(GQR40) Ordered Accessibility Invariants and Non-Commutative Aggregation (PaperX)

Time-ordered accessibility has recently been identified as a fundamental constraint on ag-gregation, transport, and response in systems with transient or sequential connectivity, for-malised via the Accessibility Ordered Invariant (AOI). Here we show that AOI itself admits ahigher-order invariant when multiple access channels are present and the ordering of access isnon-commutative. This second-order quantity, which we term the AOI-gap, is defined on AOIitself and quantifies the failure of invariant aggregation under reordering.The AOI-gap vanishes under all symmetry-preserving and unordered controls, yet emergesgenerically under accessibility ordering, even in minimal non-physical constructions. We showthat the AOI-gap is independent of specific physical realisation, arising solely from orderingstructure rather than geometry or material detail. This establishes accessibility ordering as apurely mathematical constraint, with abstract geometry and physical systems serving only asrepresentations and instantiations of the underlying invariant structure. Full 70 article programme https://github.com/jamessutton600613-png/GQR/blob/main/Programme_Index.md The programme contains both standalone and self referencing articles. These can all be read as notes in proof of GQR1.