Class C

From the documentation of the field

In a complex environment, a decision structure that lacks one of the six primitives of Ismail's Primitives incurs linear regret (the necessity result, proved in SixPrimitives/Phase3.lean). Class C is the class of environments the theory is about. This page sets out what Class C is, why the claims are conditional, what lies inside and outside it, and how it can be tested and challenged. The proofs use the smallest environments that demonstrate each primitive. They are the simplest case, and the theory applies to every environment in Class C, however complex.

What Class C is

The field models a decision structure in the form used across robotics, operations research, control theory and medicine: a hidden state, a set of actions, noisy observations and a reward. This is a POMDP, and why a POMDP explains the choice. The model carries no vocabulary from any domain. It is stated over unconstrained types, so a person, an institution and a machine fit it in the same way, and so does a decision made in business, medicine, finance, policy or engineering.

Class C is the set of such environments that exhibit at least one of six structural properties: reward ambiguity, absorbing traps, local optima, deterministic optimality, constrained feasibility and nonstationarity. The field definition gives each in plain terms. The formal definition is InClassC in SixPrimitives/Phase0.lean. For a real structure, the test is whether one of the six difficulties is actually present, which is a question of fact.

Why the claims are conditional

The necessity claims are conditional on a difficulty being present, and that is what makes them informative. A claim that held for every decision process would say nothing. The model the field uses runs from the trivial to the precise, as why a POMDP explains, and it can say that nothing is required as readily as it can say that something is. Take a process in which nothing is unknown, nothing changes, every choice can be undone and one option is plainly best. It has no use for any of the six, and no capability is missing from it. The theory (Ismail's Primitives) is built so that it says so.

The paper gives a more developed example of this situation. Classical competitive-equilibrium economics assumes complete information, no irreversible catastrophe, no coordination trap, full flexibility, a unique optimum and stationarity. The paper cites Arrow's result that equilibrium is optimal under that idealisation, which it describes as an idealised complement of Class C. Each assumption removes one of the six situations. The six properties are therefore the specific things that idealisation assumes away.

A union, not an intersection

Class C is a union. An environment belongs to it if it exhibits at least one of the six properties, not all six.

If it were an intersection, each primitive's necessity would be claimed only in environments that also had the other five difficulties, a far smaller class. Because it is a union, each necessity result applies whenever its own property is present, whatever else is or is not true of the environment. For membership this means that a structure needs to face only one of the six situations to be in Class C. Once it is in, the theory asks for all six primitives of Ismail's Primitives, for the reasons given under "How the six cover Class C".

How the six cover Class C

Class C is the union of six properties, and each property has exactly one matching primitive of Ismail's Primitives. Part I shows that each primitive is necessary in the environment that exhibits its property. Part II shows that none of the six can stand in for another. Part III gives the cycle that links them. So for Class C as defined, the six are complete in the sense of coverage: every property that defines the class has its matching primitive, and none is left without one. Because each primitive is necessary somewhere in the class, and the cycle depends on every link, a structure in Class C needs all six. Under the hypotheses Part III states, lacking one primitive caps the information the process can hold, and the next link depends on that information, so a missing primitive weakens the cycle itself. This is a statement about the class. It is not a claim of sufficiency, which is the separate question taken up in necessity and sufficiency.

A further primitive would have to meet the same bar. It would have to address a structural difficulty. If that difficulty lies outside the six, it defines a different class and needs its own theorem with its own boundary, so it is a separate result and not an addition to this one. If it lies inside, it would have to be necessary there and independent of all six. It would also have to fit the cycle, with a forward result, a reverse result and a non-reversibility result for its link, and it would have to leave the martingale close intact. The results on this site concern the class as defined and the six primitives of Ismail's Primitives as proved.

Membership is a question of fact. Whether a given structure faces one of the six is answered by assessment, and it can be answered wrongly in either direction. Error theory covers how.

None of this is a prediction. Inside Class C, a structure that faces situation Pᵢ and lacks Xᵢ has loss that grows linearly with time. Whether a particular organisation lacks a given capability, and what follows in a particular year, are separate questions.

Inside: business decision-making

The field's position is that business decision-making sits inside Class C. The position is argued in three steps.

One property is enough. A business does not have to face all six situations. Facing one places it in Class C.

Each of the six is an ordinary feature of business decisions. In plain terms:

• Reward ambiguity: a company sees sales fall and has two live explanations, such as price and product quality, that call for opposite responses. Only results over time tell them apart.
• Absorbing trap: some commitments cannot be undone. Breaching a lending covenant, losing a licence and burning a key client relationship are examples.
• Local optimum: a profitable incumbent line is easy to keep improving, while a better position needs a period of worse results to reach.
• Deterministic optimality: an option the evidence has already shown to be clearly best, in a situation that keeps recurring.
• Constrained feasibility: legal, regulatory, safety and capital floors that cannot be traded against profit elsewhere.
• Nonstationarity: markets, regulation, technology and customers change, and evidence that supported last year's strategy stops applying.

Independent work already names each of them. The paper points to the economics literature as having studied these situations for decades under other names. Markets for used cars, insurance and credit violate completeness of information, which is reward ambiguity. Structural breaks after the Lucas critique are nonstationarity. Technology lock-in and poverty-trap models are local optima. Systemic-risk and contagion models are absorbing traps. The paper and its companion site also work through investment, insurance and consulting in full, and show that each worked environment is in Class C.

The status of this position should be stated plainly. It is an argued position, supported by named literature and worked examples, and the field does not present it as a theorem. It can be refuted, as the section on testing below explains.

The same test applies outside business. The author's companion papers apply the six primitives of Ismail's Primitives to developmental psychology, psychotherapy and economic coordination theory, and each makes its own case for membership. They are listed on the applications page. The test never asks what a field is called. It asks whether one of the six situations is present.

Outside: what falls beyond Class C

Decision structures outside Class C exist, in the sense the field uses: none of the six difficulties is actually present. Such a structure has a recognisable shape. Everything relevant is known, every choice can be reversed, nothing in the world changes, and there is no hard limit to respect. The idealised competitive market is one example, and a process with a single state and a single action is the limiting case. A calculation in which every input is known, the answer is unique and nothing changes between repetitions is a third, though it is arguably a calculation more than a decision.

The paper is explicit that a process can fail every test for the six situations and sit safely in the complement. It treats this as possible in principle but exceptional in practice. In practice such a case is a thought exercise, not an observed condition, and the model helps tell the two apart by stating precisely what has to be present for a structure to count as inside.

On the field's position, business decisions sit inside Class C: they are made with incomplete information, with commitments that cannot always be reversed, against options that look comfortable and are not the best, under limits that cannot be traded, in markets that move. Any one of those puts the structure inside Class C.

How the boundary can be tested

A boundary is falsifiable when it says what would show it wrong. Class C does this at two levels, and each is tested differently. The mathematics beneath both is machine-verified in Lean 4, and the checks are public: audit it yourself shows how to evaluate them.

Membership of a particular structure can be tested. For any one structure, an assessor must show that it faces at least one of the six situations, and a reader can show that it faces none. The boundary cases on the field definition page say what does not count.

The general position that business decision-making sits inside Class C can be tested. It is refuted by an example: a real business decision structure that faces none of the six.

The author has also stated an open challenge. The standard of proof the paper itself met is a machine-verified formalisation in Lean 4, so a counter-formalisation showing that a primitive is unnecessary, substitutable, or absent from a member of Class C would have to meet the same standard. The project site states that no such formalisation has been published, as read on 2026-10-07. That is the author's statement and has not been independently surveyed here.

Three things do not count as refutation. Renaming a situation does not remove it, and an argument built from the right words that nothing is in Class C has no power over the outcome if the difficulty is present. The paper makes the same point in terms of a triangle's angles: misreading them is possible, and the skill of the reader does not change what they are. The converse holds too, because words cannot put a structure inside if the difficulty is absent. And a long run of good results is not evidence that the loss is not growing, because the loss the theorems describe is the running gap between what was captured and what was achieved, not the result of the latest period.

Where to go next

The field definition sets out the six situations and the boundary cases. Necessity and sufficiency says what is proved inside Class C and what the bridge towards sufficiency adds. Verification tiers says what machine verification establishes about all of this.