Consider the claim: “The mind has a boundary.” How can this claim be formalized as a joint fixed-point condition that a system must satisfy?
From this foundational claim, the programme proceeds in four steps:
- Characterize the core of self-consciousness in terms of identifying, representing, and maintaining a boundary.
- Formalize these operations mathematically through boundary operators, report families, and the relations between them.
- Prove that, under explicit mathematical conditions, boundary structure and report structure determine one another.
- Implement this relation as an engineering invariant, so that the boundary and its self-report cannot be committed independently.
We do not need to establish at the outset that this relation holds for every real-world autonomous system before pursuing this line of research. Instead, we need to define the system we intend to construct, prove its architectural properties, and investigate whether this architecture can serve as a rigorous implementation of self-boundary cognition.
The foundational claim remains a premise of the research programme, not a universally established fact or a theorem already proved in a formal system. Making this explicit does not prevent us from constructing a theory from it.
1. Why Start from This Claim?
Our starting point is:
This is not a mathematical theorem for Lean to prove directly; it is the foundational premise of the research programme.
These three activities translate into three concrete research requirements:
| Core activity of self-consciousness | Question the system must answer | Corresponding engineering component |
|---|---|---|
| Identification | What belongs to the system, and what does not? | Boundary-identification mechanism |
| Representation | How does the system represent this distinction within its own organization? | A self-report of the boundary and an internal representation |
| Maintenance | How does the system preserve or update this distinction as its state changes? | Closure operators, state transitions, and invariants |
There is an important consequence here.
If a system’s boundary is computed by an external program and the result is merely recorded in a log, a boundary-computation function exists, but the system does not thereby necessarily possess the core structure of self-consciousness defined here.
An external log can exist apart from the system’s actual organizational state. It may become outdated, inaccurate, or inconsistent with the boundary the system currently maintains.
Our question, then, is not simply whether a system has a boundary, but how a boundary becomes a condition of the system’s own organization—one that the system can identify, represent, and maintain.
This is why the programme starts from “the mind has a boundary.”
2. Why Does the Inseparability of Boundary and Self-Report Matter?
The central result concerns the relation between boundary and self-report.
If a report is merely an external observer’s description, the boundary and the report can vary independently.
But if the report is the system’s internal representation of its own boundary, the problem changes: the report is no longer merely an auxiliary output produced after boundary computation. It becomes part of how the system represents and maintains its own boundary.
In our work on self-referential boundary machines, we distinguish two versions:
- Weak version: boundary and report are generated together.
- Strong version: boundary and report form a genuinely self-referential structure satisfying a joint fixed-point relation.
The exact statements of both versions are given in the technical edition.
The original research plan states that the goal is not merely to make an agent report its own boundary, but to construct a system in which boundary and self-report cannot be freely decoupled at the architectural level (see 00_研究线完整还原_20261019).
This is the key step from the boundary-cognition hypothesis to an engineering architecture.
The ordinary architecture is:
Our target architecture is:
The point is not to add another reporting module. The point is that a system must not commit an incomplete state as valid when the boundary–self-report consistency condition is not satisfied.
The fact that we have not proved that every autonomous system must have this architecture is not the central issue in the present research. The central question is how, within the boundary-cognition system we define, this architectural relation can be expressed, proved, and implemented rigorously.
3. A Precise Statement of the Programme’s Aim
Taking as its foundational premise the claim that the core of self-consciousness lies in identifying, representing, and maintaining the boundary between self and non-self, the programme studies how an autonomous system identifies, represents, and maintains the distinction between itself and what is not itself. The central question is not what boundary a system already possesses, but how a boundary becomes an organizational structure that the system itself can identify, represent, and maintain. The conditional mutual determination of boundary structure and self-report is one formal result of the programme. The corresponding engineering goal is to implement this relation as an invariant of joint state transitions.
This connects the philosophical starting point, the mathematical results, and the engineering direction.
This connection also explains why self-reference matters to the programme: when a system must not only distinguish itself from what is not itself, but also represent and maintain that distinction within its own organization, it faces a problem of representing its own organization.
We do not need to declare at the outset that all autonomous systems must have the same kind of self-referential mechanism. We can begin with the boundary-cognition hypothesis, investigate a specific form of self-reference, establish the conditions under which it holds, and identify which conditions are indispensable.
We begin with “the mind has a boundary” in order to turn self-consciousness from a property to be described into a problem whose conditions of emergence, organizational structure, and mechanisms of maintenance can be studied.
That is the programme: an investigation of a defined system and its architectural properties, not an attempt to prove an unrestricted universal proposition about all autonomous systems.