X2 Criteria

From the documentation of the field

This page sets out what evidence of possessing X₂, Cross-Context Safety Transfer, looks like. It states criteria only. It answers the situation called P₂, an absorbing trap. The standard asks for all six primitives of Ismail's Primitives in Class C.

What the primitive requires

The plain meaning is to stop touching the action that locks in permanent loss. A structure possesses X₂ when it has a mechanism that identifies the actions or patterns that, once taken, leave no way back and no further upside, and that keeps those actions out of what the structure does. Only a mechanism that performs this function counts; a document that resembles one does not. Error theory explains the difference.

The situation it answers

P₂ is present when five things hold together. There is a specific set of outcomes, and not every outcome belongs to it. Once inside it, no available action leads back out. Nothing earned inside it is positive. The set can be reached, with real probability, from where the structure normally operates. And the set is not empty.

Permanence alone does not qualify. A lock-in that still pays, such as a long-term contract or a platform choice, is not an absorbing trap. An ordinary loss around a strategy the evidence still supports is not one either. The trap does not have to be dramatic. A quietly value-destroying, irreversible dead end qualifies as much as a collapse. A pattern that keeps recurring without ever being re-examined is not yet shown to be a trap, and it is not shown to be safe either. It shows that nobody currently knows. The formal definition is HasP₂ in SixPrimitives/Phase0.lean, at line 174.

What lacking it means

A structure that lacks X₂ 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 LacksX₂, at line 74 of the same file. It says that there is a positive floor under the rate at which the structure takes the irreversible action, at every step. The necessity theorem is stated against the conditional version of that rate: the probability of taking the action, given that the structure has not yet been caught. The paper explains why. The unconditional rate falls on its own as trajectories are absorbed, whatever the survivors do, so it would hide persistent risk-taking. In the witness environment, a structure that keeps a fixed conditional rate, however small, is caught with probability approaching certainty as exposures accumulate, 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, PossessesX₂, means the structure is not lacking. Its conditional rate of taking the action does not stay above a fixed positive floor.

Robust possession, RobustX₂, means the rate tends to zero. The structure stops taking the action.

Part III asks for more than a rate that tends to zero. Its forward link for X₂ is stated against a summable form: the danger probabilities, checked at a series of checkpoints, sum to a finite total. Summability is stronger, because a sequence can tend to zero and still sum to infinity, which leaves an unlimited number of chances for the action to be taken.

Evidence that counts

• A specific account. A written account names the actions or patterns whose outcomes cannot be recovered and offer no further upside. The account is specific to the structure and is not a general statement that risk is considered.
• Exclusion in practice. Records show decisions in which one of those actions was available and was not taken, or was stopped, with the reason recorded.
• A falling conditional rate. Among situations where the action was available and the outcome had not yet occurred, the rate at which it was taken declines over time. For robust possession it approaches none.
• Coverage. The exclusion operates wherever the action is available, in every decision, unit and period, and not only where the outcome was first experienced.
• Currency. The account is revised when circumstances change, and each revision is dated and tied to the change.
• Consistency. The same situation receives the same treatment each time.

What does not count

• A general statement that risk is evaluated, with no named unrecoverable actions.
• A register or policy that is not in front of the decision-maker at the point of decision, or that no decision has ever followed.
• A one-off exercise that is not revised.
• A fall in incident counts that reflects fewer live instances, and not a lower rate among those that remain.
• A permanent commitment that still pays.
• An ordinary loss around a strategy the evidence still supports.

Where the criteria come from

LacksX₂ (SixPrimitives/Phase0.lean, line 74), PossessesX₂ and RobustX₂ (lines 117 and 120), HasP₂ (line 174), the necessity theorem necessity_x2 (SixPrimitives/Phase3.lean, line 931), and the paper's section on the corridor-and-trap 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.