A measure that tracks integrated information at −0.98, across an order of magnitude

Run: gate/lineargauss.py → gate/lineargauss.json Date: 2026-09-21. Plan §6A item 2 — constructed ground truth at scale, graded by design.

Question or issue resolved

gate/RESULT-CALIBRATION-MATCHED.md established that exact integrated information is close to a step function of how complete a network's wiring is, and that the O(n⁵3ⁿ) cost of computing it forecloses going past ten or twelve nodes to escape that. The consequence was that no measure could be scored against a gradient, because over most of the range there was no gradient — only a complete/not-complete label wearing a number's clothes.

This run supplies the missing gradient. For a linear system driven by Gaussian noise, every entropy in the definition of integrated information is a log-determinant, so the quantity has a closed form and costs a few Cholesky factorizations rather than an exponential partition search. Cross-block coupling becomes a continuous knob, and the answer comes out graded instead of switched.

Two controls failed before any result existed, and both failures were real

Unnormalized integrated information is minimised by the most unbalanced cut. The first run's minimum information partition at n = 8 was a single node, at 0.4204 against the constructed module boundary's 1.2155. Severing one node costs less information than severing half the system, so the minimum was a statement about which cut had the fewest edges crossing it rather than about where the system was least integrated. Normalizing by the size of the smaller part is the standard correction and fixes it.

A construction is not a ground truth if random weights can move its boundary. With normalization applied, the control failed again — minimum at [0, 2, 3] rather than the designed 4/4 split. The cause was that within-block weights were drawn from a normal distribution, so some nodes were only weakly tied to their own module and the genuinely weakest cut was not the designed one. Every correlation computed against that would have been against whatever cut the random draw happened to produce.

The fix is uniform weights, which make the boundary the weakest cut by symmetry, with jitter reintroduced as a measured quantity rather than an assumption. The constructed boundary survives jitter up to 0.8; this run uses 0.05. All three controls now pass: a disconnected system gives Φ = 5.6 × 10⁻¹⁷, the constructed boundary is the minimiser across all 127 bipartitions at n = 8, and Φ is monotone in coupling.

The gradient exists, and it is smooth

Nine distinct levels at every size, monotone, with no step anywhere. This is what the discrete construction could not provide at any size, and it is the thing that makes the scoring below mean something.

One measure tracks it almost perfectly, with the sign inverted

Spearman rank correlation against closed-form Φ, within each size:

delta_spread holds −0.97 to −0.99 across an order of magnitude in system size. Nothing in this project has produced a relationship of that strength or that stability before.

The mechanism is legible, which is why it is worth taking seriously rather than treating as a coincidence. delta_spread is the variability across nodes of the mean response to a perturbation. In a strongly integrated system the perturbation reaches everything and the response is even, so the spread is low. In a modular system the perturbation stays largely inside the module it was delivered to, so the spread is high. The measure is tracking how unevenly a perturbation propagates, and that is the inverse of what integration means. Negated, it is the first positive candidate this line of work has produced.

participation_ratio reverses sign with system size — −0.46 at n = 10 to +0.47 at n = 100 — which disqualifies it independently of direction. A measure whose sign depends on how big the system is cannot be calibrated against anything.

Pooling across sizes inverts the answer

delta_spread correlates at −0.97, −0.99 and −0.98 within each size and at +0.11 pooled across all three. The sign reverses because Φ's magnitude falls with n — the normalizer grows — so pooling mixes a near-perfect within-size relationship with a between-size confound and returns something close to noise. This is Simpson's paradox, and a pooled figure would have hidden the strongest result here rather than revealing it.

The discrete calibration was checked for the same defect and does not have it: at n = 4 and n = 5 its within-size correlations sit within 0.1 of its pooled ones for every measure but delta_lz, whose flip is noise on a nine-system sample. gate/RESULT-CALIBRATION-MATCHED.md needs no correction. From here, correlations are reported within size and pooled figures are not reported alone.

The conclusion

The closed-form rung works, and it is now the workhorse this program needed: a graded ground truth, verified against exhaustive partition search where that is possible, costing milliseconds at a hundred nodes where the exact computation cannot run at thirteen.

The candidate is negated delta_spread, and it is not yet a result. Both Φ and delta_spread derive from the same two-block construction — one measures what the module boundary costs, the other measures how much a perturbation respects it — so a strong correlation between them may restate the construction rather than discover anything about the measure. That confound is the whole question now, and it is testable: the measure has to hold its relationship on structures that are not two blocks. Ring and small-world topologies, graded random graphs, and three or more modules are the next run, and if the correlation survives structures with no module boundary to respect, the measure is real. If it collapses, it was the construction talking.

Two of the three findings here came from controls failing rather than from the sweep succeeding. The unbalanced-cut bias and the moved boundary would both have produced confident, plausible, wrong numbers, and neither would have been visible in the output. That is the third time in this program that the informative result arrived as a control failure, which is an argument for spending the effort to write controls that can fail in interesting ways rather than ones that merely confirm the code ran.