The Algebra of Identity-Eliminator Obstruction Operations in Bilateral HoTT

This is Paper 15 in the 20 Paper PHHT Series This paper defines the algebra of obstruction operations governing guarded identity elimination in identity-eliminator-conserved bilateral homotopy type theory. A retained identity glut is positive identity evidence p: IdA (a, b) together with an explicit finite retained obstruction store R ∈ RetStore⁻_Γ (p). Such a glut may be used for positive identity transport only through the protected family-certificate premise of the guarded identity-elimination rule. The paper resolves aggregate certification into a dependent obstruction tower. An extension-complete identity-eliminator obstruction structure assigns to each retained-store glut (p, R) a sequence of pointed fibrant obstruction targets Obₖ (p, R;η_<k), obstruction operations oₖ (p, R;η_<k) ∈ Obₖ (p, R;η_<k), nullity types, filler spaces, lower-filler groupoids, and zero-preserving indeterminacy actions. Each stage after the first is defined only after compatible lower fillers have been chosen. The finite tower-certificate space is identified with the telescoping dependent sum of exact nullity types: TowerCert≤ₙ (p, R) ≃ ∑η<n: TowerCert≤ₙ₋₁ (p, R) isNullₙ (p, R;η_<n). Thus finite certification is coherent recursive nullity, not independent stagewise vanishing. Aggregate obstruction data enter guarded identity elimination only through a specified aggregate-to-family realization map. In the presence of interaction terms, realization first lands in the interaction-closed family gate Θₑ, +: TowerCert (p, R) → FamCert (p, R⁺), R⁺ = R ⊔ Int (R), and projection to the retained-store gate is controlled by the protected non-interaction certificate. Under realization completeness, exact-lifting null-reflection, and a complete certificate comparison package, aggregate, tower, and retained-family certificates agree at the declared retained dimension. Full certification is the homotopy inverse limit of the finite certificate tower. In the standard pointed fibrant Milnor range, the lim¹ term measures the obstruction to assembling compatible finite certificate components into a full certificate. Cellular, cubical, coefficient, spectral, quotient, finite detected, and completion presentations are treated as comparison presentations of the same recursive obstruction algebra. Preserving comparisons give forward detected certification, while reflecting comparisons are exactly the data needed to reconstruct retained protected certificates. This paper supplies the operation-algebra layer connecting the earlier obstruction towers, secondary and polyhedral operations, finite detected calculus, semantic certificate gates, and proof-theoretic normalization discipline. Its detected-to-retained bridge is the interface used by the later coefficient theory, spectral sequence machinery, completion/localization theory, and final synthesis of the series.