phiid_vs_phi

ΦID-based integrated information (Φ_R, CCS) vs exact IIT-4.0 Φ

The candidate_audit computed integration measures exactly from each network's TPM. This experiment asks the practical question: when integrated information is estimated the way it actually is on data — via Integrated Information Decomposition (ΦID) on a finite time series, using the phyid package — does it still track exact IIT-4.0 Φ?

It bridges the two projects this repo's author has contributed to: PyPhi (exact Φ) and phyid (ΦID).

The measure (phiid_measure.py)

Φ_R — the revised whole-minus-sum integrated information (Mediano, Rosas, Carhart-Harris, Seth & Barrett 2019). For a bipartition (A, B), from phyid's 16 ΦID atoms between the two parts' integer-coded time series:

TDMI  = Σ all 16 atoms
I_A   = rtr + rtx + xtr + xtx      I_B = rtr + rty + ytr + yty
Φ_WMS = TDMI − I_A − I_B           Φ_R = Φ_WMS + rtr

Φ_R corrects Φ_WMS's double-counting of redundancy (Φ_WMS goes negative for purely redundant systems; Φ_R adds the double-redundancy atom back). We use the CCS redundancy (common change in surprisal) rather than MMI: MMI assigns spurious synergy to independent variables, which we confirmed breaks the measure (independent systems scored as highly synergistic); CCS does not.

The system's integrated information is the minimum Φ_R over bipartitions (the minimum-information-partition analog).

Validation

The measure is checked before use:

System	Φ_WMS	Φ_R	meaning
independent	0.00	0.00	no integration ✓
redundant (copies)	−0.39	0.00	Φ_WMS double-counts; Φ_R corrects it ✓
coupled / swap	+1.42	+1.44	genuinely integrated ✓

The atom bookkeeping (I_A, TDMI from atoms) was verified to match directly-estimated mutual informations exactly.

Reproduce

python -m foundations.phiid_vs_phi.run 15 3     # -> results/phiid.csv
python -m foundations.phiid_vs_phi.analyze       # -> summary.csv, phiid_vs_phi.png

See FINDINGS.md for results.