BU78 Projection-Stratum to Higher-Gradient Interface Theorems Material Order, Consciousness Order, and Dual-Order Interface Structure

This paper develops BU78|Projection-Stratum to Higher-Gradient Interface Theorems as a structural interface file within the B_U sequence. Its central claim is that the projection stratum cannot directly pass into higher-order gradients by simple extension, accumulation, formal equivalence, or subjective elevation. The transition from projection-level readout to higher-order retained standing is governed by a dual-order interface structure: the material order and the consciousness order. These two orders are mutually dependent and mutually constraining. A purely material account remains trapped in observable substructures, measurement windows, local perturbations, and finite formal envelopes. A purely consciousness-side account risks floating in readout, meaning, narrative, or high-order experience without carrier, boundary, feedback, cost, and settlement. Higher-order standing requires both orders to return to the same total reality operator system under B_U. The first theorem is the Interface Insulation Theorem of the material order. It states that finite observation, local measurement, microscopic perturbation, and formal modeling can obtain valid projection-level readouts, but they cannot directly acquire non-observable invariants of the whole structure. In symbolic form, the observable substructure remains strictly smaller than the full admissible structure. The key material-side judgment is that perturbation covariance cannot cross strata directly. Microscopic disturbance can be measured, averaged, zeroed, filtered, modeled, or locally aligned, but it cannot become higher-order retained standing merely by extension. Interface insulation therefore prevents the projection stratum from reverse-occupying total reality through local observable fragments. The second theorem is the Interfacing-Agent Sparsity Theorem of the consciousness order. It states that high-order readout may occur, but the agents capable of stably accessing, compressing, expressing, acting upon, returning, and settling such readout into a retained chain are necessarily sparse. Ordinary readout, local experience, narrative reaction, formal expression, and identity-level interpretation can occur widely. Stable higher-order interfacing requires a much narrower chain: readout, meaning compression, direction selection, residual clearing, action binding, feedback return, and settlement. The sparse distribution of such agents prevents high-order readout from being misclassified as automatically settled standing. BU78 then establishes the mutual constraint between the two orders. The material order provides carrier, boundary, measurement, cost, feedback, and constraint. The consciousness order provides readout, meaning, direction, selection, amplification, and return. Each order fails when isolated. Material order without consciousness-side retention produces local measurement without higher-order direction. Consciousness order without material-side carrier produces ungrounded readout without settlement. The higher-order retained chain must therefore be read through the total reality operator system, where matter, information, consciousness, feedback, causality, residual stratification, and settlement unfold as one closure process. The final judgment is that the projection stratum reaches higher-order gradients only through disciplined interface conditions. Material-side interface insulation blocks illegitimate cross-stratum projection. Consciousness-side agent sparsity blocks automatic elevation of high-order readout. Together, they define the dual-order gate through which higher-order retained standing becomes possible.