Admissible Projection, Nonlinear Memory, and Weak Observation

ithin the Universal Grid Mechanics (UGM) framework, admissible states are finite, continuous, bounded under update, and memory-bearing. This paper constructs the conditional observation-sector closure by defining the admissibility-intersection operator that maps hidden admissible state structure into rank-three observable coordinates on certified resolved channel domains. We define the pointwise admissible fiber D0, the admissible state-structure domain D_Omega, the finite reachable update tube R_Delta(X), internally generated observation-channel constraints C_i, the intersection core I_C(X), the minimal-burden selector Sel_C, and the rank-three coordinate map Q^(3)_C. The resolved observation operator is: Pi^(3)_obs(X; C) = Q^(3)_C(X_C), where X_C in Sel_C I_C(X), with single-valuedness holding exactly when the admissible selector has a unique minimizer and rank DQ^(3)_C(X*_C) = 3. The construction explicitly builds the channel admissibility topology tau_C and marks selector existence as conditional on compact-regularity of the reachable tube. It classifies three regimes: inadmissible joint channel, resolved rank-three observation, and admissible but unresolved observation. The non-generation property holds by construction because selection occurs only inside the finite reachable admissible tube. Version 1.04d restores the corpus-continuity title while preserving the hardened conditional scope. It hardens the paper as the corpus observation-interface reference. The admissible selector class Q_adm remains owned by this paper and every admissible selector is scoped as q_i = q_i[S, M, Pi_M, L6, R_Delta]. The required selector-admissibility conditions are boundedness preservation, memory compatibility, non-generation, L6 support compatibility, projection continuity, and domain certification compatibility. The revision adds explicit export rules for channel-certificate targets Pi^(L)_obs, Pi^(M)_obs, and Pi^(G)_obs. AIO exports the certification grammar only; it does not close any named channel certificate inside this paper, and no downstream channel becomes canonical unless its nonempty intersection, singleton selector, rank-three readout, and coordinate-status certificate are separately supplied. Version 1.04d also formalizes relay certification and burden grammar for conditional observation-chain applications. Relay counts, burden decompositions, and the projection factor Lambda_3 ≈ 0.97 remain application-layer objects. They are forbidden from entering the admissibility-intersection core, selector geometry, pseudometric, or rank-three readout. The bibliography and reference audit now use concept DOIs from the supplied corpus ledger whenever a Zenodo latest-version DOI differs from a displayed uploaded-record DOI by one in the final suffix.