id(𝒞,𝒫)). Thus models of the bilateral theory are precisely identity-eliminator-conservative structure-preserving realizations of the universal classifying object. The guarded identity-elimination gate is store-level. For a retained identity glut (p,R), the primitive premise is the bare retained-family certificate BareFamCert(p,R) = ∏_{q∈R} Cert(p,q). Aggregate certificates belong to the conical aggregate enrichment. In that enriched setting, AggCert(p,R) ≃ BareFamCert(p,R⁺), R⁺ = R ⊔ Int(R), and the projection to BareFamCert(p,R) is an equivalence precisely under the protected non-interaction certificate. The StoreCompat-inclusive retained-family convention of the surrounding series is obtained from this classifying object only through a complete compatibility comparison or by a trivial or contractible compatibility factor. The paper proves algebraic validity relative to admissible generated no-bypass targets by validity in 𝔹Id^univ. It also derives positive-core conservativity and non-explosion by a support-graded no-bypass construction: raw identity gluts generate arbitrary positive evidence only through the certified no-bypass gate, and positive identity transport from retained obstruction is generated only by a certified guarded identity-elimination gate with retained family-certificate premise. Finite stages are classified by truncated universal models. Full retained classification requires compatible finite threads together with the stated limit-coherence data. The paper therefore supplies the classifying and presentation-invariance layer needed for later completion, localization, and synthesis comparisons in the series.