This is Paper 10 in the 20 Paper PHHT Series This paper constructs the finite-retained-store dependent type-former layer of identity-eliminator-conserved paraconsistent homotopy type theory. The calculus is a closed state-indexed bilateral dependent type theory with signed judgments 𝔖; Γ ⊢⁺ t: T, 𝔖; Γ ⊢⁻ t: T, ordinary positive rules for Π, Σ, +, ×, 𝟘, 𝟙, and Id, a predicative cumulative universe hierarchy, higher identity, equivalence evidence, and obstruction-gated univalence. Negative evidence is retained obstruction rather than classical complement. It propagates only through explicitly displayed negative constructors and never supplies an unrestricted bridge into positive derivability. A judgmental state 𝔖 assigns to each formed positive type T a finite retained-negative store RetStore⁻🕕, ⏒ (T). Positive identity elimination along p: IdA (a, b) is derivable exactly when every retained negative witness at IdA (a, b) is discharged by a protected finite certificate family ∏q∈RetStore⁻{𝔖, Γ (IdA (a, b) ) } Cert≤ₙ (p, q). The empty-store case recovers ordinary path induction. Nonempty retained stores require the corresponding certificate family before positive transport, substitution, or identity elimination can proceed. The paper extends the bilateral identity core from Paper IX to dependent products, dependent sums, finite products, coproducts, empty and unit types, universes, universe identity, equivalence evidence, and guarded transport. Universe transport and idtoequiv are derived bridge operations obtained from the same certified identity eliminator. Univalence is two-stage and obstruction-gated. Certified equivalence evidence produces a raw universe path, but that path supports universe transport only after the retained universe-identity store is also certified. Thus univalence does not bypass the retained obstruction gate: equivalence-to-identity and identity-to-transport remain separated by explicit certificate requirements. The main theorem is polarity separation for the closed normal-form signature. Every positive dependence on negative or glutty evidence factors through a named certified bridge carrying its finite certificate family. Consequently raw gluts do not create arbitrary positive derivations, ordinary positive absurdity remains isolated in 𝟘-elimination, and the positive non-glutty fragment is presentation-equivalent to the selected ordinary intensional dependent type theory, or to the corresponding selected ordinary univalent fragment when positive univalence is included. This paper is the syntactic basis for the semantic, higher-categorical, classifying, and normalization papers that follow. It fixes the finite-retained-store rules, dependent type-former discipline, guarded transport mechanisms, universe-identity gates, and obstruction-gated univalence principles used by the later layers of the series.
David Betzer (Mon,) studied this question.