Calibration against exact integrated information, with baseline activity matched

Run: gate/calibration_matched.py → gate/calibration_matched.json Supersedes the correlation claims in gate/RESULT-CALIBRATION.md. Date: 2026-09-20. Method version at time of run: 0.3.5.

The question or issue resolved

gate/RESULT-CALIBRATION.md reported that five of six candidate measures anti-correlated with exact integrated information, and named one confound it could not remove. Those networks used an OR update rule, so densely wired networks sat at high baseline activity and had little headroom left for a perturbation to change. Coupling density was confounded with baseline activity, and baseline activity with response size. The reported anti-correlation was therefore consistent with two explanations — a real property of the measures, or an artifact of where each network happened to be operating — and that run could not separate them.

This run separates them. Every network is driven to the same operating point before anything is measured, so density can no longer act through activity.

The findings

The matching is exact, not approximate. The update rule is a logistic threshold whose midpoint is continuous, which makes stationary mean activity a smooth monotone function of one knob. A bisection drives every network to mean activity 0.500. The activity is the exact stationary value — the eigenvector of the full 2^n x 2^n state transition matrix — not a simulation estimate. Spread across all 66 networks is 3 x 10^-14. Two earlier attempts failed this control honestly and were discarded: an integer threshold could only take a handful of values at these in-degrees and left a spread of 0.811 against a tolerance of 0.08.

Phi is close to a step function of how complete the wiring is. Sweeping the number of edges between the two modules from none to the full complement:

Phi is exactly zero until the wiring is roughly two thirds complete, then rises steeply. This explains a result in the earlier run that had looked like noise: its nine networks with Phi above zero were exactly its nine fully connected ones, and every network with any module structure at all scored exactly zero. The reason is that the minimum information partition finds the module boundary, and a sparse bridge across it is cheap to cut.

The consequence is methodological and it is the more important half of this run. A calibration set of networks this small cannot produce a Phi gradient over most of its range, so a correlation computed across the whole set is largely a comparison against a near-binary complete/not-complete label rather than against a graded quantity. The sweep here reaches the full complement precisely so that a real gradient — 0.25 to 2.77 at n=5 — exists to correlate against.

Once activity is matched, the earlier anti-correlation mostly disappears, and the one measure that had looked promising reverses sign.

Four of six measures lose their relationship with Phi entirely. participation_ratio — the only measure the earlier run identified as tracking Phi positively, and the one it named as the candidate worth pursuing — inverts from +0.44 to −0.49. Only delta_dimensionality keeps a strong relationship, and that relationship is negative: it scores the less integrated networks higher.

Restricting to the 22 networks that have a genuine Phi gradient does not rescue any of them. Within that subset the rank correlations are −0.41, −0.56, −0.62, +0.15, −0.58 and −0.69. None tracks Phi positively.

Controls. PyPhi reproduces its documented example at exactly 2.3125. A disconnected network returns Phi = 0. Activity matching is verified rather than assumed, and the run raises instead of returning a number when the target activity falls outside the bisection bracket.

The conclusion

The earlier anti-correlation was largely an artifact of baseline activity, and should not be cited. The corrected finding is stronger and more negative: none of the six candidate measures tracks exact integrated information once the networks are compared at a common operating point. The measure previously nominated as the positive candidate is anti-correlated once the confound is removed.

Three things follow.

First, gate/RESULT-CALIBRATION.md carries a superseded notice, and the participation_ratio recommendation is withdrawn before it reached anything citable. This is the validation ladder in the plan's §6A working as intended — the claim was caught at the Exact rung, which is the only rung where it could have been caught.

Second, the near-step behavior of Phi is a real constraint on this whole line of work, not a defect of this run. Calibrating anything against exact Phi requires networks large enough for Phi to vary smoothly, and at n ≤ 5 it does not. The next calibration needs either larger networks — which the O(n^5 3^n) cost forecloses beyond about 10 to 12 nodes — or a constructed ground truth that is graded by design rather than discovered to be binary after the fact.

Third, the open question the plan should now carry is not "which of these six measures tracks Phi" but "does any measure in this family track Phi at all." On the evidence here the answer is provisionally no, and a measure in a genuinely different currency — surrogate-normalized compressibility, or the Loschmidt echo — is the more promising place to spend the next run.