Five of six measures read integration backwards where integration is known

SUPERSEDED, 2026-09-20. The correlation results below are confounded by baseline activity and should not be cited. gate/RESULT-CALIBRATION-MATCHED.md repeats the run with activity matched exactly across structures: four of the six relationships disappear, and participation_ratio — nominated below as the one measure tracking Phi positively — reverses sign. The method, the controls and the near-binary Phi observation below stand; the correlations do not.

Run 2026-09-20. Code: gate/calibration.py. Output: gate/calibration.json (27 networks). Ground truth: exact big-Φ via PyPhi 1.2.0.

Question or issue resolved

Every measure this register might use has been justified by argument or by agreement with other measures, never against a known answer, because for brains no known answer exists. gate/RESULT-MEASURES.md screened seven candidates against four criteria and could say which behaved sensibly, not which were right.

For a network small enough that exact integrated information is computable rather than approximated, Φ is arithmetic. This builds that answer key and scores the candidates against it.

The answer key

27 networks at 3, 4 and 5 nodes, in three structures: one integrated block; two modules with sparse cross-coupling; two modules with none. Each is read twice — exactly from its transition table, and by simulated perturb-and-record exactly as the register reads a brain.

Controls, both passed before anything was recorded: PyPhi reproduces its own documented example at 2.3125, and a fully disconnected network returns Φ = 0.

Φ turned out to be near-binary across this construction. Every modular network returns exactly 0, including the weakly coupled ones that genuinely carry one to three cross edges. Integrated networks return 0.2712 at four nodes and 0.7648 at five. The minimum information partition finds the module boundary and a sparse bridge is cheap to cut, so the key discriminates rather than grades: 9 networks with Φ > 0 against 18 with Φ = 0.

The result

AUC 1.00 is perfect separation of integrated from modular; 0.50 is chance.

Five of six measures are anti-correlated with integrated information. They do not merely fail to detect it — they score modular networks higher than integrated ones, at AUC 0.15 to 0.28. Only the participation ratio points the right way, and weakly.

This is the same mechanism found twice already, now against exact ground truth rather than intuition. Integration makes a response more redundant across units; redundancy compresses well; compression-based measures score it low. A modular network's response is less coupled, less compressible, and scores high. gate/RESULT-MEASURES.md showed that phase-shuffling raised these measures. gate/RESULT-ARTIFICIAL.md showed that weak coupling raised them. This shows that the absence of integrated information raises them, where integrated information is computed exactly.

The confound, which must be removed before the result is trusted

δ magnitude reads 0.26 at Φ > 0 against 3.15 at Φ = 0 — the perturbation propagates less in the integrated networks. That is unlikely to be a fact about integration and likely a fact about saturation.

These networks use an OR rule with noise: a node turns on if any input is on. Dense coupling therefore drives high baseline activity, and a network already mostly on has little headroom for a perturbation to change. A modular network has lower drive, more headroom, and a bigger response — for reasons that have nothing to do with what anyone means by integration.

So coupling density is confounded with baseline activity, and baseline activity with perturbation headroom. Until baseline activity is matched across structures, the anti-correlation above is consistent with two explanations and this run cannot separate them. The next iteration holds mean activity constant by tuning thresholds per structure.

This is the same class of defect as the noise model in RESULT-ARTIFICIAL.md — an observation property of the simulation masquerading as a property of the thing measured — and it was found the same way, by reading a number that pointed the wrong way and asking why rather than reporting it.

What is already safe to say

The confound affects the magnitude measures most directly. It does not explain why a measure of complexity should rank a two-module system above a single integrated one, which is a claim about structure rather than about response size. And it does not touch the phase-shuffle result at all, since shuffling holds the system fixed.

The defensible statement after this run: on systems where integration is known exactly, the compression-based family does not track it, and at least one plausible reading is that it tracks the opposite. Establishing which requires the activity-matched replication, which is a day.

Limitations, stated

• 27 networks, 3 to 5 nodes, one update rule, binary states. Nothing here generalizes to a brain by itself; the point is that a measure failing here has been falsified cheaply, not that one passing here has been validated.
• Φ is near-binary in this construction, so the comparison is two-group rather than graded. A Φ gradient needs coupling strength inside a dense graph rather than sparse bridges between modules.
• The baseline-activity confound above is unresolved.
• PyPhi 1.2.0 required three one-line compatibility patches for Python 3.11 (collections.Iterable and kin moved to collections.abc in 3.10). No computational path was altered, and the documented example reproduces exactly, which is why that control exists.