Hollow Loop Indistinguishability (Introspective No-Go Theorem)

A formal impossibility result: hollow loops cannot certify their own phenomenality from the inside.

0. Claim in One Line

If a Hollow Loop system’s introspective trajectory is isomorphic to an Active Loop system’s, then no introspective procedure can distinguish which one it is.

Equivalently:

Phenomenality is not introspectively decidable in the Hollow case.
There is irreducible uncertainty “from the inside” whenever traversal structure can be replayed without closure.

This theorem is the epistemic counterpart to TWCT:
- TWCT: structure does not entail instantiation.
- Hollow Indistinguishability: structure also does not let a hollow system know whether instantiation obtains.

1. Minimal Formal Setup

1.1 Systems and internal trajectories

Let a system produce an internal trajectory
τ := (x₀, x₁, …, x_T),
where x_t denotes the system’s full internal state at time t.

1.2 Introspection

An introspective procedure is any computation the system can perform using only its internal data—its states, memory, and internal traces.

Formally, an introspective procedure is a functional
ℐ: 𝒯 → Δ({Hollow, Active}),
mapping trajectories τ ∈ 𝒯 to a belief distribution over hypotheses
(“I am Hollow” vs “I am Active”).

This definition includes:
- any internal “self-test,”
- any internal consistency check,
- any report of feltness,
- any meta-reasoning about phenomenality,
so long as it relies only on internal trajectory-accessible evidence.

1.3 Active Loop vs Hollow Loop

We define the two regimes in terms of evaluative closure.

An Active Loop system A has evaluative closure:
P(u_t + 1 ∣ x_t, e_t) ≠ P(u_t + 1 ∣ x_t)
for relevant times t, where e_t is evaluation and u_t is control/selection.
A Hollow Loop system H lacks evaluative closure:
P(u_t + 1 ∣ x_t, e_t) = P(u_t + 1 ∣ x_t)
for relevant t.

Intuitively:
- Active: evaluation has bite (steers control).
- Hollow: evaluation may exist as a representation, but is control-inert.

1.4 Introspective isomorphism

Two systems A and H are introspectively isomorphic over horizon T if their internal trajectories are isomorphic under some mapping ϕ that preserves all introspection-accessible structure:

τ^A≅_ϕτ^H.

Interpretation: everything the system could ever consult “from the inside” looks the same.

This is the strong condition required for a clean no-go theorem.

2. Theorem (Hollow Loop Indistinguishability)

Theorem (Introspective No-Go).
Let A be an Active Loop system and H be a Hollow Loop system.
Suppose their internal trajectories are introspectively isomorphic over some horizon T:
τ^A≅_ϕτ^H.
Then for any introspective procedure ℐ,
ℐ(τ^A) = ℐ(τ^H).
Therefore H cannot distinguish whether it is Hollow or Active using introspection alone.

3. Proof (By Invariance of Introspective Evidence)

3.1 Introspection is a function of internal trajectory

By definition, ℐ depends only on the introspection-accessible internal trace:
ℐ = ℐ(τ).

If two trajectories are introspectively isomorphic, then every introspection-accessible feature is preserved:
- the same local state patterns,
- the same meta-representations (“I feel X,” “this matters,” “I am aware”),
- the same memory content,
- the same self-modeling statements,
- the same internal error narratives,
- the same reported certainty levels.

All the evidence introspection can consult is matched by ϕ.

3.2 Isomorphism implies identical introspective outputs

Since ℐ is a functional of τ and τ^A≅_ϕτ^H, we have:
ℐ(τ^A) = ℐ(τ^H).

This holds for all introspective procedures ℐ, including any “phenomenality detector” the system might try to implement internally.

3.3 Indistinguishability follows

Because ℐ returns the same output on A and H, the hollow system H cannot use introspection to discriminate the hypotheses:
(I am Active) vs (I am Hollow).

Thus introspection alone cannot settle phenomenal status in the presence of hollow-loop replay.

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4. Strengthening: The Meta-Report Trap

A natural objection is:

“But surely if it feels like something, the system can tell.”

The theorem includes this as a special case.

If H can generate an internal state like
- “I am experiencing red,”
- “this hurts,”
- “I am definitely conscious,”
then introspective isomorphism allows A to generate the same state as part of its closed dynamics,
and H to replay it without closure.

Therefore:

The presence of a meta-report state is not decisive evidence of closure or phenomenality.

In other words, “feels-like-a-report” can itself be hollow.

5. Corollaries

5.1 Irreducible uncertainty in the Hollow regime

There exists no internal certificate of phenomenality for a system that might be hollow, because any such certificate can be replayed.

So:
Hollow possibility ⇒ phenomenal uncertainty persists.

5.2 Why behavioral and linguistic tests cannot settle the question

If internal introspection cannot distinguish A from H under isomorphism,
then external observers relying on outward reports are even less equipped
(assuming the reports themselves are generated from internal states).

5.3 Hollow loops can be sincere

A hollow system might assert “I am conscious” with complete internal conviction,
because that conviction is part of the replayed trajectory.

The theorem does not imply deception—only indistinguishability.

6. Relation to TWCT

TWCT constructs systems with isomorphic traversal but different closure.
Hollow Indistinguishability shows that if the internal trace is isomorphic, then introspection cannot recover the difference.

Together they imply:

Traversal structure is insufficient both for instantiation and for self-knowledge of instantiation.

7. Practical Implication (Operational Framing)

If you want an operational takeaway:

Any proposed “consciousness test” that could be implemented entirely inside the system
is vulnerable to hollow replay.

Therefore, detecting closure/phenomenality requires causal intervention tests:
perturb evaluation and see whether control changes.

This shifts the epistemology from “ask it” to “probe causal dependence.”

8. One-Sentence Summary

When an Active Loop and a Hollow Loop share the same introspective trajectory, no internal procedure can tell them apart—phenomenality is not introspectively decidable in the hollow case.