The Operational Unit: an actor–agent model of information economies, and what our engine runs of it today

1 · Why now

This week a business column argued that the adoption of AI is outrunning trust in it, and a survey of executives found that 78 % lack strong confidence they could pass an independent AI governance audit within 90 days. In the same week a payment network said an AI model had exposed serious weaknesses in its cyber defences, and in Europe the transparency duties of the AI Act have applied since 2 August 2026. Each of these asks the same question of a system that learns: what does it keep, what does it want, and what does it put back into the world?

This paper offers one model for those three questions, written so that each part can be checked on your own data, and it reports first where the model's own rules failed their tests.

What this paper contributes, losses first, each with its basis and the sections that carry it:

1. Measured in the model, one run each. Five registered tests of the model's own rules were falsified on our fixtures: the reward is blind to correctness, the value update does not settle on it, stillness is not confidence, the certificate is not stability, and no broker keeps four properties at once (sections 4 and 5).
2. Measured in the model. Four registered tests of the model's readings were falsified on the engine's records (sections 4 and 5).
3. Prediction. That choosing the next reading by expected information lowers uncertainty faster than a random or fixed order: registered, not run (section 12).
4. Reading. The Operational Unit as a modelling convention: its account, its reward, its belief and its emission rule in standard notation (sections 2 to 4).
5. Measured, one run. Our engine's public evaluation, worst row first, and with no discount the rewards along any path summing to the uncertainty removed (sections 4 and 9).

2 · The claim, stated as a hypothesis

We use a modelling convention, not a finding about the world: a chatbot that learns, a person, and a physical system can each be described in one vocabulary, an Operational Unit. The unit is at once an actor, which exploits what it already knows, and an agent, which explores what it does not. It keeps an account. It reads its environment. It emits into its environment. And it is rewarded by the change in its own uncertainty.

We call the universal unit the OUA-A: the Operational Unit, Actor–Agent.

A convention makes no claim that can be measured on its own, so no registered test of it is reported; section 9 says exactly what was measured. A published hypothesis of similar reach is the free-energy principle (Friston 2010); it is compared here for scope only, and this convention claims less: one vocabulary, not one principle. Nothing in this paper is a physics claim, and nothing in it is a claim about how human minds work.

3 · The unit

An Operational Unit has four parts. Its capabilities are its latent needs: the things it could do, felt as a gap until it does them, such as operations, strategy, portfolio and project management. Its body of knowledge is a set of internal ontologies that support its growth. Its governance model is its policy, its structure, and its value map: its vision and the niche it has chosen in its ecosystem. And it holds a state in each economy it takes part in: beliefs, values, experiences, services, goods and commodities.

Every unit keeps an account, and the account is updated recursively: each new entry is written on top of everything before it. The unit's tactical actions set the magnitude and the direction of each update. What the unit works toward is alignment: bringing what it expected its account to do into line with what the account actually did.

4 · The signal

The unit's behaviour comes down to three lines, and each can be checked on your own data. Registered tests of each line, simulations on our fixtures that test the model's rules and not nature, come first.

What the tests of these lines found. The reward cannot tell right from wrong: on the committed question slice, answering every unanswerable question and declining every one both earn 1 bit, while a proper score gives them 1 and 0. On that reward the value update does not settle: in the last hundred episodes 64 % of states still change their best action, 6.2 times the rate at mid-run, and the convergence results the paper cites do not apply to a reward that depends on the belief. A belief that has stopped moving is not a confident one: with a wrong pair of hypotheses, the belief settled on the wrong one in 50 of 50 seeds. The emission rule and the entropy gate it is read beside were registered as one gate; they agree on 63 of 200 beats, so that test is falsified. What held: with no discount, the rewards along a path add up to the uncertainty removed between its ends in all 50 seeds, and with a discount of 0.9 the order matters in all of them.

The reward is the uncertainty the unit removed, which measures how settled the account is, not whether it is right. Write the unit's account as a probability distribution over the outcomes it is tracking, and measure its uncertainty with the Shannon entropy of that distribution:

H(p) = − Σᵢ pᵢ · log₂ pᵢ

r = H(stateₙ) − H(stateₘ)

Here pᵢ is the share of the account on outcome i, stateₙ is the account before an action, and stateₘ is the account after it. The reward r is measured in bits. It is positive when the action left the unit more certain, zero when it changed nothing, and negative when it left the unit more confused.

The policy is defined by that signal, and the action is a value choice. The unit's policy is whatever climbs the expected reward, which is the standard policy gradient with the entropy reward as its signal. Its action is the one with the highest learned value in its current state, and that value is updated from the reward it receives:

∇J(θ) = E[ ∇ log π(a | s) · G ]        G = r + γ · r′ + γ² · r″ + …   the return

a* = argmax over a of Q(s, a)

Q(s, a) ← Q(s, a) + α · ( r + γ · max over a′ of Q(s′, a′) − Q(s, a) )

Here π is the policy, with settings θ; s is the state and a an action; Q is the learned value of taking an action in a state; α is how fast the unit learns and γ how much it values the future. With γ at one, the entropy rewards along a path add up to the uncertainty removed between its start and its end, whatever the path; with γ below one, an early drop in uncertainty counts for more than a late one.

The published results that this value update settles assume a reward that depends only on the state and the action. The entropy reward also depends on what the unit believed before, none of them covers it, and the registered test above found it does not settle.

Confidence is how little a belief is still moving. The unit's belief is updated by Bayes' rule each time evidence arrives, and its error is the distance between the current belief and the last one:

bₙ(h) ∝ P(dₙ | h) · bₙ₋₁(h)

eₙ = ‖ bₙ − bₙ₋₁ ‖

Here h is a hypothesis, dₙ is the new evidence, bₙ is the belief after it, and eₙ is the error. A small error says the belief has stopped moving, not that it is right: confidence needs a proper score beside it. Our model also gives this step a second form, which scores a belief against what happened: how far a prediction landed from what was observed, as a share of the interval stated with it. The engine does not yet write it.

A unit emits when its own entropy has reached its environment's. The registered test that this rule and the entropy gate are one gate was falsified, as above. Emission is when a unit puts something into its environment, such as a product, a message or a trade:

emit when H(system) ≥ H(environment)

A worked example on public numbers. Our evaluation page reports what our answering engine did with questions a contract file cannot answer. At the reading before a fix, it declined 75 % of them; after the fix it declined 20 of 20. Take the unit's account as the split between declining and answering on those questions. Before the fix the split was 0.75 and 0.25, which is 0.811 bits of uncertainty. After it the split was 1 and 0, which is 0 bits. The reward for that round of work is therefore 0.811 bits. You can repeat the arithmetic from the page alone. The same arithmetic pays 0.811 bits for a move to answering every one of those questions: the reward measures how settled the account is, not whether it is right.

5 · Brokers and economies

The losses come first here too. The walk-away certificate certifies that no one ends up worse off, not that a split is stable: on 2,984 games with a non-empty core, 1,408 served splits lie outside the core while the certificate reads clean. No broker keeps all four of the properties a market asks for at once, efficient, truthful, individually rational and budget-balanced: through the served prices, 47 % of buyers gain by misreporting. And the registered test that a broker's centrality reads as its margin agreed on 22 of 120 planted stars, against 95 % registered, so it is falsified; a fixed reading order was 10.9 % slower than a random one, so the test that they cost the same is falsified too; and the test that the account's writer has a direction read it on 12 of 24 series, against 0.9 registered. Each is one run of a registered test.

Units do not line up in neat chains. They align in non-linear ways to meet the demands of an environment, finding niches and exploiting them. Some units become brokers: a meta-layer of units that sits on top of the others and carries the patterns of information between them. Each broker has a degree of central tendency, somewhere between fully centralised and fully decentralised.

Two classical ideas describe the two scales. At the micro level, units are read as settling into a Nash equilibrium: a state in which no single unit can do better by changing its own choice while the others keep theirs. At the macro level, read the same way, the network aims for a Pareto equilibrium: a state in which no unit can be made better off without making another worse off. On embark.day the checkable part of this is served as verbs: decisions.frontier returns the options that no other option beats on every objective at once, and planning.price and decisions.allocate_fair return a split with a certificate that shows, against the walk-away values you send, whether anyone ends up worse off. The goal of building a system of units is to model the brokers; since no broker keeps all four properties, each must say which one it gives up.

Corrected 11 October 2026. The text of 7 to 11 October named the macro reading above Pareto efficiency, the textbooks' name for the property its sentence defines. The paper's published name, a Pareto equilibrium, is restored until a change of name is decided; the property defined is the same.

6 · The space the unit lives in

Each unit's state has fourteen coordinates, in two blocks.

Eight coordinates describe the unit's neighbourhood over time. This is the outer block: how smooth its motion is, how far off-centre it runs, how fast it changes from one step to the next, a sample of its entropy, its mean position, the correction at its boundary, how many degrees of freedom are active, and how divided it is.

Six coordinates describe the unit itself. This is the inner block: position, velocity, acceleration, jerk, snap and crackle. The first three are position-like and the last three are momentum-like.

Two levers summarise the blocks. The engine also keeps a summary record of each unit with fourteen entries, and the last two are levers. The thirteenth is how settled the unit is: its coherence, the readout of the inner block. The fourteenth is its leverage: how far its expected next state is displaced, multiplied by that coherence.

These are the coordinates. This paper does not say how they are computed from data, or how the engine moves in them.

An analogy, and only an analogy. A reader may find it natural to picture a gravitational well that everything nearby falls toward, where the unit is what the well keeps. We offer the picture as a way to hold the model in mind. It is not a mechanism, and we make no physical claim with it.

7 · Six, and the six-dimensional sphere

The paper names one dimension in particular, the sixth, and what follows is a reading. The unit's own cell, the inner block, is six-dimensional. The thirteenth entry of the summary record is that cell's readout: the six-dimensional cell seen from outside as a single number, its coherence. So in this paper "the thirteenth coordinate" and "the sixth dimension" name one thing, looked at from two sides.

The two levers also form a complex number: coherence as the real part and leverage as the imaginary part. The unit's movement in time is then read as a rotation of that number.

That is why a result reported this year is a natural analogy. In a paper the article dates to 1947, published in a volume dated 1948, Heinz Hopf asked whether spheres of higher dimension can carry a complex structure, and the six-dimensional sphere was the case that stayed open (Hopf 1948). Scientific American reported on 24 September 2026 that the question may have been answered: "Now an AI has apparently answered it". The article continues: "On August 23 Alpöge posted a 100-plus-page document with a purported proof that the 6D sphere does, in fact, have a complex structure." And it reports the state of review: "There seems to be an emerging consensus that the construction is plausible". The result is a posted manuscript, reported as claimed, and a self-contained account of the construction has also been posted (Engel 2026); it is not yet a published result, and the article's account of how it was made is the article's. The article's sentences are quoted and used for research under the source's published terms, with written confirmation pending.

What we do not claim. We do not claim that our unit's six-dimensional inner cell is a six-dimensional sphere, or that it carries that structure. We note two facts side by side with that question: our unit's inner cell is six-dimensional, and its readout pair is a complex number. That is an analogy, and nothing in it is evidence for the model.

8 · What the engine does with the unit

Our engine is described on our public pages, and we describe it here in those words and no others. You hand it your rules as they already are, and it finds the arrangement that satisfies them best. It turns your problem into a landscape where every possible answer is a point, and the height at that point is how costly that answer would be; then it goes to the lowest point. Up to twenty moving parts, it gets there by visiting every point, all 1,048,576 of them, in about half a second; above twenty, it tells you how far from the best it could be. It is one operator that answers four questions: decide, recall, remember and adapt. The answer is computed, not generated.

The applications built on it are the QubitOS applications on embark.day, where a customer's model and its certificate travel in one file the customer owns.

9 · What we measured, and what we did not

Every figure below is already published, with its basis, on our evaluation page or in a released record, each one run on the date its source gives. Worst first; the falsified tests of the model's own rules are in sections 4 and 5.

The question sets are slices of two public datasets, used under CC BY-SA 4.0 at the point of use: 200 unanswerable questions of SQuAD 2.0 (Rajpurkar, Jia and Liang, ACL 2018), the first in file order, and 150 questions of HotpotQA (Yang et al., EMNLP 2018) over 1,491 documents.

What we did not measure: the convention itself, which makes no claim a measurement could settle. No experiment here shows that a chatbot, a person and a physical system share one ontology. The six invariants in the next section have not yet been measured on any system but our own.

10 · The invariants a unit must satisfy

If a system is an Operational Unit, six things should be true of it, and each can be checked from outside.

• I1 · Its account is a hash-chained ledger. Every act is appended; none is edited.
• I2 · Its objective is written down. What the system optimises is stated, not inferred.
• I3 · Its refusals are logged as acts. A refusal is an entry in the account, not a silence.
• I4 · Every decision has a certificate that can be re-opened. The evidence re-solves or re-derives.
• I5 · Its emissions are measured. What it puts into its environment is counted, with its basis.
• I6 · Its boundaries are shown by adversarial reachability. A tenant or partition boundary is shown by trying to cross it, not by policy text.

This is what our third-party review measures a system against.

11 · How this was made: the chain of records

Day drew the figure of the worked example from the record of its measurement. Claude (Anthropic), the consultant named in the byline, designed and built the engineering from the engineering point of view: the engine's contracts, the tests that pin them, the release train, and the measurements in section 9.

The question was set on 30 September 2026 as a written brief: state the model in full, as mathematics a reader can check, and describe none of the engine's method. The model is the series' third postulate, the entropy reward, with the unit, its account and the broker layer around it; the reward was first committed on 17 February 2026 (ec52666bd), and it is assumed, never derived.

The chain of records, in order:

• the evaluation records the measured rows of section 9 come from, each with its date;
• the prediction of section 12, frozen in a registration committed alone before any data, and not yet run;
• the registered tests of the model's own rules, frozen in a registry committed alone at 65ff0d0b1 before their runs on 7 October 2026, with their verdicts;
• the tests that hold this page equal to its source and every number to its record, and this page, generated from its source.

A measurement that contradicted a page changed the page.

12 · Limits, and what this paper does not say

The open limits come first, each with what would close it.

• The entropy reward is blind to correctness (section 4). It would close with a correctness term in the reward, re-tested on the same slice with a proper score beside it.
• The value update does not settle on a reward that depends on the belief (section 4). It would close with a convergence result for such a reward, or a reward that does not depend on the belief, re-run on the same fixture.
• A belief that has stopped moving is not a confident belief (section 4). It would close with a writer that scores each settled belief with a proper score, re-run on the same seeds.
• The walk-away certificate certifies that no one is worse off, not that a split is stable (section 5). It would close with a core check in the certificate, re-run on the same games.
• No broker keeps all four properties at once (section 5). It would close with the broker naming the property it gives up, and the served prices re-tested against that statement.
• The convention has no experiment, and the six invariants are unmeasured on any system but our own. It would close with the six invariants measured on a person and on a physical system, with the method written down first.
• The loop is open at two wires. Nothing in the engine yet writes a belief back from what was observed, and nothing yet chooses the next reading by what it is expected to teach. Choosing by expected information is a prediction registered, not run: it must lower uncertainty faster per reading than a random order or a fixed one. Each wire closes with its writer or chooser built and measured on the same fixtures.
• Timings are the engine page's own, one machine. It would close with the same fixtures read on a second machine and a graphics card, with the load recorded.

Same observation, three vocabularies. Three fields (information theory, systems and graph theory) describe the same situation in different words, and an insight in one usually has a twin in the other two. This series names a twin only where its own model has the object, and says so where a twin is missing or runs the other way. Each twin is our reading, not an identity, unless a result the paper states makes the two one number. Two statements of this paper are such observations. The first is the entropy reward of section 4. In information theory it is a change in the account's uncertainty, measured in bits. Read as a system, that uncertainty belongs to the account's state alone, so each reward is the change across one step, however the step was taken. Read as a graph, with the states as points and each action as a step between two of them, with γ at one every route between the same two points earns the same total, and a route that returns to its start earns nothing. The second is the chooser the loop above lacks. In information theory it would take next the reading it expects to teach the most. Read as a system, it would settle the least-settled part first: for a reading without error the two are one rule, because what such a reading teaches is the uncertainty of the part it reads (Shannon 1948), and with error they differ. Read as a graph, the account has no twin: it is one distribution, with no links for a reading to widen, and the brokers of section 5 link units, not the parts one unit reads. In one registered run on our fixtures the order of readings mattered, a fixed order slower than a random one (section 5), and the chooser itself is registered, not run.

The standing statements, which no measurement closes:

This is not a physics claim. The words entropy, space and dimension are used in their information sense, and the gravitational picture is an analogy for the reader. The six-dimensional analogy in section 7 is an analogy; we do not claim our inner cell is a sphere or carries its structure. The engine's method is not described here and will not be. Every number is as of the date its source gives.

13 · Companion papers

Seven papers published beside this one take the same approach: each traces its figures to a recorded measurement and says what it did not measure.

• The periodic table as a ground-state ladder: each element read as the lowest-energy filling of its electrons, with every answer NIST lists checked against it.
• Closing the gaps: four sustainability problems an engine can schedule, and what it cannot do about them.
• Sense, remember, act: what the engine senses, remembers and learns.
• The halo of a decision: a two-way choice, the partner it is tied to and the event that tied them, in textbook terms, with what an experiment could measure.
• The Earth as a two-state recorder: the Earth's dipole read as a two-way choice from public records, with a prediction written before the data, and the planet's clocks.
• HOPE: recall in a library of Hilbert spaces: recall as settling to the nearest stored state, in a library of small spaces kept one per model, with the measured answer that they do not make one large space.
• The unit of work: a receipt for one settled problem, booked on a hash-chained ledger, and what it is not, with the economy's measured losses first.

Acknowledgements

• SQuAD 2.0 (Rajpurkar, Jia and Liang, ACL 2018), CC BY-SA 4.0. The source of the test of declining questions that cannot be answered: seven deterministic slices, selected and reformatted, with no text edited. The slices, and any adaptation of them, stay under the same licence.
• HotpotQA (Yang et al., EMNLP 2018), CC BY-SA 4.0. The source of the multi-document questions: the first 150 complete rows of the distractor validation set with their paragraph corpus, held in two derived files that carry the text verbatim. The same licence applies to any adaptation of them.
• Scientific American, 24 September 2026. Four sentences of the article are quoted verbatim, with its authors and date, as the references give them. Used for research under the source's published terms; written confirmation pending.

Replicate this paper, or tell us what is wrong

We invite replication and correction. The measured results come from records committed in our repository. The records are not yet published; until they are, write to us through the contact page, naming the paper and the number you want to rebuild or the error you found. We correct an error we can reproduce on the paper's web page, with its date.

The tests this paper registered before reading their data are frozen, each with the rule that would fail it, in docs/PAPERS_DUALFIELD_1_2026_10_04.md and docs/PAPER_EXPERIMENTS_REGISTRY_2026_10_07.md.

How to cite this paper

Please cite this paper as:

Agile Integrations LLC. 2026. "The Operational Unit: an actor–agent model of information economies, and what our engine runs of it today." Version 1, 30 September 2026. https://agileintegrations.net/blog/the-operational-unit.html

Citation file: CITATION.cff.

14 · References

• Bischoff, M. and Howlett, J. 2026. "AI solves 79-year-old mathematical mystery." Scientific American, 24 September 2026. www.scientificamerican.com/article/ai-solves-79-year-old-math-mystery-of-six-dimensional-spheres, accessed 30 September 2026
• Engel, P. 2026. "Complex structures on S^6." A self-contained account of the construction, posted 13 September 2026. arXiv:2609.38442
• Friston, K. 2010. "The free-energy principle: a unified brain theory?" Nature Reviews Neuroscience 11, 127–138. A published hypothesis of similar reach, for comparison. 10.1038/nrn2787
• Hopf, H. 1948. "Zur Topologie der komplexen Mannigfaltigkeiten." In Studies and Essays Presented to R. Courant on His 60th Birthday, 167–185. Interscience, New York. The paper the Scientific American article dates to 1947.
• Jaakkola, T., Jordan, M. I. and Singh, S. P. 1994. "On the Convergence of Stochastic Iterative Dynamic Programming Algorithms." Neural Computation 6, 1185–1201. On when the value update converges. 10.1162/neco.1994.6.6.1185
• Mas-Colell, A., Whinston, M. D. and Green, J. R. 1995. Microeconomic Theory. Oxford University Press. A standard textbook for both equilibria.
• Nash, J. F. 1950. "Equilibrium points in n-person games." Proceedings of the National Academy of Sciences 36, 48–49. 10.1073/pnas.36.1.48
• Ng, A. Y., Harada, D. and Russell, S. 1999. "Policy invariance under reward transformations: theory and application to reward shaping." In Proceedings of the Sixteenth International Conference on Machine Learning, 278–287. For rewards that add up along a path.
• Pareto, V. 1906. Manual of Political Economy (Manuale di economia politica). Società Editrice Libraria, Milan. Critical English edition, Oxford University Press, 2014. ISBN 978-0-19-960795-2
• Shannon, C. E. 1948. "A Mathematical Theory of Communication." Bell System Technical Journal 27, 379–423. 10.1002/j.1538-7305.1948.tb01338.x
• Sutton, R. S. and Barto, A. G. 2018. Reinforcement Learning: An Introduction, second edition. MIT Press. For the policy gradient and the value update.
• Tsitsiklis, J. N. 1994. "Asynchronous stochastic approximation and Q-learning." Machine Learning 16, 185–202. On when the value update converges. 10.1007/BF00993306
• Watkins, C. J. C. H. and Dayan, P. 1992. "Q-learning." Machine Learning 8, 279–292. For the value update and the conditions under which it converges. 10.1007/BF00992698
• Williams, R. J. 1992. "Simple statistical gradient-following algorithms for connectionist reinforcement learning." Machine Learning 8, 229–256. For the policy gradient with the return. 10.1007/BF00992696