In February 2043, K. Solheim (Department of Informatics, University of Oslo) and T. Amankwah published the theoretical paper that supplied the missing piece of the architecture two years before any of it was built. It takes the objection Amankwah and Gustafsson (2041) had left unanswered, that local rules cannot produce globally coherent structure, and argues that the objection holds only for graphs whose resources are unconstrained. Impose a fixed global budget and the coupling a global objective would have provided appears as a consequence of the constraint itself.
The Objection, Restated
The paper opens by granting the objection fully rather than softening it. A rule depending only on quantities available at the site of the update has no channel by which one region of a graph learns anything about another. Accumulated over a developmental run, the expected result is a collection of locally sensible arrangements that do not compose.
The authors then identify an assumption inside the objection that had not been noticed, because it is true of nearly every learning system the field had built:
“The objection assumes that the resources a system can allocate are unbounded, or bounded only by cost. Under that assumption it is decisive. We do not think the assumption is necessary, and when it is dropped the objection does not weaken. It inverts.”
The Fixed-Budget Constraint
The paper’s central claim is that a conserved total converts every local decision into a global one. If the sum of allocation across the graph is constant, then strengthening any pathway necessarily starves another, and no region can reorganize itself without altering the allocation available everywhere else. Connectivity, in this framing, is not the coupling mechanism. Scarcity is.
The authors credit the observation to the substrate they had both been working with rather than to themselves. Physarum polycephalum tube reinforcement, the dynamics Qian et al. (2037) had grafted onto Lenia-class morphology, conserves total tube volume as a property of the organism’s physics, not as a modelling convenience. The paper’s argument is that this conservation is not incidental to why the composition produced adaptive foraging behaviour. It is the reason.
“The conserved budget does the work that a global objective is usually introduced to do, and it does it without an objective, without an output, and without any signal entering the graph from outside it. We had been reading the tube-diameter constraint as a detail of the biological model. It is the mechanism.”
Two Timescales
The architecture the paper proposes has two, and the separation between them is what keeps them from interfering.
The fast rule is where Amankwah and Gustafsson (2041) left it: an update at each site depending only on the recent covariance of the activity on either side of it, continuous, with no target value and no error signal. The slow rule samples the graph’s activity at a fixed interval and reallocates the conserved budget in proportion to recent activation covariance, strengthening pathways that carry signal and withdrawing from those that do not.
The paper argues the separation is required rather than convenient. The fast rule’s covariance estimate is only meaningful over a window in which the graph’s gross allocation is stationary; if both rules moved on the same timescale they would each destroy the other’s precondition. It gives a worked stability analysis and concludes that the ratio between the two intervals, not either interval’s absolute value, is the parameter that matters.
Composition with a Connectomic Prior
The paper’s closing section takes up the step Qian et al. (2037) had named and left unexplored: composition with connectomic constraint data. Solheim and Amankwah show that a conserved-budget allocator composes with a static connectivity prior without either displacing the other, because the prior sets the initial distribution of the budget and the allocator only moves it. They predict that a physical implementation initialized from a biological scaffold would retain the scaffold’s gross topology while reallocating freely within it, and that the retention would be a consequence of the initial distribution rather than of any constraint on the allocator.
They note the limits of what they are claiming. The result is analytic, assumes idealized cells, and depends on the conserved quantity remaining conserved, which they flag as the assumption most likely to fail in hardware.
Predictions
Four, stated as checkable:
- A system built to this architecture develops without a loss function present anywhere in it.
- Its developmental path is a function of the sequence it is exposed to, and is not reproducible across instances. The paper gives a logical argument for this and notes that a physical substrate would add a second, independent source.
- Switching speed is not the constraint. Elapsed developmental time will be set by how many slow-timescale cycles have to complete, each of which requires the fast rule to settle into a stable distribution first, and not by the throughput of the fast rule. Compression, where it occurs, comes from the architecture and not from faster hardware.
- Composition with a specific connectomic prior is not neutral: two systems initialized from different biological scaffolds and exposed to the same sequence should diverge in gross structure, not merely in detail.
Institutional Context
The work was supported by a Norwegian research council grant and the Ålife Working Group’s open-release arrangement, with no industrial partner. The paper’s acknowledgements record that the collaboration began after Amankwah read the Qian et al. future-work note and wrote to Gustafsson, who introduced him to Solheim.
The 2045 Connection
The 2045 neural core implements this paper’s proposal directly. Its decisive hardware component is a co-processor that reallocates a fixed global budget at a configurable interval, default 100 μs, in proportion to recent activation covariance, while node-level conductances move continuously on the fast rule. Fennimore et al. (2045) describe the arrangement in the terms this paper established.
Prediction 2 was given its physical basis in Bhaskaran et al. (2036), which had already documented drift and read noise in MORSe arrays: two instances developed under an identical protocol on identical hardware cannot occupy identical states, whatever the logical argument says. Prediction 3, that compression is architectural, is the reading the 2045 disclosure’s power and load figures support.
Both authors appear on the Fennimore et al. (2045) preprint byline. The relation between K. Solheim and the later xenobiologist of the same surname is not recorded.