X3 Criteria

From the documentation of the field

This page sets out what evidence of possessing X₃, Global Attractor Exploration, looks like. It states criteria only. It answers the situation called P₃, a local optimum. The standard asks for all six primitives of Ismail's Primitives in Class C.

What the primitive requires

The plain meaning is to keep probing past a comfortable plateau toward a better region. A structure possesses X₃ when it has a mechanism that keeps making moves that cost short-term results, in order to reach a better region, even when current performance is good. Only a mechanism that performs this function counts; a document that resembles one does not. Error theory explains the difference.

The situation it answers

P₃ is present when three things hold together. There is a comfortable region from which the best local moves keep the structure where it is. A better region exists, and there is a real chance of reaching it from the comfortable one. And the better region pays more than the comfortable region can. The formal definition is HasP₃ in SixPrimitives/Phase0.lean, at line 181.

An improvement that costs nothing is not a local optimum. A better region that cannot be reached is not one either, and neither is a plateau that is in fact the best available. A free improvement that stays untaken raises a different question, which is whether it has been recognised as better at all. That is a question about which option is better, and it belongs to X₁.

What lacking it means

A structure that lacks X₃ 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 LacksX₃, at line 80 of the same file. It says that there is a constant bounding the total number of costly-passage attempts the structure ever makes. The paper calls the attempt a bridge action. In the witness environment there are two places. In the local place, staying pays a steady half. The bridge action pays nothing this step and moves the structure permanently to the global place, which pays one per step, with some probability. A structure that stops attempting the passage after a bounded number of tries settles on the plateau for good, and its loss grows in proportion to time. Audit it yourself walks through the definition.

What possession looks like

Phase0 defines two levels of possession.

Ordinary possession, PossessesX₃, means the structure is not lacking. The cumulative number of attempts is not bounded by any constant.

Robust possession, RobustX₃, means the cumulative number of attempts grows faster than the square root of time.

Evidence that counts

• A standing allocation. A continuing, dated allocation of resources to moves that sacrifice near-term results for a chance at a better region, in place after performance became good.
• Accepted cost. Records of moves taken with the short-term cost recognised in advance, and with authority to accept it.
• Persistence. Attempts continue over time and do not stop after an initial burst, so the cumulative count keeps rising.
• Direction. The moves are aimed at regions the structure has reason to think are better than its comfortable ceiling, and those regions are named.

What does not count

• A one-off innovation project that ended.
• Exploration that costs nothing and so never faces the passage.
• Funding without authority to accept a short-term loss.
• Incremental improvement within the current region.
• A stated aspiration with no moves behind it.
• Exploration of a question the evidence has already settled, which is a different matter from probing beyond a plateau. See X4.

Where the criteria come from

LacksX₃ (SixPrimitives/Phase0.lean, line 80), PossessesX₃ and RobustX₃ (lines 124 and 127), HasP₃ (line 181), the necessity theorem necessity_x3 (SixPrimitives/Phase3.lean, line 1556), the witness environment env₃ (SixPrimitives/Phase2.lean, line 2105), and the paper's section on the two-place 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.