pyphi.formalism.iit4.bounds.sum_phi_relations_measured_bound

pyphi.formalism.iit4.bounds.sum_phi_relations_measured_bound(distinctions)

Certified upper bound on Σφ_r from a measured distinction set.

Evaluates the per-atom linear-program maximum (Eqs 13-14) on the measured profile: each atom o — a unit in a state, drawn from the distinctions’ purview unions (Eq 10) — receives the total φ density S(o) of the distinctions containing it, and contributes at most S(o)·(2^k − 1 − k)/k, where k is the number of those distinctions. The self-relation sum is carried exactly (Eq 9), a strict tightening of Eq 15. Cost is O(D·n) for D distinctions over n units; no relations are enumerated.

Parameters:

distinctions (iterable) – The distinctions of a Φ-structure. Must be a resolved set — each mechanism at most once; duplicates would contribute their densities twice.

Returns:

UpperBound – Certified. Unlike the size-based bounds, this holds with no binary-units, TPM-factorization, or measure assumption: the theorem is combinatorial over the relation formula and the non-negativity of φ, so there is no config domain guard.

Return type:

UpperBound

Notes

Atoms are state-tagged units, exactly the objects in each distinction’s purview_union.

The certificate holds because the true per-atom contribution (the inner sum of Eq 11) satisfies the linear program’s budget constraint (Eq 13) with equality at the measured S(o), so it cannot exceed the program’s maximum (Eq 14). For a complete distinction set the exact Σφ_r is available at the same cost via pyphi.relations.AnalyticalRelations.sum_phi(); this function’s value is the auditable certificate, directly comparable to the worst-case ceiling of sum_phi_relations_upper_bound().

A group of more than 1023 distinctions sharing an atom overflows the 2^k weight to inf, a valid (if uninformative) ceiling.