The measure survives structures it was not built on, up to a nameable limit

Run: gate/topologies.py → gate/topologies.json Date: 2026-09-21. Plan §6A item 3 — break the construction confound.

Question or issue resolved

gate/RESULT-LINEARGAUSS.md found delta_spread tracking closed-form integrated information at −0.97 to −0.99 across an order of magnitude in system size, and refused to call it a result for one reason: both quantities were computed on the same two-block construction. One measures what the module boundary costs; the other measures how far a perturbation crosses it. A strong correlation between them could be a discovery about the measure or a restatement of the construction, and nothing in that run separated the two.

This run separates them. The same measure is scored against the same quantity on seven topology families, five of which have no module boundary anywhere for a measure of boundary-crossing to exploit.

The answer is conditional, and the condition is nameable

Spearman correlation of delta_spread against integrated information, within family:

It was not the construction. A chain and a ring have no modules at all, and the relationship holds on both — at n = 60 the chain gives −0.99, the strongest figure anywhere in this program. That disposes of the confound the previous run could not exclude.

It is also not universal. On an Erdős–Rényi graph the relationship vanishes to +0.08, and on a hub topology it reverses to +0.49. Those two failures share a cause, and it is the same cause as the successes. delta_spread measures how unevenly a perturbation propagates. That is informative only where the substrate has structure for propagation to be uneven across: a chain has distance, a ring has distance, a small-world graph has distance with shortcuts. A random graph has none — every node is roughly equidistant from every other, so spread carries no information about integration. A hub is worse than uninformative: a perturbation delivered into a star reaches everything in one step regardless of how integrated the system is, so evenness stops tracking integration and starts tracking the hub.

The measure therefore has a domain of validity, and it is statable in one line: delta_spread inverts integrated information on substrates where propagation is distance-structured, and fails on substrates where it is not. Cortex is distance-structured. A star network is not.

Corrected 2026-09-21 by gate/RESULT-CELEGANS.md. The last two sentences were an assertion about real substrates drawn from constructed ones, and they do not survive measurement. The measured causal graph of a complete nervous system — 23,433 stimulated pairs in C. elegans — is as concentrated in emitted influence as this run's hub family (87th percentile of the whole ensemble) while having no hub at all (top_share 0.032 against the hub family's 0.348), and its response decays with wiring distance at −0.10 against a chain's −0.39. It does not place in this ensemble: five families sit equidistant from it. The domain sentence stands only in its first clause. Whether cortex is distance-structured is not established, and nothing in this file established it.

Size helps rather than hurts, except where the relationship is absent. Every family that works works better at n = 60 than at n = 12 — chain −0.91 → −0.99, ring −0.63 → −0.91, small-world −0.38 → −0.79 — while random_er and hub stay flat near zero and near +0.45. A relationship that strengthens with scale in exactly the families where the mechanism applies, and does not appear in the families where it does not, is the shape a real effect has.

The controls, including the one that limits the result

The positive control reproduces. On two_block this harness returns −0.96 against the previous run's −0.97 to −0.99. Had it disagreed, its verdict on the other six families would have been worthless, and the run refuses to report at all if that check fails.

A disconnected system gives Φ = 0, as before.

The spectral partition is an approximation, and its error is measured rather than assumed. At n = 12 every one of the 2047 bipartitions is enumerated, so the minimum information partition is exact. At n = 60 that is impossible and the Fiedler cut of the coupling graph supplies a candidate instead. It matches the exhaustive minimum in 13 of 21 test cases and overestimates Φ by 21% on average when it misses. The n = 60 column is therefore evidence about a slightly-too-high Φ, not about Φ — which matters least for a rank correlation and is stated rather than buried. The n = 12 column is exact throughout and carries the same verdict.

The conclusion

The program has its first measure with a real, survivable relationship to integrated information and a stated boundary on where it holds. Negated delta_spread tracks integration on distance-structured substrates across seven families, three sizes and two partition methods, and the failures are explained by the same mechanism as the successes rather than by an unexamined confound.

Three things follow.

This is now testable on the register's own data, which is the point. delta_spread is computable from evoked responses the register already holds. The next run scores it on the human deposits against the PCIst values already issued — not as a validation of either, since neither is a ground truth, but to see whether the two disagree anywhere, and where.

The domain of validity has to travel with the measure. A measure that inverts on star topologies cannot be handed to anyone without that sentence attached, and an artificial system was assumed more likely than a brain to be hub-dominated — an assumption gate/RESULT-CELEGANS.md has since removed the evidence for. If this measure ever reaches METHOD.md, the hub failure goes in the same section, not a footnote — and it is a concrete argument for the measurement modality tier of §3.4, which exists to stop exactly this kind of quiet transfer between substrates.

The different-currency work is no longer urgent, and should not be dropped. It was promoted to the main line when nothing worked; something now works on a bounded domain. Surrogate-normalized compressibility and the Loschmidt echo stay on the list as the candidates most likely to cover the substrates where delta_spread fails, which is a more useful reason to build them than having no alternative.