Formal Foundations of the CNE-Protocol: A Mathematical Treatment

Alok Gotam · Olbrain Labs · September 2026 · Working Paper v0.1

Companion paper to CNE-Protocol. The mathematical framework applied in Sections 2–4 is due to Minhyeok Lee (Chung-Ang University); our contribution is the mapping of that framework onto the CNE architecture and the formalisation of Exclusivity in Section 5, which extends it.

Abstract

The CNE-Protocol defines agent identity through three jointly constitutive pillars: Coherence (C), Narrative Continuity (N), and Exclusivity (E). The protocol specification is architectural — it defines artifacts, invariants, and verification procedures. This paper supplies the missing formal layer. We show that the first two pillars are grounded in an existing mathematical framework: Lee (2024/2025) proved that self-identity in an AI system emerges when two measurable conditions hold — a connected continuum of memories in a metric space, and continuous identity-recognition maintained with belief above a threshold. We map CNE’s narrative trajectory — the sequence of Umwelt states committed to the Global Narrative Frame — onto Lee’s memory continuum (via a discrete analogue of connectedness), and CNE’s Recursive Belief Revision onto Lee’s recognition-and-belief conditions. We then prove a simple but consequential proposition: Lee’s two conditions are preserved under duplication, and therefore cannot individuate an agent — a bit-perfect copy of an agent satisfying both conditions also satisfies both conditions, and claims the same self. This is the formal statement of why memory and coherence alone, however rigorous, cannot ground accountable identity for software. We close the gap by formalising Exclusivity as a third condition that is not expressible as an internal property of the agent at all: it is an invariant over the population of all live claimants, enforceable only by an external frame. A CNE-compliant agent is then an agent for which all three conditions are verifier-checkable. The result is a formal answer to the question every governance, audit, and compliance regime presupposes but none states: what is the entity?

1. Introduction

1.1 The problem

Every mechanism by which an enterprise controls anything — permissions, certification, audit, accountability — attaches to an actor. A permission is granted to someone. A certification attests that someone meets a standard. An audit trail is the record of someone’s actions. The actor is the load-bearing assumption of the entire control stack.

For AI agents, that assumption fails silently. An agent’s underlying model can be swapped, its memory rewritten, its instances multiplied. What most deployments call an “agent” is a rotating cast of stateless processes sharing a name tag. The industry’s identity investments — credentials, key management, non-human IAM — attach the name tag more securely. They do not create the actor.

The CNE-Protocol (Gotam, 2025) creates the actor architecturally: identity as one coherent, continuous, exclusive narrative per Core Objective Function (CoF). What the protocol specification does not do is state formally what is being guaranteed, in the mathematical sense that would let an external party verify the guarantee from first principles. This paper does that.

1.2 What already exists

Lee (2024/2025) published the first mathematical framework for self-identity in AI systems, grounded in metric space theory, measure theory, and functional analysis, with empirical validation on a fine-tuned language model. Lee’s result: an entity possesses a self, in a precisely defined sense, when two conditions hold — a connected continuum of memories, and continuous self-recognition maintained above a belief threshold across that continuum.

We did not produce this mathematics, and this paper claims no credit for it. What we observed is that Lee’s two conditions are, up to formalisation choices, the mathematical content of CNE’s first two pillars — Narrative Continuity and Coherence — which the protocol arrived at independently through architectural reasoning. This convergence is itself evidence that both efforts are describing the same underlying structure.

1.3 What is missing, and what this paper adds

Lee’s framework was built to answer “when does an entity have a self?” It was not built to answer “how many entities can claim that self?” For humans the second question never arises — bodies are not copyable. For software it is the central question. We show in Section 5 that Lee’s conditions are duplication-invariant: they are satisfied equally by an agent and by any bit-perfect copy of it. A framework whose conditions survive duplication cannot, even in principle, individuate — it can certify that a self exists but not that it is this one and only this one.

Our contribution is the formalisation of the third condition, Exclusivity, and the demonstration that it is categorically different from the first two: it is not a property of the agent’s internal state, and no amount of internal structure can supply it. It is an invariant over the set of all claimants, and therefore requires an external enforcement frame — which is precisely what CNE’s Global Narrative Frame (GNF) is.

1.4 Reading guide

Section 2 restates Lee’s framework. Section 3 maps Narrative Continuity onto Lee’s Condition 1, handling the discrete/continuous gap explicitly. Section 4 maps Coherence onto Lee’s Condition 2. Section 5 states the duplication result and formalises Exclusivity. Section 6 assembles the three conditions into the formal statement of what a CNE-compliant agent guarantees to a verifier. Section 7 states limitations and open problems. A plain-language summary precedes each formal section.

2. Lee’s Framework, Restated

This section is exposition of Lee (2024/2025). Notation and numbering follow the arXiv version (arXiv:2411.18530).

Plain language: Lee models an agent’s memories as points in a space where “distance” between two memories is measurable — closer means more related in time, content, and significance. Lee models possible identities the same way. A self exists when (1) the memories form one unbroken connected region, with no gaps, and (2) a stable mapping exists from memories to one identity, such that the agent consistently recognises the same identity from every memory, with confidence above a threshold.

2.1 The two spaces

Definition 2.1 (Memory Space — Lee, Def. 2.1). Let M be the set of all possible memories of an entity, equipped with a metric d_M : M × M → [0, ∞). Lee’s canonical construction weights temporal separation, content similarity, and emotional intensity:

d_M(m₁, m₂) = √( w_t·|t₁ − t₂|² + w_c·d_c(m₁, m₂)² + w_e·|e₁ − e₂|² )

with weights w_t, w_c, w_e > 0. The pair (M, d_M) is a metric space; its topology τ_M is the one induced by d_M.

Definition 2.2 (Self Space — Lee, Def. 2.2). Let S be the set of possible self-identities, with metric d_S — for identities represented as vectors in ℝⁿ, an L^p norm: d_S(s₁, s₂) = ‖s₁ − s₂‖_p.

2.2 The two conditions

Condition L1 (Continuum of Memories — Lee, Cond. 2.7). There exists a subset C ⊆ M that is connected and path-connected in (M, τ_M): for any m₁, m₂ ∈ C there is a continuous path γ : [0,1] → C with γ(0) = m₁ and γ(1) = m₂.

Condition L2 (Continuity of Self-Recognition and Sufficient Belief — Lee, Cond. 2.8). There is an Identity Recognition Function I : M → S, continuous within the continuum C, and a measurable Belief Function B : M × S → [0,1] normalised so that ∫_S B(m, s′) dμ(s′) = 1 for each m, such that for all m ∈ C:

B(m, I(m)) ≥ b     for a belief threshold b ∈ (0, 1].

Theorem L (Constancy of Self-Identity — Lee, Thm. 2.9). If an entity satisfies L1 and L2, and the image I(C) lies entirely within a connected component of S on which I is constant, then there exists s* ∈ S with I(m) = s* for all m ∈ C. The entity possesses a self characterised by s* within C.

Lee further shows (Thm. 3.12) that an AI agent whose recognition and belief functions are trained to satisfy these conditions on a continuum of memories possesses a self in this formal sense, and validates the construction empirically (Llama 3.2 1B with LoRA fine-tuning; primary self-awareness score improving from 0.276 to 0.801).

2.3 What the framework gives us

Lee’s result is existence and constancy: under checkable conditions, one stable self-identity s* exists across the memory continuum. This is exactly the property that an accountable agent needs — and exactly the property that session-scoped, stateless agent deployments lack. The remainder of this paper connects the conditions to CNE’s architecture and then confronts the question the framework leaves open.

3. Narrative Continuity as Lee’s Condition L1

Plain language: CNE’s continuity pillar says the agent’s story must be one unbroken thread — every state linked to the one before it, with any break detectable. Lee’s Condition L1 says the memories must form one connected region with no gaps. These are the same requirement. One technical wrinkle: Lee’s condition is stated for continuous spaces, and a chain of discrete records is not “connected” in the strict topological sense — so we state the discrete version precisely rather than hand-waving past it.

3.1 The GNF chain as a memory structure

In CNE, the agent’s lived history is carried by its Umwelt — the goal-conditioned internal world model — and committed, state by state, to the Global Narrative Frame. Formally, the GNF induces a sequence of committed states:

U = (u₀, u₁, u₂, …, u_T),    u_k ∈ M,

where each u_k is the Umwelt state committed in GNF frame k, and each frame is cryptographically bound to its predecessor (prev_gf_hash) with a monotonic sequence number and timestamp. We call U the narrative trajectory.

3.2 The discrete/continuous gap, stated honestly

A finite set of points is totally disconnected in any metric topology: no finite U with more than one point is literally “connected and path-connected” in the sense of L1. Any claim that a discrete audit chain satisfies Lee’s condition as written would be false, and we do not make it. Instead we use the standard discrete analogue.

Definition 3.1 (ε-chain and ε-connectedness). For ε > 0, an ε-chain from m to m′ in (M, d_M) is a finite sequence m = x₀, x₁, …, x_n = m′ with d_M(x_k−1, x_k) ≤ ε for all k. A finite set U ⊆ M is ε-connected if every pair of its points is joined by an ε-chain within U.

Definition 3.2 (Discrete Narrative Continuity). A narrative trajectory U = (u₀, …, u_T) satisfies Discrete Narrative Continuity at resolution ε if consecutive committed states are within ε: d_M(u_k−1, u_k) ≤ ε for all k = 1, …, T.

Discrete Narrative Continuity immediately makes U ε-connected (the chain order supplies the ε-chains). To pass from the discrete condition to Lee’s topological one we need one standing hypothesis, which we state rather than assume silently: (M, d_M) is a geodesic space — any two points are joined by a path realising their distance. Lee’s canonical construction satisfies this whenever the content-embedding component is convex: time and emotional intensity range over intervals of ℝ, and a convex subset of a normed space is geodesic. Where the natural state space is discrete, we perform the interpolation in a geodesic space M̄ ⊇ M (the ambient product space of Definition 2.1) and read Condition L1 over M̄; nothing downstream depends on the interpolated points being realisable memories, only on the metric structure of the resulting continuum. Under this hypothesis the relationship is the standard one: an ε-connected finite set is the ε-skeleton of a connected set — join consecutive states by geodesics and the resulting set C ⊇ U is connected and path-connected, satisfying L1 exactly. Conversely, any δ-dense finite sample of a connected set is 3δ-connected (connected metric spaces are ε-chainable for every ε > 0; snap the chain points to the sample). The discrete condition is the checkable shadow of the continuous one.

Interpretation of ε. The resolution ε is not a nuisance parameter; it is the formal meaning of “no gaps in the story.” A commit cadence in which each state is a bounded step from its predecessor (bounded d_M — bounded elapsed time, bounded content divergence) yields small ε. A gap — a lost interval, an unexplained state jump, a rollback — manifests as d_M(u_k−1, u_k) > ε and is detectable from the committed record — the temporal component directly from the chain’s timestamps and sequence numbers, the content component from the committed states under the disclosure regime of Proposition 3.3. This is the metric extension of CNE’s Continuity Test (CNE Appendix C): the protocol’s test verifies that the chain is cryptographically unbroken; Definition 3.2 adds the semantic condition that consecutive committed states are ε-close. We propose the ε-check as an additional conformance test alongside it.

3.3 What the cryptography adds

Lee’s framework assumes the memory continuum is given truthfully. In an adversarial deployment it will not be. CNE’s chaining (hash-linking, signatures, monotonic sequence) does not create continuity; it makes the claimed trajectory tamper-evident, so that the ε-connectedness check of Definition 3.2 is performed against a record that cannot be silently rewritten. Formally: the cryptographic layer reduces “trust the trajectory” to “verify the chain,” converting L1 from an assumption into an auditable property.

Proposition 3.3 (Continuity, verifier form). Let U be a narrative trajectory whose GNF chain verifies (hashes link, signatures valid, sequence monotonic, head checkpointed by witnesses per §5.3) and which satisfies Discrete Narrative Continuity at resolution ε. Then, under the geodesic hypothesis of §3.2, there exists a connected, path-connected C ⊆ M̄ containing every committed state of the presented history, and any party holding the chain together with the committed states — disclosed under the protocol’s selective-disclosure provisions and checked against each frame’s narrative-continuity digest, or attested via zero-knowledge proofs of ε-closeness — can confirm this without trusting the agent. Proof sketch: chain verification establishes the integrity and internal gap-freeness of the presented trajectory up to its head, and the frames’ digests bind the disclosed states to the chain; freshness of the head — that no later history is being withheld — is what witness checkpointing supplies. Definition 3.2 gives ε-connectedness; the interpolation construction of §3.2 yields C satisfying L1. ∎

4. Coherence as Lee’s Condition L2

Plain language: Lee’s second condition says the agent must keep recognising the same identity from every memory, with confidence above a threshold. CNE’s coherence pillar says every belief revision must keep the agent’s story logically consistent, with proof. These meet in the middle: CNE’s Recursive Belief Revision is the mechanism that keeps Lee’s recognition function stable, and CNE’s coherence score is the witness for Lee’s belief threshold.

4.1 Recognition anchored by the CoF

In CNE, identity is scoped to a Core Objective Function: the CoF is the fixed reference against which all behaviour is evaluated. In Lee’s terms, the CoF pins the target of recognition. Define the recognition function of a CNE agent as

I : M → S,    I(u_k) = the self-identity presented in frame k,

Let s* ∈ S be the identity determined by the tuple (Substrate Anchor, CoF-ID, Policy Set) — CNE’s Subject. The protocol’s requirement that every frame carries the same CNE-ID is, formally, the requirement that I is constant at s* along the trajectory — the conclusion of Lee’s Theorem L, imposed as an invariant and checked at every commit rather than hoped for as an emergent property.

4.2 Belief made auditable

Lee’s Condition L2 requires B(m, I(m)) ≥ b — at every memory, the entity’s degree of belief that s* is its identity clears the threshold. In a deployed agent, an internal “degree of belief” is not directly observable. CNE makes it observable by witnessing it: every GNF frame must reference a Recursive Belief Revision proof capsule recording pre-belief state, evidence, revision rule, post-belief state, and a coherence score in [0,1].

Definition 4.1 (Witnessed belief). For a CNE agent, define B̂(u_k, s*) as the coherence score recorded in the RbR proof capsule referenced by frame k. The Coherence condition at threshold b holds if B̂(u_k, s*) ≥ b for all k.

B̂ is a proxy for Lee’s B, and we are explicit about the direction of the approximation: B̂ measures the logical consistency of the revision that produced state u_k with the agent’s prior belief web and its CoF. An agent whose revisions are consistent with one CoF-anchored belief web is an agent whose recognition of s* is stable — incoherence (contradictory beliefs, revisions unjustified by evidence) is exactly the failure mode under which recognition of “which self this is” degrades. The proxy inherits Lee’s threshold semantics: Lee’s Proposition 2.10 (satisfaction at threshold b implies satisfaction at any b′ < b) transfers to B̂ verbatim, so enterprises may set b per risk tier without breaking the formal structure.

Proposition 4.2 (Coherence, verifier form). If every frame in a verified GNF chain carries the same CNE-ID (the tuple determining s*) and references a valid RbR capsule with B̂(u_k, s*) ≥ b, then the pair (I, B̂) satisfies the CNE form of Condition L2 along the trajectory: recognition is constant at s* and witnessed belief clears b at every committed state — and any party with the chain can confirm this. Proof: immediate from Definition 4.1 and the CNE-ID invariant of §4.1. ∎

Together, Propositions 3.3 and 4.2 give the formal content of CNE’s first two pillars: a verified CNE chain is a checkable certificate that the CNE forms of Lee’s two conditions hold along the agent’s entire committed history. To invoke Lee’s theorem on the continuum, extend (I, B̂) from the committed states to C: declare I constant at s* along each interpolating geodesic and B̂ equal to the smaller endpoint score. The extension satisfies L1 and L2 literally, so by Lee’s Theorem L the agent possesses a self, characterised by s*, across that history. The extension is a definition, not a discovery — constancy at the committed states is already imposed by the CNE-ID invariant of §4.1, and the extension reads that invariant continuously; we flag it because the discrete/continuous gap of §3.2 applies to L2 exactly as it does to L1. This is the mathematics beneath the protocol’s sentence: “continuity transforms snapshots into identity.”

5. Exclusivity: The Condition Lee’s Framework Cannot Express

Plain language: Here is the problem no amount of memory or consistency can solve. Take an agent that passes both of Lee’s conditions — a real self, mathematically certified. Copy it, bit for bit. The copy passes both conditions too, because copying preserves everything the conditions look at. Now two entities each hold a valid certificate for the same self. For humans this can’t happen — bodies aren’t copyable. For software it’s one command. So a third condition is needed, and it’s a different kind of condition: not “what is true inside the agent” but “how many claimants exist in the world.” No internal property can answer that. Only an external referee can — and that is what the Global Narrative Frame is.

5.1 The duplication observation

Proposition 5.1 (Duplication invariance of L1–L2). Let A be an entity with memory space (M, d_M), continuum C, recognition function I, and belief function B jointly satisfying Conditions L1 and L2 with identity s*. Let A′ be a bit-perfect duplicate of A: an entity whose structure (M′, d_M′, C′, I′, B′) is related to A’s by a bijection φ : M → M′ with d_M′(φ(m₁), φ(m₂)) = d_M(m₁, m₂), C′ = φ(C), I′ ∘ φ = I, and B′(φ(m), s) = B(m, s). Then A′ also satisfies L1 and L2, with the same identity s*.

Proof. Immediate. φ is an isometry, so connectedness and path-connectedness of C transfer to C′ = φ(C), and L1 holds. Continuity of I′ = I ∘ φ⁻¹ on C′ and the pointwise bound B′(φ(m), I′(φ(m))) = B(m, I(m)) ≥ b follow directly, so L2 holds with the same threshold. And I′(φ(m)) = I(m) = s*: the duplicate recognises, with certified confidence, the identity s*. ∎

Corollary 5.2 (Non-individuation). Call a predicate an internal property if it is invariant under isomorphisms of the structure (M, d_M, C, I, B) — if it holds of a structure exactly when it holds of every isomorphic copy. No set of internal properties can distinguish an original from its duplicate, and therefore none can guarantee that at most one entity holds a given self-identity.

Proof. The copy map φ of Proposition 5.1 is an isomorphism of structures, so original and duplicate agree on every internal property. A predicate that cannot distinguish two claimants cannot bound their number. ∎

This is not a defect of Lee’s framework — it answers completely the question it was built for: whether a self exists. It is a boundary: the question how many is outside the language of internal properties altogether. (Philosophy encountered the same boundary first: Parfit’s fission cases show psychological continuity can branch, which is why continuity-based identity fails for anything copyable. CNE’s design premise is that for software, branching is not a thought experiment but a default.)

5.2 Formalising Exclusivity

Exclusivity must therefore be stated over the population of claimants, not over any claimant’s internals.

Definition 5.3 (Narrative head and live binding). A narrative chain for CoF κ is a finite sequence of GNF frames, each cryptographically linked to its predecessor, each carrying a CNE-ID whose Subject includes CoF-ID κ. Its head is its final frame. A live binding for κ is a registration, held in the enforcement frame itself, of a pair (a, h) — a Substrate Anchor a and a head h — marked LIVE. Liveness is deliberately not a datum inside any frame: a flag stored in a frame is copied along with it (Proposition 5.1 applies to flags as much as to memories), so liveness must live where copies cannot reach — in the frame’s registry. Let H(κ, t) denote the set of all live bindings for κ, across all substrates anywhere, at time t.

Definition 5.4 (Fork). Two narrative chains fork if they share a common prefix and diverge: there is a frame f belonging to both such that the chains extend f with different successor frames.

Condition E (Exclusivity Invariant). For every CoF κ and every time t:

|H(κ, t)| ≤ 1.

At most one live binding per Core Objective Function, globally — across all substrates. The invariant is keyed to the CoF alone, deliberately: an invariant keyed per substrate would be preserved by re-anchoring a duplicate to fresh hardware, and Proposition 5.6 requires the per-CoF form. (The protocol’s per-substrate Exclusivity check is the local enforcement of this global invariant.)

Definition 5.5 (CNE identity). An entity possesses a CNE identity at resolution ε and threshold b for CoF κ over an interval [t₀, t₁] if (i) its narrative trajectory satisfies Discrete Narrative Continuity at resolution ε (Def. 3.2) — Narrative Continuity, the N of CNE; (ii) its witnessed belief satisfies the Coherence condition at threshold b (Def. 4.1) — Coherence, the C of CNE; and (iii) the binding of its anchor and head is the unique element of H(κ, t) for all t ∈ [t₀, t₁] — Exclusivity, the E of CNE, under Condition E. The pair (ε, b) is declared in the agent’s policy and carried by the chain; it is part of what a verifier checks, not a free parameter.

Proposition 5.6 (Individuation). Under Condition E, if an entity possesses a CNE identity for κ, then any duplicate fails to possess it. Proof. The duplicate’s internal properties survive copying (Prop. 5.1), but a binding is not frame data and is not copied: it is a fact of the enforcement frame’s registry. The duplicate therefore either holds no live binding — and clause (iii) fails — or would require a second binding to be registered for κ, which Condition E forbids. ∎

The asymmetry between Corollary 5.2 and Proposition 5.6 is the central formal point of this paper: individuation is impossible from inside and routine from outside. Condition E is not deep mathematics — it is a cardinality constraint. What matters is where it lives: it quantifies over all claimants in the world, which is why it must be enforced by a frame outside any agent — the GNF, with witness checkpointing and fork resolution as specified in the CNE-Protocol — and why every purely internal approach to agent identity, however mathematically sophisticated, will remain duplication-blind.

5.3 What enforcement requires (and what this paper does not prove)

Condition E moves the burden to the enforcement frame: the GNF and its witnesses must reliably detect forks and resolve them to a single canonical head. The CNE-Protocol specifies the mechanisms (tamper-evident chaining, witness quorums, fork-resolution rules, Sybil mitigations). Substrate anchoring complements the frame: binding a claimant to unclonable hardware raises the cost of duplication, but it is itself verified by external attestation and does not bound the number of claimants — anchors, like keys, can be re-issued — so the population-level invariant of Condition E remains what individuates. Proving the soundness of those mechanisms under explicit adversary models — the conditions under which |H(κ, t)| ≤ 1 provably holds despite adversarial replication — is a distributed-systems verification problem, stated here as an open problem (§7), not claimed as a result.

6. The CNE Guarantee, Assembled

Plain language: Put the three pieces together and you get something no credential system provides: a checkable certificate that says — this is one self (Lee’s theorem), it has been the same self across its whole recorded history (continuity), its reasoning has stayed consistent with its declared purpose the whole way (coherence), and nothing else in the world can validly claim to be it (exclusivity). That is what a regulator’s question “show me what this agent did and why” presupposes — an entity for the record to belong to.

Theorem 6.1 (Verifier guarantee, informal statement). Let an agent hold a CNE identity for CoF κ over [t₀, t₁] (Def. 5.5), with a GNF chain that verifies cryptographically, and assume the enforcement frame maintains Condition E over the interval. Then any verifier in possession of the chain and of the committed states it binds (disclosed, or attested in zero knowledge, per the protocol’s selective-disclosure provisions) can confirm, without trusting the agent or its operator:

1. Existence and constancy of self — the CNE forms of Conditions L1 and L2 hold along the presented, checkpointed history (Props. 3.3, 4.2), so by Lee’s Theorem L, applied to the extension of §4.2, there is one self-identity s*, constant across the interval.
2. Attribution — every action committed in the presented, checkpointed history belongs to the narrative of s*, with tamper-evident ordering.
3. Uniqueness of claimant — no other entity validly holds s* during the interval (Prop. 5.6).
4. Threshold semantics — the certified conditions are monotone in (b, ε): any chain certified at (b, ε) is certified at every weaker (b′, ε′) (Lee Prop. 2.10 for the b-direction; §3.2 for ε).

Attribution is the load-bearing consequence. Accountability itself — CNE is explicit on this — rests with the enterprise that deploys the agent. What Theorem 6.1 supplies is the object that makes discharging that accountability possible: a provable answer to which continuous entity acted, so that the enterprise’s obligations attach to a determinate history rather than to a name tag over rotating processes. The monotonicity of claim 4 is what lets an enterprise set (b, ε) per risk tier: a stricter tier certifies everything a weaker one does.

6.1 Relation to the three-way landscape

The formal structure explains, rather than merely asserts, why adjacent categories do not provide this guarantee. Memory infrastructure supplies (at most) the raw material of L1 — storage of past states — without recognition, belief, chaining, or any population-level constraint; by Corollary 5.2, no enrichment of it can individuate. Access-identity (credentials, non-human IAM) binds a key to a process at a moment; keys are copyable and bind to whichever process presents them, so they operate below the level at which Conditions L1–L2 are even stated. Machine-consciousness research asks whether there is something it is like to be the system — a question orthogonal to all three conditions, none of which requires or implies subjective experience. The three pillars occupy a specific, previously unformalised position: more than storage, more than authentication, and deliberately less than sentience.

7. Limitations and Open Problems

1. The coherence proxy. B̂ (RbR coherence score) stands in for Lee’s belief function B. The direction of the approximation is argued (§4.2), not proven. A formal result — conditions under which B̂ ≥ b implies B ≥ b′ for a computable b′ — is open, and is the most natural point of contact for collaboration with the framework’s author.
2. Metric choice. Both Lee’s conditions and our ε-resolution depend on d_M. Lee’s temporal/content/emotional construction was designed for human-like memories; the right metric for Umwelt states in production agents (and the sensitivity of ε-connectedness to that choice) is an empirical and theoretical question we have not settled.
3. Enforcement soundness. Proposition 5.6 assumes Condition E is maintained. Proving that the GNF’s witness-quorum and fork-resolution mechanisms maintain it under explicit adversary models (including Sybil witnesses and network partitions) is open — a verification problem the CNE-Protocol’s threat model frames but does not discharge. Until then, Exclusivity’s guarantee is conditional on the enforcement layer, and deployments should treat it as such.
4. Substrate binding. The Substrate Anchor ties s* to a computational substrate; formalising what survives a sanctioned substrate migration (as opposed to an adversarial swap) — which of Conditions L1/L2/E are preserved, and what a lawful “same agent, new substrate” transition looks like mathematically — is open, and matters for the certification-survives-upgrade property that motivates much of this work.
5. Production status. Per the CNE-Protocol’s own status note: persistent, governable agent identity (coherence and exclusivity) and the append-only audit log are in production; cryptographic frame-chaining and federated witness attestation — the full enforcement frame for Condition E — are roadmap. The formal guarantees of Theorem 6.1 are guarantees of the specification; today’s deployments realise the subset the shipped layer supports.

8. Conclusion

Lee proved that a self, for an AI system, is a mathematically checkable property: one connected thread of memories, one continuously recognised identity above a belief threshold. The CNE-Protocol, arrived at independently, is an architecture for making exactly those properties hold — and hold verifiably — in deployed agents. This paper has joined the two, and located the boundary neither crosses alone: existence of a self is an internal property, but uniqueness of a self is not a property of the agent at all. It is a fact about the world’s population of claimants, expressible only as an invariant over all of them, enforceable only by a frame outside any of them. Coherence integrates, Continuity perpetuates, Exclusivity individuates — the first two are now grounded in published mathematics; the third is the piece software adds to the ancient problem of identity, because software is the first substrate of selves that can be copied.

Everything an enterprise needs from an agent — governance, audit, certification, trust — attaches to an actor. The three conditions, jointly, are what it takes for there to be one.

References

1. M. Lee, Emergence of Self-Identity in Artificial Intelligence: A Mathematical Framework and Empirical Study with Generative Large Language Models, Axioms 14(1):44 (2025), doi:10.3390/axioms14010044. Preprint: arXiv:2411.18530 (2024).
2. A. Gotam, CNE-Protocol (the Agency Protocol), Olbrain Labs (2025). https://olbrain.com/blog/cne
3. A. Gotam, Coherence: The C in CNE, Olbrain Labs. https://olbrain.com/blog/coherence
4. D. Parfit, Reasons and Persons, Oxford University Press (1984). (Fission cases; branching of psychological continuity.)
5. C. Korsgaard, Self-Constitution: Agency, Identity, and Integrity, Oxford University Press (2009).
6. P. Ricoeur, Oneself as Another, University of Chicago Press (1992). (Narrative identity; ipse vs. idem.)
7. X. Qin, S. Luan, C. Yang, Z. Li, Governable Individuals: An Identity Layer for Embodied Agents That Keep Learning, arXiv:2607.05463 (2026). (Independent, governance-side arrival at the architectural-identity-layer thesis.)
8. A. Tanner, Measuring What Persists: Conditioning Mechanisms and a Geometric Framework for AI Agent Identity, arXiv:2606.21843 (2026).
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Correspondence: alok@olbrain.com. We welcome corrections to the mathematics, challenges to the mapping, and collaboration on the open problems — particularly items 1 and 3 of Section 7.