Normalization and Cut Elimination for Identity-Eliminator-Conserved Paraconsistent HoTT

This is Paper 14 in the 20 Paper PHHT Series This paper proves relative normalization and cut elimination for identity-eliminator-conserved bilateral paraconsistent homotopy type theory. The positive layer is an ordinary HoTT fragment with identity elimination, while the negative layer records retained obstruction data over identity witnesses. The proof-theoretic constraint is that retained obstruction over an identity datum remains explicit during internal identity transport. For a positive identity witness p: IdA (a, b) and a finite retained obstruction store R ∈ Ret⁻_Γ (p), internal bilateral transport is permitted only through a protected proof-theoretic product family certificate η: FamCertᵖt (p, R) = ∏ₐ∈ₑ Cert (p, q) and the guarded eliminator IdElimᵍC (d;p, R, η): C (a, b, p). Raw retained obstruction does not by itself license positive transport. Aggregate certificates refine the product-family gate only through the displayed projection ρR: AggCert (p, R) → FamCertᵖt (p, R). Detected, spectral, cellular, quotient, tower, and finite-stage certificate presentations enter the proof theory only through explicit reflecting comparisons into FamCertᵖt (p, R), or through complete equivalences with that proof-theoretic gate. Transfer to the full StoreCompat-inclusive retained-family gate requires the corresponding compatibility comparison. Under the stated relative hypotheses on the positive HoTT fragment, negative obstruction closure, protected certificate calculus, support grading, subject reduction, commuting conversions, and certified-block reductions, every derivation reduces to a separated block-normal form. Such a normal form consists of a normalized positive spine, negative obstruction normal forms, protected certificate normal forms, and localized product-family-certified identity-elimination blocks. The resulting proof theory establishes syntactic non-explosion and no-bypass. Raw identity gluts are inert for positive transport; arbitrary positive conclusions cannot be derived from simultaneous positive and negative evidence. Every effective negative dependency in a positive derivation factors through a certified identity-elimination block with an explicit family certificate. The support grading gives positive-core conservativity. If a projectable positive judgment has support zero, its bilateral derivation projects to an ordinary positive HoTT derivation. Thus support-zero positive judgments are conservative over the ordinary positive fragment. This paper supplies the proof-theoretic normalization and cut-elimination layer for the bilateral obstruction framework. Its results are used by the later obstruction-operation algebra, protected certificate completion and localization theory, and final synthesis architecture of the series.