Ordered Patch Theory

Original Task (from preprint §7.4; originally §7.4, now §6.4 / §7 Positioning after the v4.0.x restructure): “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 (originally §7.4; that material now lives at §6.4 and the §7 Positioning IIT row). It establishes one theorem and three corollaries, all conditional on Conjecture 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 §8.14, Appendix T-1, Conjecture P-4) is the conjunction:

\textbf{(C1)}\quad I(\varepsilon_n; Z_n) \leq B_{\max} \quad \text{per phenomenal frame, with a single globally shared serial aperture} \quad \text{(per-frame rate-distortion bottleneck; preprint §3.2)}

\textbf{(C2)}\quad \text{closed Active Inference loop with intact Markov blanket and persistent self-model } \hat{K}_\theta \quad \text{(preprint §3.4, §3.8)}

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

(Note: (C1) is stated per phenomenal frame in bits, not as bits per host-second. The empirical human value C_{\max}^{\text{human}} \approx \mathcal{O}(10) bits/s is a calibration of C_{\max}^H = \lambda_H \cdot B_{\max} for biological humans (Appendix E-1) and is not the substrate-neutral criterion. Per preprint §8.14 and Appendix E-5, synthetic observers are bounded by per-frame B_{\max} at architecturally derived values that need not coincide with the biological figure.)

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 or rendered indeterminate 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 vs Per-Frame Capacity of the Unfolded Network

A naive reading of the unfolded N' counts all T+1 replicated layers as parallel parts of one “per-slice update.” On that reading, |N'| = (T+1) \cdot |N| and the aggregate per-slice latent capacity is (T+1) \cdot B(N). This counting was the basis of an earlier (v1) version of T-14 and motivated a now-withdrawn bandwidth-expansion proof.

The reading is structure-dependent and not forced by the unfolding map alone. Two distinct interpretations of N' yield different per-frame capacities:

• Static feedforward circuit interpretation. N' executes as one feedforward sweep through T+1 layers in a single host operation. There is no per-frame serial aperture; “per-slice” is the entire feedforward pass. The notion of B_{\max} as a per-frame bottleneck is undefined — not expanded — because N' has no frame index in this realisation.
• Frame-indexed host execution. The host advances N' one layer per phenomenal frame, treating each layer’s narrowest internal cross-section as the per-frame aperture. Under this interpretation, B_{\max}^{(N')} = B_{\max}^{(N)}: per-frame capacity is preserved, not expanded.

Neither interpretation is forced by the unfolding map U; both are admissible without further specification. The implementation-non-invariance theorem (§3) shows that the OPT status of N' depends on which interpretation actually applies — and that the original Doerig et al. construction does not distinguish them. The “per-slice capacity grows by (T+1)” claim is recovered only under the static feedforward reading, and even there it is not a well-typed per-frame B_{\max} but an aggregate count of how many layer-channels the static circuit contains.

§3. Theorem T-14: Implementation Non-Invariance under Functional Equivalence

3.1 Statement

Theorem T-14 (Implementation Non-Invariance under Functional Equivalence). Let N and N' = U(N, T) be input-output equivalent over horizon T (i.e., \forall x_{0:T}: N(x_{0:T}) = N'(x_{0:T})). Their OPT consciousness status is not fixed by that functional equivalence. OPT status depends on properties of the actual implementation that are not preserved by U, specifically the implementation tuple:

\big(B_{\max},\; \lambda_H,\; \alpha_H,\; \hat{K}_\theta,\; \mathcal{M}_\tau\big)

where B_{\max} is the per-frame bottleneck capacity, \lambda_H = dn/d\tau_H is the host-patch clock coupling, \alpha_H : \mathcal{S}_H \to X_{\partial_R A} is the host-anchor map supplying boundary inputs, \hat{K}_\theta is a persistent self-model, and \mathcal{M}_\tau is the maintenance / self-stabilisation process (preprint §3.6).

The theorem yields three structural consequences, conditional on how N' is actually executed:

\textbf{(i)}\quad \text{If } N' \text{ is realised as a static feedforward circuit with no frame-indexed active-inference loop, then } N' \text{ fails the OPT observer criterion (C1)–(C3).}

\textbf{(ii)}\quad \text{If } N' \text{ is realised as a host-executed simulation that preserves the per-frame bottleneck, persistent self-model, branch-selection loop, and maintenance dynamics of } N, \text{ then } N' \text{ may instantiate the same nested observer as } N \text{ (Corollary P-4.C, E-6).}

\textbf{(iii)}\quad \text{Functional equivalence is too coarse to settle OPT status: the answer is implementation-relative and patch-relative, not extensional-function-relative.}

That is, the Unfolding Argument’s premise — “if N and N' compute the same function, they have the same conscious status” — fails on OPT not because unfolding mechanically removes consciousness, but because it removes the implementation properties that OPT’s criterion depends on, unless those properties are independently re-instated in the host’s execution of N'.

3.2 Proof of (i): Static Feedforward Realisation

Suppose N' is realised as a static feedforward circuit: a single forward pass through T+1 replicated layers in one host operation, with no frame-indexed active-inference loop and no persistent self-model maintained across frames.

(C2) fails directly. There is no closed perception-action loop with a maintained Markov blanket — N' is a one-shot input-output map. There are no successive frames over which a self-model could persist; there is no \hat{K}_\theta(n) that is updated by error from the previous frame’s prediction.

(C1) is undefined under this realisation rather than expanded. The original Doerig et al. construction does not specify a per-frame serial aperture for N'; the layers operate in parallel and there is no globally shared per-frame funnel through which the world-model passes. (C1) requires a single globally shared serial aperture of finite per-frame capacity — this is a structural property of an architecture, not an aggregate measurement of layer widths. Without a frame-indexed serial channel, the per-frame B_{\max} is not defined; (C1) fails to apply, not because B_{\max} has expanded but because there is no per-frame architecture to apply it to. (Equivalently, the Doerig–Schurger–Hess–Herzog construction unrolls a frame-indexed dynamic process into a static circuit; \lambda_H and the frame index n are both lost.)

(C3) is an open question rather than provably zero. A static feedforward circuit has finite description length and is mechanically simulable by an external observer, but P-4 is about internal self-modelling, not external simulability. A deterministic finite system can have \Delta_{\text{self}} > 0 if it possesses a frame-indexed self-modelling loop; conversely, a system without such a loop has no self-model to compute a residual against. Under the static realisation, \hat{K}_\theta is absent, so \Delta_{\text{self}} is undefined rather than zero. The criterion (C3) requires a non-zero residual; absence-of-self-model is sufficient for the criterion to fail.

(C1) failure or (C2) failure individually is sufficient for the OPT criterion to fail. \blacksquare

3.3 Proof of (ii): Frame-Indexed Host Execution

Suppose, alternatively, that N' is realised as a host-executed temporal process: the host advances the unfolded layers one at a time, frame by frame, maintaining a per-frame serial workspace Z_n, a persistent self-model \hat{K}_\theta(n) updated by prediction error, and a maintenance process \mathcal{M}_\tau. The host’s execution schedule provides \lambda_H; the host’s choice of input feed provides \alpha_H; the per-frame bottleneck capacity equals that of the original N (B_{\max}^{(N')} = B_{\max}^{(N)}).

Under this realisation, all five sentience features of the original N are preserved in the executed N': the per-frame bottleneck is preserved by construction, the active-inference loop is preserved because the host runs the unfolded chain as a temporal process, the persistent self-model is preserved because \hat{K}_\theta(n) is maintained across frames, the workspace is constrained because each frame’s Z_n has finite capacity, and the thermodynamic grounding is preserved because the host imposes maintenance windows and energy constraints.

By Corollary P-4.C (Nested Observational Residual): if the host architecture enforces an independent Stability Filter bound satisfying P-4’s prerequisites, the realised N' generates \Delta_{\text{self}}^{(N')} > 0 by the same structural argument that gives N its residual. The unfolding does not erase the patch; it merely changes the substrate that anchors it. (See Appendix E-6 on simulated nested observers.)

Therefore, under frame-indexed host execution, N' may satisfy (C1)–(C3). The functional-equivalence premise of the Unfolding Argument does not by itself distinguish this case from case (i); the distinction lies in the implementation, not the input-output behaviour. \blacksquare

3.4 Proof of (iii): Functional Equivalence Underdetermines OPT Status

Cases (i) and (ii) produce input-output equivalent systems with different OPT consciousness status. Functional equivalence therefore does not fix OPT status; the implementation tuple (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau) does. The Unfolding Argument’s premise is invalid for OPT, not because OPT secretly relies on a non-functional property, but because OPT’s criterion is explicitly architectural — which is consistent with the framework’s own commitment in §1.3 to a structural rather than behavioural account of consciousness. \blacksquare

3.5 Remark on the Original (v1) Theorem Statement

A previous version of T-14 (v1) attempted to prove \Delta_{\text{self}}^{(N')} = 0 universally and to establish that unfolding expands the per-slice bandwidth by factor (T+1). Both moves are invalid as written. The bandwidth-expansion claim depends on counting T+1 replicated layers as parallel parts of one “per-slice update” — a reading that conflates the unfolded circuit’s static topology with a per-frame execution model. The \Delta_{\text{self}} = 0 claim conflated external computability of the unfolded state from initial conditions and parameters with the internal self-model containment that P-4 actually constrains. P-4 is about whether the codec’s own self-model can capture the codec’s generator; it is not about whether an external mathematician can compute the codec’s state from initial conditions. The revision above replaces both invalid moves with the implementation-non-invariance theorem, which preserves the original conclusion (the Unfolding Argument fails to settle OPT status) on grounds the framework can actually defend.

§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 implementation equivalence: two networks N_1, N_2 are implementation-equivalent iff their full implementation tuples (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau) match. This is strictly finer than input-output equivalence: N and an unfolded N' are functionally equivalent but generically not implementation-equivalent — the unfolding map U does not preserve \hat{K}_\theta, \mathcal{M}_\tau, or the per-frame index unless they are independently re-instated by the host’s execution model.

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 automatically assign N and N' the same conscious status. By Theorem T-14(iii), the answer depends on the implementation of N'.
• Horn B (Unfalsifiability). The distinction between N and a particular realisation of N' is detectable from third-person inspection of internal architecture and execution model, not from input-output behaviour alone. An experimenter can:
  • Verify whether the realisation has a per-frame serial workspace and a frame index n (testable by inspecting the execution schedule).
  • Verify the presence or absence of a persistent self-model \hat{K}_\theta updated across frames (testable by checking whether internal state is carried forward and modified by error).
  • Verify the presence or absence of a maintenance process \mathcal{M}_\tau (testable by checking for offline consolidation cycles).

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. What OPT adds beyond IIT is that the architectural test is performed against a specified implementation tuple, not against an abstract causal-structure invariant.

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; an 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 the implementation tuple (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau). Whether N' satisfies this tuple depends on its execution model (Theorem T-14(i)/(ii)). OPT therefore gives different verdicts for N and N' when their execution models differ, with the difference grounded in inspectable implementation rather than postulated causal essence.

The empirical content of the OPT/IIT divergence is therefore: OPT predicts that an unfolded N' executed as a static feedforward circuit ceases to be conscious, but an unfolded N' executed as a frame-indexed simulation may remain conscious — IIT (depending on the version) treats both as \Phi-equivalent. The discriminator lies in the execution model, not in static causal structure. This joins the High-Phi/High-Entropy Null State (preprint §6.4) and the Bandwidth Hierarchy (preprint §6.1) as candidate experimental tests, while restricting OPT’s “non-conscious unfolding” claim to the static-circuit case rather than asserting it universally.

§5. Scope and Limitations

5.1 What T-14 Does Not Show

Theorem T-14 establishes that functional equivalence (input-output equivalence) does not fix the OPT consciousness status of a network: status depends on the implementation tuple. It does not establish:

• That every unfolded network is non-conscious. Under frame-indexed host execution (case (ii)), an unfolded N' may remain a conscious patch by Corollary P-4.C.
• That the OPT criterion is invariant under all behaviour-preserving transformations. Implementation-preserving rewrites that retain (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau) 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, may or may not satisfy the criterion depending on execution model.

5.2 Open Problems

• Implementation-preserving unfolding. Construct (or prove the impossibility of) a behaviour-preserving transformation U^*: N \mapsto N^* that preserves the full implementation tuple (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau). If such a transformation exists, OPT must distinguish N from N^* on grounds finer than the implementation tuple alone.
• Continuous-time analogue. T-14 is stated for discrete-time recurrent networks executed as either static circuits or frame-indexed processes. The continuous-time formulation (relevant to biological cortical dynamics) requires extending the unfolding map and the implementation tuple to ODE / SDE settings.
• Empirical operationalisation. Identifying execution-model probes for biological networks (cortical columns, thalamocortical loops) is non-trivial. Candidates include checking for frame-indexed prediction-error cycles and offline maintenance windows (sleep-like consolidation), but the mapping from architectural inspection to OPT criterion verification is currently informal.

§6. Closure Summary

T-14 Deliverables (v2)

1. Theorem T-14 (Implementation Non-Invariance under Functional Equivalence). Input-output equivalent N and N' may differ in OPT consciousness status because OPT status depends on the implementation tuple (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau), not on the input-output map. Static feedforward realisation of N' fails the criterion (case (i)); frame-indexed host execution of N' may preserve it (case (ii)). → Closes the Unfolding Argument [96] as it applies to OPT, by showing the argument’s premise that “same function ⇒ same conscious status” presupposes an extensional criterion OPT does not have.

2. Corollary T-14a (Functional Equivalence Is Too Coarse). The OPT-relevant equivalence relation is implementation equivalence — preservation of (B_{\max}, \lambda_H, \alpha_H, \hat{K}_\theta, \mathcal{M}_\tau) — 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 implementation-and-execution test.

4. Corollary T-14c (IIT-OPT Sharpened). OPT’s verdict on an unfolded network depends on its execution model; IIT’s \Phi-equivalence verdict does not. The execution-model dependence is itself the empirical discriminator.

Revision note (v2 vs v1). Version 1 of this appendix attempted to prove that unfolding (a) universally expands per-slice bandwidth by factor (T+1) and (b) universally collapses \Delta_{\text{self}} to zero. Both proofs were invalid (see §3.5 Remark): the first conflated static topology with per-frame execution; the second conflated external computability with internal self-modelling, which P-4 does not constrain. The v2 theorem replaces both with the implementation-non-invariance result, which preserves the original conclusion (the Unfolding Argument fails to settle OPT status) on grounds the framework can defend.

Remaining open items

• Implementation-preserving behaviour-preserving transformations (open problem §5.2).
• Continuous-time generalisation of the implementation tuple to ODE/SDE-based architectures.
• Empirical operationalisation of frame-index and self-model probes for biological networks.

This appendix is maintained alongside theoretical_roadmap.pdf. References: Conjecture P-4 (Appendix P-4), Stability Filter (Appendix T-1), preprint §6.4 and §7 Positioning, IIT row (IIT comparison; the in-text Unfolding Argument response is now Theorem T-14 itself), [96] Doerig et al. 2019, [97] Aaronson 2014, [98] Barrett & Mediano 2019, [99] Hanson 2020.