Ordered Patch Theory

Original Task (from preprint §7.4): “Address the Doerig–Schurger–Hess–Herzog Unfolding Argument [96] against causal-structure theories of consciousness, and demonstrate that OPT’s consciousness criterion is not vulnerable to it.” Deliverable: Formal theorem that OPT’s bandwidth-bottleneck plus \Delta_{\text{self}} criterion is not invariant under functional equivalence; corollaries identifying the precise structural property the Unfolding Argument fails to preserve.

Closure status: DRAFT STRUCTURAL CORRESPONDENCE. This appendix formalises the response sketched discursively in preprint §7.4. It establishes one theorem and three corollaries, all conditional on Theorem P-4 (Algorithmic Phenomenal Residual) and Appendix T-1 (Stability Filter rate-distortion specification). No equations of T-1 or P-4 are altered; this appendix derives a structural invariance property from them.

§1. Background and Motivation

1.1 The Unfolding Argument

Doerig, Schurger, Hess & Herzog [96] advance the following dilemma against any causal-structure theory of consciousness — explicitly Integrated Information Theory (Tononi [8]) and Recurrent Processing Theory (Lamme), and by extension any framework asserting that consciousness is fixed by the network’s recurrent causal organisation.

The argument. For any recurrent network N with bounded compute and any finite horizon T, there exists a feedforward network N' — the temporal unfolding of N — such that:

1. N and N' are functionally equivalent over T: they produce identical input-output mappings for every admissible input sequence of length \leq T.
2. N' contains no recurrent connections: every layer feeds strictly forward to the next.
3. N' is constructible by mechanical procedure (the standard “unrolling” of N across T time steps).

If consciousness is identical to causal structure, then either:

• (Horn A — Falsity). N and N' have the same conscious status, so feedforward networks are conscious whenever functionally-equivalent recurrent ones are. This contradicts the central claim of causal-structure theories that recurrence is constitutive of consciousness.
• (Horn B — Unfalsifiability). N is conscious and N' is not, despite identical input-output behaviour. Then consciousness is undetectable from any third-person observation of system behaviour, and the theory cannot be tested.

The dilemma is sharp because the construction of N' from N is mechanical and behaviour-preserving; no causal-structure theorist has succeeded in identifying a behaviourally observable property that distinguishes the two.

1.2 Why OPT Is Not a Direct Target — and Why a Formal Reply Is Still Needed

OPT is not a causal-structure theory in Doerig et al.’s sense: it does not assert that consciousness supervenes on recurrence per se. The OPT consciousness criterion (preprint §7.8, Appendix T-1, Theorem P-4) is the conjunction:

\textbf{(C1)}\quad B_{\max} \leq C_{\max} \approx \mathcal{O}(10) \text{ bits/s} \quad \text{(rate-distortion bottleneck)}

\textbf{(C2)}\quad \text{closed Active Inference loop with intact Markov blanket} \quad \text{(preprint §3.4, §3.8)}

\textbf{(C3)}\quad \Delta_{\text{self}} > 0 \quad \text{(Phenomenal Residual; Theorem P-4)}

None of (C1)–(C3) is a property of recurrence in isolation. However, an honest engagement with [96] requires showing that the OPT criterion is not invariant under the unfolding map U: N \mapsto N' — i.e., that some component of (C1)–(C3) is broken by unfolding even though the input-output mapping is preserved. Otherwise the dilemma migrates: if (C1)–(C3) were invariant under U, OPT would reduce to a behaviourist theory and inherit Horn B regardless of its surface formalism.

This appendix establishes the non-invariance directly.

§2. Formal Setup

2.1 The Unfolding Map

Let N = (V, E, f, h_0) be a discrete-time recurrent network with vertex set V, edges E (including self-loops and within-layer recurrent edges), update function f, and initial hidden state h_0. Let |N| = |V| denote its node count, and let B(N) denote the per-cycle latent-channel capacity of N’s narrowest internal cross-section, measured in bits per update.

Given a finite horizon T \geq 1, the unfolding U(N, T) = N' is the feedforward network obtained by:

1. Replicating the substrate of N once per time step: V' = \bigsqcup_{t=0}^{T} V_t, with V_t a copy of V at time t.
2. Replacing every recurrent edge u \to v in N with a forward edge u_t \to v_{t+1} in N' for each t < T.
3. Removing all self-loops and intra-layer connections.

The standard result (Goodfellow, Bengio, Courville, Deep Learning, ch. 10) is that N' computes the same input-output mapping as N over horizon T:

\forall x_{0:T}: \quad N(x_{0:T}) = N'(x_{0:T}) \quad \text{(functional equivalence over } T\text{)}.

This is the construction Doerig et al. invoke.

2.2 Per-Slice Capacity of the Unfolded Network

Crucially, |N'| = (T+1) \cdot |N|. The unfolded network has T+1 disjoint layers, each containing a full copy of the recurrent substrate. The per-time-slice latent channel of N' has capacity:

B(N')_{\text{per-slice}} = (T+1) \cdot B(N) \quad \text{(in the worst case: all replicated layers active in parallel)}.

The per-slice capacity grows linearly with the unfolding horizon T. There is no value of T at which B(N') remains equal to B(N) at the per-slice level: unfolding expands the latent channel by replication.

§3. Theorem T-14: Bandwidth-Structure Non-Invariance

3.1 Statement

Theorem T-14 (Bandwidth-Structure Non-Invariance under Functional Equivalence). Let N be a recurrent network satisfying the OPT consciousness criterion (C1)–(C3). Let N' = U(N, T) be its unfolding for any T \geq 1. Then:

\textbf{(i)}\quad B(N') \;>\; B(N) \quad \text{by a factor of at least } (T+1) \text{ at the per-slice level};

\textbf{(ii)}\quad \Delta_{\text{self}}^{(N')} = 0;

\textbf{(iii)}\quad N' \text{ does not satisfy the OPT consciousness criterion.}

That is, functional equivalence does not preserve the OPT consciousness criterion. The premise of the Unfolding Argument — “if N and N' compute the same function, they have the same conscious status” — fails on OPT for structural reasons internal to (C1)–(C3).

3.2 Proof of (i): Capacity Expansion

By construction (§2.2), |N'| = (T+1)|N| and the latent channels of the T+1 replicated layers operate in parallel within a single per-slice update of N'. The aggregate per-slice capacity is therefore (T+1) \cdot B(N). For any T \geq 1, this strictly exceeds B(N). \blacksquare

Remark. A more refined analysis can lower-bound the effective bottleneck of N' to B(N) in the limit where N'’s layers are forced to communicate only through a single shared channel, but such an analysis requires adding a fresh bandwidth constraint to N' that was not present in the unfolding map. The unmodified output of U has capacity (T+1) \cdot B(N), which violates (C1) for any T such that (T+1) \cdot B(N) > C_{\max}. Since N saturates (C1) by hypothesis, any T \geq 1 produces a violating N'.

3.3 Proof of (ii): Collapse of \Delta_{\text{self}}

Theorem P-4 establishes that the Phenomenal Residual \Delta_{\text{self}} > 0 obtains exactly when the codec K_\theta admits no exact internal description \hat{K}_\theta such that K(\hat{K}_\theta) = K(K_\theta), where K denotes Kolmogorov complexity. The non-vanishing residual depends on self-reference within a single update cycle: \hat{K}_\theta(t) is required to model K_\theta(t) during the same time-slice t.

In the unfolded network N', no layer at time-slice t contains recurrent self-reference. The state of layer V_t is fully determined by the input layer V_0 (and the externally-supplied input sequence x_{0:t}) via a finite composition of t feedforward maps. Concretely, the Kolmogorov complexity of the state at V_t given V_0 and the network parameters is bounded by:

K\big(\text{state}(V_t) \mid V_0, \theta_{\text{params}}\big) \leq O(\log t) + \text{const},

since the state is computable in O(t) time by mechanical evaluation of the feedforward chain. Each layer admits an exact internal description from data already present in earlier layers; there is no residual algorithmic gap of the kind required for \Delta_{\text{self}} > 0.

Formally: let \hat{K}_\theta^{(N')} be the model of K_\theta^{(N')} formed by composing the layer-to-layer feedforward maps of N'. Then K\big(\hat{K}_\theta^{(N')}\big) = K\big(K_\theta^{(N')}\big) by construction, so by Theorem P-4 the Phenomenal Residual vanishes:

\Delta_{\text{self}}^{(N')} = 0. \qquad \blacksquare

3.4 Proof of (iii): Criterion Failure

(C1) fails by (i) for any unfolding horizon T \geq 1 (capacity exceeds C_{\max}); (C3) fails by (ii). The OPT consciousness criterion requires the conjunction of (C1)–(C3); failure of either is sufficient. \blacksquare

§4. Corollaries

4.1 Corollary T-14a: Functional Equivalence Is Too Coarse

Corollary T-14a. Input-output functional equivalence is too coarse a relation to fix the OPT conscious status of a network. The relevant equivalence relation is bandwidth-structure equivalence: two networks N_1, N_2 are bandwidth-structure equivalent iff (a) their narrowest per-cycle latent channels have equal capacity, and (b) their internal self-reference graphs admit the same Phenomenal Residual under Theorem P-4.

This is strictly finer than functional equivalence: an unfolded N' is functionally equivalent to N but not bandwidth-structure equivalent.

4.2 Corollary T-14b: The Unfolding Dilemma Does Not Apply to OPT

Corollary T-14b. OPT is positioned on neither horn of the Doerig et al. dilemma:

• Horn A (Falsity). OPT does not assign N and N' the same conscious status. By Theorem T-14, N' violates the criterion.
• Horn B (Unfalsifiability). The distinction between N and N' is detectable from third-person inspection of internal architecture, not from input-output behaviour alone. Specifically, an experimenter can:
  • Measure the per-cycle latent-channel capacity of the narrowest internal cross-section (testable via information-bottleneck probes; cf. Tishby et al.).
  • Verify the presence or absence of within-cycle self-reference (testable by inspecting the network’s connectivity graph).

OPT therefore evades the dilemma by granting that input-output behaviour underdetermines conscious status — this is not a bug, because OPT’s criterion is explicitly an internal-architectural one, not a behavioural one.

4.3 Corollary T-14c: The IIT-OPT Distinction Sharpens

Corollary T-14c. Theorem T-14 yields a clean structural distinction between OPT and IIT under the Unfolding Argument:

• IIT’s \Phi is computed over the system’s transition probability matrix; the unfolded N' has a different transition matrix than N (because connectivity differs), but Doerig et al. argue that the causal structure relevant to function is preserved, leaving IIT on Horn A or Horn B.
• OPT’s criterion is bandwidth + within-cycle self-reference; both quantities are demonstrably altered by unfolding (per (i) and (ii) above), so the OPT criterion legitimately gives different verdicts for N and N', with the difference grounded in inspectable internal structure rather than postulated causal essence.

The empirical content of the OPT/IIT divergence is therefore: OPT predicts that an unfolded high-\Phi network ceases to be conscious, while IIT (depending on the version) predicts either that it remains conscious (Horn A) or that the question is empirically meaningless (Horn B). This is a candidate experimental discriminator joining the High-Phi/High-Entropy Null State (preprint §6.4) and the Bandwidth Hierarchy (preprint §6.1).

§5. Scope and Limitations

5.1 What T-14 Does Not Show

Theorem T-14 establishes that functional equivalence (input-output equivalence) does not preserve the OPT consciousness criterion. It does not establish:

• That the OPT criterion is invariant under all behaviour-preserving transformations (e.g., bandwidth-preserving graph rewrites that retain within-cycle self-reference may preserve consciousness; this is left open).
• That consciousness is exhausted by (C1)–(C3); these are necessary conditions and the framework does not claim they are individually or jointly sufficient absent the broader Stability Filter context.
• That every recurrent network satisfying (C1)–(C3) is conscious; the appendix only shows that the unfolded counterpart of one that is satisfies the criterion strictly less.

5.2 Open Problems

• Bandwidth-preserving unfolding. Construct (or prove the impossibility of) a behaviour-preserving transformation U^*: N \mapsto N^* that preserves both per-cycle bandwidth and within-cycle self-reference. If such a transformation exists, OPT must distinguish N from N^* on grounds finer than (C1)–(C3) alone.
• Continuous-time analogue. Theorem T-14 is stated for discrete-time recurrent networks. The continuous-time formulation (relevant to biological cortical dynamics) requires extending the unfolding map to ODE / SDE settings; the per-slice capacity argument (§3.2) generalises, but the Phenomenal Residual argument (§3.3) requires a continuous analogue of within-cycle self-reference that has not yet been formalised.
• Empirical operationalisation. Identifying bandwidth-bottleneck and within-cycle self-reference probes for biological networks (cortical columns, thalamocortical loops) is non-trivial; candidates include per-area mutual-information bottlenecks (Tishby), but the mapping from architectural inspection to OPT criterion verification is currently informal.

§6. Closure Summary

T-14 Deliverables

1. Theorem T-14 (Bandwidth-Structure Non-Invariance). Functional equivalence under unfolding does not preserve the OPT consciousness criterion: per-cycle bandwidth is expanded by factor (T+1), and the within-cycle self-reference required for \Delta_{\text{self}} > 0 is destroyed. → Closes the Unfolding Argument [96] as it applies to OPT.

2. Corollary T-14a (Functional Equivalence Is Too Coarse). The OPT-relevant equivalence relation is bandwidth-structure equivalence — strictly finer than input-output functional equivalence.

3. Corollary T-14b (No Dilemma for OPT). OPT is positioned on neither horn of Doerig et al.’s dilemma: it grants that behaviour underdetermines conscious status (because its criterion is architectural) and supplies an inspectable architectural test.

4. Corollary T-14c (IIT-OPT Sharpened). Unfolded high-\Phi networks provide a candidate experimental discriminator between IIT and OPT, joining the High-Phi/High-Entropy Null State and Bandwidth Hierarchy.

Remaining open items

• Bandwidth-preserving behaviour-preserving transformations (open problem §5.2).
• Continuous-time generalisation of the within-cycle self-reference argument.
• Empirical operationalisation of bandwidth and self-reference probes for biological networks.

This appendix is maintained alongside theoretical_roadmap.pdf. References: Theorem P-4 (Appendix P-4), Stability Filter (Appendix T-1), preprint §7.4 (IIT comparison and Unfolding Argument response), [96] Doerig et al. 2019, [97] Aaronson 2014, [98] Barrett & Mediano 2019, [99] Hanson 2020.