11.10 Counterfactual hyperemergence-dependence criterion

A rigorous criterion can be expressed through controlled counterfactual removal. Let C be a particular crossfunctional output and let ActiveInputs(C) denote the task-time neural inputs actually used to produce it. C qualifies as Hyperemergent when at least one active input e is a Higher-Hyperfunction-generated emergent dimension and removing or downgrading that dimension, while holding the rest of the relevant architecture and task conditions fixed as far as conceptually possible, removes or changes at least one defining property of C.

HEC(C)=1 iff exists e in ActiveInputs(C) such that e in E_phi,h for some h>=2 AND C_without_e != C

The inequality C_without_e != C does not require total disappearance of the whole crossfunction. Hyperemergence may concern one irreducible property of an otherwise recognizable output. If removing e preserves the broad output category but deletes its novel precision, adaptability, reliability, multidimensionality, integration-rule, or other defining higher-order property, then that property is hyperemergent.

The same criterion can be applied by order-downgrade rather than single-dimension deletion. Replace an H_h contribution with a lower H_j contribution, j<h, while preserving the same function-kind. If a relevant crossfunctional property vanishes because the lower order lacks the required emergent dimensionality, that property is order-dependent Hyperemergent Crossfunctionality.

C[Expanded(H_h(phi)), X] != C[Expanded(H_j(phi)), X] for some j<h -> h-dependent hyperemergent contribution