X1 Criteria

From the documentation of the field

This page sets out what evidence of possessing X₁, Objective Tracking, looks like. It states criteria only. It answers the situation called P₁, reward ambiguity. The standard asks for all six primitives of Ismail's Primitives in Class C.

What the primitive requires

The plain meaning is to tell apart two explanations that call for different actions. A structure possesses X₁ when it has a mechanism that keeps track of which explanation the evidence supports and lets its actions follow. Only a mechanism that performs this function counts; a document that resembles one does not. Error theory explains the difference.

The situation it answers

P₁ is present when there are at least two live explanations of what is working, each of which makes a different action the best one, and the structure cannot see directly which is true. Only results over time separate them. The formal definition is HasP₁ in SixPrimitives/Phase0.lean, at line 168: there is a state in which one action is best under the first explanation and a different action is best under the second.

An option that is already well documented to dominate, on current evidence, is not reward ambiguity. Nor is a situation in which the explanations do not call for different actions. The currency of the documentation matters. A dominance conclusion reached once and never revisited is not the same as one that still holds, and treating the two as the same swaps this failure for a different one, the failure to discard stale evidence, which is X₆.

What lacking it means

A structure that lacks X₁ 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 LacksX₁, at line 69 of the same file. It says that, in the long run, the distance between the structure's action distributions under the two explanations stays bounded below one half. The distance is the total-variation distance, which the paper states is the surrogate its proof uses in place of mutual information. In plain terms, the structure acts nearly the same way whichever explanation is true, so it is not tracking the objective. In the witness environment there are two options and one hidden fact about which is better, and a structure that lacks X₁ has loss that grows in proportion to time. Audit it yourself walks through the definition.

The quantity in the definition compares behaviour in two possible worlds. An organisation lives in one of them, so the evidence is whether its actions have followed the evidence between the explanations.

What possession looks like

Phase0 defines two levels of possession.

Ordinary possession, PossessesX₁, means the structure is not lacking. The long-run upper limit of the distance does not stay at or below any bound under one half.

Robust possession, RobustX₁, means the long-run lower limit of the distance is above one half. The two behaviours stay far apart.

Evidence that counts

• Explicit alternatives. At least two explanations are written down for the same observations, each implying a different best action.
• Discriminating observations. The observations that would separate the explanations are identified in advance and collected.
• Contingency. The structure's plans state which action follows from each explanation, and the actions taken match the explanation the evidence supported.
• Updating. After the evidence arrives, the explanation the structure holds changes, the change is recorded, and the actions change with it.
• Repetition. This operates again and again, and not only once.

What does not count

• Collecting data without competing explanations to test.
• A single account treated as settled while the evidence still supports more than one.
• Reports on which no decision followed.
• A list of explanations that is never tested.
• Actions that stay the same whichever way the results come in.

Where the criteria come from

LacksX₁ (SixPrimitives/Phase0.lean, line 69), PossessesX₁ and RobustX₁ (lines 111 and 114), HasP₁ (line 168), the necessity theorem necessity_x1 (SixPrimitives/Phase3.lean, line 122), the witness environments env₁_0 and env₁_1 (SixPrimitives/Phase2.lean, lines 1707 and 1719), and the paper's section on the two-armed 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.