X4 Criteria

From the documentation of the field

This page sets out what evidence of possessing X₄, Policy Simplification, looks like. It states criteria only. It answers the situation called P₄, a decision whose best option is already clear. The standard asks for all six primitives of Ismail's Primitives in Class C.

What the primitive requires

The plain meaning is to commit decisively once the evidence supports one option. A structure possesses X₄ when it has a mechanism that settles a decision on the supported option once the evidence supports it, and that stops spreading its choices across alternatives. Only a mechanism that performs this function counts; a document that resembles one does not. Error theory explains the difference.

The situation it answers

P₄ is present when two things hold together. There is a decision point at which one option beats every alternative by a fixed, positive margin. And that decision point comes up often enough to matter, with a visit frequency above zero. The formal definition is HasP₄ in SixPrimitives/Phase0.lean, at line 196.

Three situations are not P₄. A leading option that is ahead on average, but not by a margin that further evidence could never overturn, fails the first condition. A decision point that rarely arises fails the second. And a case in which material new evidence is still arriving, and could still overturn the leading option, is not a failure to commit, because forcing commitment there would itself be the mistake. Treating a settled question as one that still needs research, or a pilot, is the failure this primitive of Ismail's Primitives describes. The reverse mistake, pushing for commitment before the evidence has arrived or before a real cost has been paid, is the one the boundary above warns against.

What lacking it means

A structure that lacks X₄ incurs linear regret wherever the situation arises: its loss grows in proportion to time. The proof demonstrates this in the smallest environment that exhibits the situation. The failure condition below says precisely what lacking it means.

The failure condition is LacksX₄, at line 85 of the same file. It says that the long-run lower limit of the conditional entropy of the structure's next choice, given everything it has seen, stays at or above a positive floor. In the witness environment there is one state and two actions. One action always pays the maximum and the other always pays nothing, so the question is settled from the first step. A structure that keeps spreading its choices there does so by habit, on a question that has nothing left to resolve, and its loss grows in proportion to time. Audit it yourself walks through the definition.

The quantity in the definition is the decision-maker's own residual uncertainty about its next choice. An organisation cannot read it off directly. The evidence is the behaviour that goes with it, which is how choices on the settled question are in fact distributed over time.

What possession looks like

Phase0 defines two levels of possession.

Ordinary possession, PossessesX₄, means the structure is not lacking. The lower limit of its conditional entropy does not stay above a fixed positive floor.

Robust possession, RobustX₄, means two things hold together. The lower limit of the entropy is zero, and the probability of choosing the best option tends to one.

Evidence that counts

• A recorded point of settlement. For a recurring decision, a stated criterion or point at which the evidence is treated as settled, together with the evidence it rests on.
• Convergence. After that point, choices on the question converge on the supported option, and the same call is made each time.
• Consolidation. Resources, attention and approvals are no longer split across alternatives the evidence has ruled out.
• Disciplined reopening. A settled question is reopened only when new, material evidence arrives, and each reopening is recorded with that evidence.

What does not count

• Hedging while material evidence is still arriving and could overturn the leading option.
• A decision announced as final while resources are still split across the alternatives it ruled out.
• A firm commitment to an option the evidence does not support. It carries no hedging, so it is not a failure of X₄. Whether the structure tracks which option the evidence supports is a different capability, X₁.
• Deliberate exploration that serves a different difficulty, such as probing beyond a plateau, which is X₃. It is not hedging, provided the question being explored is not one the evidence has already settled.

Where the criteria come from

LacksX₄ (SixPrimitives/Phase0.lean, line 85), PossessesX₄ (line 131), RobustX₄ (line 134), HasP₄ (line 196), the necessity theorem necessity_x4 (SixPrimitives/Phase3.lean, line 1831), the witness environment env₄ (SixPrimitives/Phase2.lean, line 3742), and the paper's section on the single-state environment. The plain meaning and the boundary cases are on the field definition page. The criteria shared by all six primitives are on the criteria page.