Higher Categorical Semantics for Full Identity-Eliminator-Conserved Paraconsistent HoTT

This is Paper 12 in the 20 Paper PHHT Series This paper constructs higher-categorical semantics for full identity-eliminator-conserved paraconsistent homotopy type theory. A model consists of a positive comprehension (∞, 1) -category 𝒞, an attempted positive completion Ĉ, and a pointed cartesian obstruction fibration π: 𝒫 → Ĉ. Positive validity in the bilateral theory is not ordinary interpretability in 𝒞. It is the least exposed-positive relation generated by the selected positive rules and the certified bridge rules. Negative evidence is interpreted as obstruction data in the fibres of π, possibly over attempted data that are not exposed positively. The syntax-faithful identity-elimination gate is the retained family certificate. For an exposed identity witness p: IdA (a, b) with retained obstruction store R = RetStore⁻_Γ (p), guarded identity elimination through dimension n is exposed exactly when R = ∅ or when the retained family certificate FamCert≤ₙ (p, R) = (∏ₐ∈ₑ Cert≤ₙ (p, q) ) × StoreCompat≤ₙ (p, R) is exposed inhabited. Aggregate obstruction Q_Γ (p) = Agg (R) is treated as a higher semantic refinement of this gate. Aggregate certification implies retained family certification only through a displayed aggregate-to-family comparison map. It becomes equivalent to retained-family certification only under realization-completeness, null-reflection, and complete aggregate-family comparison data. Finite truncations form a tower of identity-eliminator-conserved models. The full obstruction fibration is the fibrewise homotopy inverse limit 𝒫∞ = holimₙ 𝒫≤ₙ, and the full bilateral model is 𝔹Id (𝒞, 𝒫∞). Full retained certification is not the same as levelwise finite certification. It requires a compatible thread of finite retained family certificates together with coherent lifting to the homotopy-limit certificate space. The compatibility obstruction CompObs records failure of a compatible finite certificate thread, while the limit coherence obstruction LimObs records the remaining homotopy-limit lifting obstruction. The resulting semantics proves no-bypass for uncertified identity gluts, non-explosion of negative evidence, positive reflection on the exposed non-glutty core, exact obstruction-gated univalence, and invariance of identity-elimination admissibility under identity-eliminator-conservative equivalence. The paper also shows that overlay, cellular, and external cubical models are validated presentations of the same obstruction-fibrational semantics. Directed preserving comparisons give implications, while complete certificate-family comparisons give equivalences. This higher-categorical layer supplies the coherent ambient setting for the later classifying, completion, and synthesis comparisons in the series.