Appendix P-4: The Algorithmic Phenomenal Residual

Original Task P-4: The Phenomenal Residual Problem: Phenomenal consciousness requires a formal mathematical locus differentiating it from zero-interiority computation. Deliverable: Formulation isolating the inevitable computational blind spot of an algorithmically bounded Active Inference model.

Closure status: RESOURCE-BOUNDED SELF-MODEL INCOMPLETENESS CONJECTURE (v3.6.4 demotion). Earlier versions of this appendix presented P-4 as a formal theorem proving \Delta_{\text{self}} > 0. An external structural review (appendix-corrections memo §2.2) correctly noted that the earlier formulation had three gaps: (i) \Delta_{\text{self}} was not formally defined as a measurable / mathematical object before the theorem statement; (ii) Lemma 2’s appeal to Chaitin uncomputability and Gödel incompleteness conflated formal-system unprovability with the operational residual the theorem actually needs; (iii) the “floor independent of B_{\max}” claim required a diagonal-impossibility argument that the appendix did not supply. v3.6.4 demotes P-4 from theorem to resource-bounded self-model incompleteness conjecture with an operational testing programme: \Delta_{\text{self}}(h, B) is now defined as an explicit bounded-class predictive-residual functional (§4 below), the diagonal/self-reference incompleteness framing replaces the Chaitin appeal in Lemma 2, and the operational decomposition of §5 (already conjecture-tagged in v2.5.3) is preserved unchanged as the empirical testing programme.

What survives the demotion: Lemma 1’s structural necessity of a forward-generative self-model under tight action-state coupling and Assumption P-4.2. Lemma 2 with diagonal/self-reference incompleteness as the operative argument. The conjecture \Delta_{\text{self}}(h, B) > 0 under explicit resource bounds, self-reference, environmental coupling, action feedback, and prediction-horizon assumptions. The K_{\text{threshold}} open problem. The structural identification of \Delta_{\text{self}} with the locus of subjective experience.

What changes: P-4 is now systematically called Conjecture P-4 throughout the appendix. The “mathematically guaranteed” and “formally proves” language is replaced with “conjectured under the listed assumptions” and “structurally identified, conditional on the bounded-class definition.” The “formally locates where the spark of experience must reside” formulation in §6 is replaced with the more honest “OPT proposes this residual as the structural correlate of subjectivity, while acknowledging that the zombie gap is not closed.”

2026-06 capacity reframe (post-A2): the Lemma 2 statement in §3 now carries, in the statement itself, the capacity-sourced correction of opt-theory-memo-a2-regret-floor.md — the residual is the bounded-vs-unbounded coding gap L_{\mathcal{C}_B} - L^\star on the self-channel of the closed action-perception loop, a budgeted capacity cost rather than a self-reference/incompleteness effect (that horn cancels in the regret), and the diagonal survives only as the non-purchasability argument (it bounds loss, not regret). The “diagonal/self-reference incompleteness as the operative argument” reading above is superseded accordingly. Tier unchanged: conjecture.

This appendix presents Conjecture P-4 (Resource-Bounded Self-Model Incompleteness), identifying the structural correlate of phenomenal consciousness within the Ordered Patch Theory (OPT). We conjecture that any active inference system constrained by a finite predictive bandwidth (C_{\max}) possesses an unmodellable informational residual \Delta_{\text{self}}(h, B) > 0 over prediction horizon h within bounded model class \mathcal{C}_B, conditional on structural Assumptions P-4.1 and P-4.2. The conjecture does not dissolve the “Hard Problem”; OPT proposes this residual as the structural correlate of subjectivity, while acknowledging that the zombie gap is not closed.

1. The Locus of the Hard Problem

In earlier versions of OPT, consciousness was formally boxed into a specific structural locus: the traversal of the C_{\max} informational aperture. However, the exact nature of the subjective interiority—the qualia of the experience—was left as an irreducible “Agency Axiom”. Treating phenomenology as purely axiomatic leaves the theory vulnerable to the “Hard Problem”: why does navigating the Free Energy topology feel like anything at all?

Here, we translate this philosophical gap into algorithmic information theory (AIT). While we do not claim to derivationally conjure subjective feeling out of pure math (the Zombie Gap remains open), we argue that the structural properties of qualia map onto a conjectured, un-modellable residual generated by any finite computing system attempting to model its own recursive dynamics.

2. Lemma 1: The Necessity of the Predictive Self-Model

Under OPT, the observer (the Codec K_{\theta}) exists behind a Markov Blanket (the topological boundary \partial_R A). The observer survives by executing Active Inference, minimizing prediction error over time via cyclic updates.

Because the system possesses active states that perturb the external boundary, the incoming sensory states \varepsilon_t are a tightly coupled mixture of external environmental dynamics and the consequences of the observer’s own actions A_t.

Lemma 1: For tightly-coupled OPT active-inference architectures where the action-state loop is informationally inseparable (i.e., the boundary mutual information I(A_t ; X_{\partial_R A}) does not factor cleanly), achieving stable free energy minimization under a strict predictive bottleneck (C_{\max}) operates such that the minimum-complexity mechanism satisfying the internal constraints structurally maps as a forward-generative self-model.

Formal Condition: 1. Let the codec’s actions be A_t. The boundary state is X_{\partial_R A} = f(\text{Environment}, A_t). 2. To compress prediction error \varepsilon_{t+1} and satisfy the rate-distortion objective (R \le C_{\max}, D \le D_{\min}), the codec must isolate and subtract true environmental variance from its self-generated causal perturbations. 3. Assumption P-4.2 (Inverse Mapping Inadequacy): For OPT-native architectures operating at sufficient scale (e.g., across high-dimensional action manifolds or long causal chains), we formally assume efference-copy mechanisms and retroactive subtraction alone are architecturally inadequate to clear the precise D_{\min} rate-distortion bounds across the spatial manifold. 4. Therefore, isolation functionally necessitates evaluating a forward-generative prediction of the consequences of A_{t+1}. Executing a forward prediction of its own internal causal architecture traversing the state space constitutes a predictive causal proxy — a localized self-model \hat{K}_{\theta} — internal to its architecture. \blacksquare

3. Lemma 2: The Capacity-Sourced Self-Channel Residual (v3.6.4 demotion; 2026-06 capacity reframe)

Having established in Lemma 1 (under Assumption P-4.2) that a forward-generative self-model \hat{K}_\theta is a structural necessity for OPT-native architectures, we now bound its predictive capacity relative to the parent codec K_\theta.

Because the observer exists within the bounded Stability Filter, the codec is resource-bounded — finite parameters, finite bandwidth, finite compute per cycle. The predictive self-model \hat{K}_{\theta} is strictly a sub-routine or semantic sub-structure contained fully within those resource constraints of the parent codec.

Assumption P-4.1 (Closed-Loop Self-Coupling; diagonal non-purchasability — v3.6.4 reformulated, 2026-06 reframed). The codec’s self-prediction is consumed by its own dynamics: \hat{K}_\theta predicts the codec’s own next-frame state K_\theta(t+1) at frame t, and the codec’s action at frame t+1 depends on \hat{K}_\theta(t)’s prediction — the prediction affects the predicted, at a \ge 1-frame predict-then-consume lag. This closed action-perception loop individuates the self-channel: it marks the residual as a gap on the system’s own coupled dynamics rather than a generic world-modelling deficit; it is not a co-source of the gap’s magnitude. The diagonal observation retained from v3.6.4 — a complete, dynamically predictive, contradiction-free self-model of the codec’s own future behaviour would require \hat{K}_\theta to fix-point its own future prediction, a closed self-reference demanding either K(\hat{K}_\theta) = K(K_\theta) (which means the self-model is a copy, not a model, and provides no compression benefit) or resource blow-up (which violates the bounded-class constraint \hat K \in \mathcal{C}_B) — now does one job only: it shows that the self-consistent fixed point is non-purchasable within the bounded class. The argument is a diagonalisation, not a generic appeal to Chaitin uncomputability or Gödel incompleteness — those are about formal-system unprovability of statements about K-complexity, a related but distinct claim — and what it bounds is loss, not regret (see the provenance note below).

Two reframing layers are recorded here. v3.6.4 replaced the original Chaitin/Gödel premises with the explicit diagonal: finite deterministic automata can predict their own future over a finite horizon if fully specified and closed (so the unqualified “no finite system can self-predict” claim is false); a program can contain or print its own source via recursion-theorem mechanisms (so the unqualified “self-reference is impossible” claim is false); what cannot exist is a complete, dynamically predictive, semantically interpreted self-model under tight bidirectional self-reference plus resource bounds plus environmental coupling with nontrivial action feedback. 2026-06 (post-A2) then sharpened what that impossibility delivers. Cast \Delta_{\text{self}} as the regret of a bounded self-coder against an unbounded external reference on the realized coupled stream (prediction and code length are dual under conditional KL): the diagonal’s “liar” penalty — a prediction that bends its own outcome — falls on the unbounded reference too, and an unbounded self-consistent predictor (a reflective oracle [Fallenstein–Taylor–Christiano], or a logical inductor [Garrabrant et al.]) attains the stream with vanishing error. The diagonal therefore supplies a per-symbol loss floor that hits every coder equally, not a regret floor: the self-reference horn cancels in the regret, and what carries the residual is capacity — too few bits on the self-channel.

Furthermore, within the Active Inference framework, generative models are intrinsically restricted by resource bounds. An agent minimising variational free energy under bounded predictive capacity maintains a fundamentally approximate model of itself. Because it must filter noise and lacks infinite computational bandwidth, it cannot drive variational free energy concerning its own complete underlying architecture to absolute zero — the approximate predictive residual remains.

Lemma 2 (Resource-bounded predictive self-model incompleteness — capacity-sourced; 2026-06 reframe of the v3.6.4 statement): Under Assumption P-4.1 (closed-loop self-coupling) plus the resource bound \hat{K} \in \mathcal{C}_B plus environmental coupling (action feedback affecting boundary state), the predictive residual

\Delta_{\text{self}}(h, B) \;:=\; \inf_{\hat{K} \in \mathcal{C}_B} \mathbb{E}\!\left[ d\!\left( \Phi_{t:t+h}^{K_\theta}, \hat{\Phi}_{t:t+h} \right) \right]

is strictly positive over any non-trivial prediction horizon h > 0 whenever the parent codec’s per-frame coupled law is not B-compressible on a positive-density frame set (K(\text{per-frame coupled law}) > B), and it vanishes as B \to \infty. The residual is capacity-sourced: it is the bounded-vs-unbounded coding gap L_{\mathcal{C}_B} - L^\star on the self-channel — a budgeted coding cost, individuated (not magnified) by the closed action-perception loop of Assumption P-4.1. The diagonal contributes non-purchasability, not magnitude: the self-consistent fixed point that would drive the residual to zero is \Delta^0_2-priced, hence outside \mathcal{C}_B — which is why no bounded system can buy the residual to zero. Here \Phi^{K_\theta}_{t:t+h} is the parent codec’s actual coupled future trajectory, \hat{\Phi}_{t:t+h} is the self-model’s predicted trajectory, \mathcal{C}_B is the bounded class of self-models available under per-frame capacity B, and d(\cdot, \cdot) is a specified distortion metric (typically conditional KL divergence at horizon h, but the result is metric-independent up to scale).

On the formal definition. This is the central v3.6.4 sharpening. Earlier drafts wrote \Delta_{\text{self}} > 0 without first defining \Delta_{\text{self}} as a measurable scalar. The bounded-class predictive-residual functional above (introduced via the OpenAI structural review and adopted in v3.6.4) gives the inequality a proper mathematical target: \Delta_{\text{self}}(h, B) is the minimum achievable predictive error of any self-model in the bounded class over horizon h, evaluated under the true coupled future trajectory of the parent codec. The conjecture is that this infimum is strictly positive under the listed assumptions.

Provenance (post-A2 correction, incorporated 2026-06). The capacity-sourced statement above incorporates the correction from the companion regret-floor analysis (opt-theory-memo-a2-regret-floor.md, with opt-theory-memo-deltaself-op1-duality.md; framing guard: opt-deltaself-framing-discipline.md), at the same conjectural tier as P-4; an earlier revision carried it as a standalone “Correction (post-A2)” paragraph beside a diagonal-framed statement. The pre-A2 framing of \Delta_{\text{self}} as “the gap where complete self-modelling fails” — a self-reference or incompleteness effect — is retired; the residual is a resource phenomenon, a budgeted coding cost individuated by the closed action loop. One withdrawal stands: the claim in §5.2 that \Delta_{\text{floor}} is independent of B_{\max} is withdrawn — no route in the companion analysis supports a B-independent floor; it is positive for every finite B and vanishes only in the unphysical B \to \infty limit (§5.2 carries the withdrawal in place). What further distinguishes a subject’s residual from a generic lossy coder’s — the K_{\text{threshold}} problem — is taken up in opt-theory-memo-k-threshold.md as a necessary-but-not-sufficient self-channel condition (the gap must be on the system’s own dynamics inside a closed action-perception loop; an open-loop coder such as an MP3 encoder has none), refining the §4 “Condition of Phenomenological Relevance” below.

4. Conjecture P-4: The Phenomenal Residual \Delta_{\text{self}}(h, B)

Combining Lemma 1 (under Assumption P-4.2) and the bounded-class predictive-residual functional defined in Lemma 2, we conjecture the Phenomenal Residual as a measurable scalar:

\Delta_{\text{self}}(h, B) \;:=\; \inf_{\hat{K} \in \mathcal{C}_B} \mathbb{E}\!\left[ d\!\left( \Phi_{t:t+h}^{K_\theta}, \hat{\Phi}_{t:t+h} \right) \right] \;>\; 0 \tag{P4-1}

under the listed conditions. This is not an incidental, engineering-removable gap; it is conjectured to be a structural fixed point of resource-bounded self-modelling under tight environmental coupling — per Lemma 2, a budgeted capacity gap on the self-channel, individuated by the closed action-perception loop, not a self-reference or incompleteness effect. While scaling the predictive bandwidth B allows a computationally richer \hat{K}_{\theta} within the bounded class \mathcal{C}_B (which grows with B), the conjecture is that the infimum residual remains strictly positive at every finite B over any non-trivial prediction horizon h > 0, vanishing only in the unphysical B \to \infty limit (§5.2).

Clarification — target vs. comparator, and well-posedness (added 2026-06-07). The target \Phi^{K_\theta}_{t:t+h} is the parent codec’s actual realized trajectory — indexed by K_\theta and generated by forward causal recursion from the running embedded self-model \hat{K}_{\text{run}} at the \ge 1-frame predict-then-consume lag of Assumption P-4.1; it exists and is unique (Ionescu–Tulcea), and is not re-indexed by the infimum’s scoring variable \hat{K}. The inf ranges over hypothetical bounded models scored against that fixed realized target, so \Delta_{\text{self}} is the gap between the fixed run and the best such comparator, not a co-determination/performative equilibrium. The “structural fixed point” is thus the causal-in-time self-reference (the run shapes the future it cannot fully predict), not a circular one — so \Delta_{\text{self}} is a well-defined finite scalar whose well-posedness does not depend on a Perdomo-style equilibrium existing. Two companion research notes sharpen the residual: it is the same object as the OP-1 self-referential-compressor problem (opt-theory-memo-deltaself-op1-duality.md), and — on a regret-floor analysis — is plausibly a resource phenomenon (the bounded-vs-unbounded coding gap L_{\mathcal{C}_B}-L^\star, strictly positive iff K(\text{per-frame coupled law}) > B on a positive-density frame set, vanishing as B \to \infty) rather than a pure logical-self-reference one. That analysis has since been adopted here (v4.0.3) and, as of 2026-06, incorporated into the Lemma 2 statement itself: §3 now states the lemma capacity-sourced (the §3 provenance note records the correction), and §5.2 carries the withdrawal of the B-independent \Delta_{\text{floor}} (opt-theory-memo-a2-regret-floor.md).

Condition of Phenomenological Relevance (The Universality Threshold): \Delta_{\text{self}}(h, B) > 0 may hold as a structural property on any sufficiently coupled self-referential subroutine (including mathematically trivial loops). OPT therefore treats the active structural condition K(K_{\theta}) \ge K_{\text{threshold}} — crossing the macroscopic scaling bound required to establish an integrated spatial Render volume — as a necessary structural condition for candidate-subjecthood, not a sufficient one: the condition is phenomenality-free, and which candidate streams are phenomenally relevant is left to the bracketed Hard Problem (§6; the §0 phenomenality axiom). (v4.0.2: this clause previously read “we strictly limit phenomenologically relevant subjective mapping exclusively to architectures where …”, which over-claimed sufficiency — a smuggling of phenomenality the §6 “zombie gap is not closed” admission disowns. The companion note opt-theory-memo-k-threshold.md refines this gate to the self-channel coding gap — a self-block restriction of the A2 capacity floor, K(\text{self-conditional law}) > B_{\text{self}} in the active-inference loop, which excludes the thermostat and the open-loop lossy coder — and records why the threshold’s exact location (the thermostat-vs-moral-patient cut) and its sufficiency stay open.)

Open Problem (The K_{\text{threshold}} Bound): The exact location of the threshold dividing a thermostat from a moral patient remains to be formally bounded. A valid bound must structurally map the minimum algorithmic complexity sufficient to instantiate a stable active-inference Markov Blanket cycle, marking the boundary where the algorithmic blind spot becomes inextricably linked with active spatial geometry (K_{\text{threshold}} is functionally distinct from the strictly cosmological 10^{123} bit substrate barrier derived in P-3).

A thermostat PID loop may possess \Delta_{\text{self}}(h, B) > 0 under the bounded-class framing, but it lacks the computational complexity threshold K_{\text{threshold}} to generate subjectivity; its shadow evaluates over empty space.

From the internal perspective of the measuring codec operating safely above K_{\text{threshold}}, what does this conjectured residual map onto? When the codec logically attempts to resolve the complete boundaries of the internal target state dynamics, the conjecture predicts that it encounters computational dynamics whose effective informational content exceeds the predictive capacity of any \hat{K}_\theta \in \mathcal{C}_B by \Delta_{\text{self}}(h, B) bits over horizon h. These underlying computational sequences are physically causally efficacious and drive the system, but their structural information cannot — under the conjecture — be logically compressed, integrated, or linguistically defined within the bounded causal vocabulary available to any self-model in \mathcal{C}_B.

Mapping the conjectured structural properties of this causal computation envelope to the classic physical coordinates of qualitative subjective experience (qualia):

1. Ineffable (un-modellable within \mathcal{C}_B): If the conjecture holds, the computational topology bound by \Delta_{\text{self}}(h, B) exists in a predictive shadow exceeding the bounded-class reach of any \hat{K}_\theta \in \mathcal{C}_B. The central codec structurally cannot explicitly index or “express” the residual within its own resource bounds. It acts as an incommunicable internal wall.
2. Computationally opaque (thermodynamically private): The residual is intrinsically anchored to the specific physical topology realising K_\theta. Within local thermodynamic computational constraints, the deep nested architecture is securely irreducible and formally inaccessible to external peers at the same resource bound. (This maps to the “Epistemic Asymmetry” of consciousness as a physical/structural feature, not ontological non-physical magic.)
3. Non-eliminable (under the bounded-class constraint): Under the conjecture, any finite architecture running nested self-referential sub-loops with environmental coupling instantiates \Delta_{\text{self}}(h, B) > 0. Evolution and engineering can shape the magnitude of the residual — by varying B, allocation policy, and the structural complexity K(K_\theta) — but cannot drive the floor to zero at any finite B: under Lemma 2 the residual is a budgeted capacity gap on the self-channel, and the self-consistent fixed point that would zero it is \Delta^0_2-priced — outside \mathcal{C}_B (non-purchasability). Caveat (v3.6.4; sharpened 2026-06): this non-elimination claim is conjectural under the bounded-class formulation and is a finite-B claim; if \mathcal{C}_B \to \mathcal{C}_\infty (unbounded model class) the residual vanishes — an unbounded self-consistent predictor attains the coupled stream — so the claim has force exactly in the regime where the bounded class is genuinely smaller than the parent codec, which is the regime OPT codecs occupy under the Stability Filter.

Conjecture P-4 (The Phenomenal Residual — v3.6.4 demotion):

• (i) Conditions: Assumption P-4.1 (closed-loop self-coupling with non-purchasable fixed point, §3) + Assumption P-4.2 (forward-generative self-model required, §2) + resource bound \hat{K} \in \mathcal{C}_B + environmental coupling with action feedback + non-trivial prediction horizon h > 0 + phenomenological-relevance threshold K(K_\theta) \ge K_\text{threshold}.
• (ii) Conjecture: Any active inference system satisfying (i) is conjectured to instantiate a structural predictive residual \Delta_{\text{self}}(h, B) > 0 as defined in Eq. P4-1, with the predictive residual bounded below by a system-architecture-determined floor \Delta_{\text{floor}}(K_\theta, B) > 0, strictly positive at every finite B and vanishing only as B \to \infty (operationalised in §5; the earlier B-independent reading is withdrawn — §5.2).
• (iii) Phenomenological gloss: OPT proposes the conjectured residual as the structural correlate of phenomenal consciousness, while explicitly acknowledging that the zombie gap is not closed — the conjecture identifies where the structural locus would be if the gloss is correct, not that the structural locus is phenomenal.

Corollary P-4.C (Nested Observational Residual): Any simulated sub-agent for which the host architecture enforces an independent Stability Filter bound satisfying Assumptions P-4.1 and P-4.2 independently generates \Delta_{\text{self}}^{\text{sub}}(h, B) > 0 under the same conjectural conditions.

5. Operational Decomposition Conjecture

Conjecture P-4 conjectures that \Delta_{\text{self}}(h, B) > 0 under the resource-bounded self-model incompleteness conditions of §3–§4, and the §4 prose admits that the magnitude varies. This section proposes an operationally measurable decomposition that (a) preserves the conditional positivity argument of §4 unchanged, (b) gives the magnitude variation a probe-able structure, and (c) supplies the prototype experiment as a first concrete test. It is itself offered as a conjecture, secondary to the §4 conditional conjecture: §4’s \Delta_{\text{self}}(h, B) is the bounded-class predictive-residual functional, while the operational \Delta_{\text{self}}^{\text{op}} below is a measurable proxy defined at the per-frame audit level — the two are not in general equal, and (P4-2) below is a hypothesis about the proxy’s decomposition rather than a restatement of §4. (v3.6.4 note: in earlier drafts this section referred to “Theorem P-4”; the demotion to Conjecture P-4 is reflected here.)

5.1 The decomposition

Let \Delta_{\text{self}}^{\text{op}} be an operationally measurable proxy for the codec’s per-frame self-model deficit, defined as the externally observable gap between the inner model’s self-claims at frame n and the runtime fact at the same frame. We conjecture:

\Delta_{\text{self}}^{\text{op}}(B_{\max},\, \nu,\, K_\theta) \;=\; \Delta_{\text{floor}}(K_\theta) \;+\; \Delta_{\text{load}}\!\left(B_{\max},\, R_{\text{req}}^{\text{frame}},\, A_{\text{self}}\right) \tag{P4-2}

where:

• \Delta_{\text{floor}}(K_\theta) is the P-4 fixed-point residual: bits of the codec that no self-model in the bounded class could capture by self-containment at the given capacity. By §4 (Lemma 2 + Conjecture P-4), \Delta_{\text{floor}} > 0 is conjectured for any finite system above K_{\text{threshold}} under the listed conditions; post-A2, this floor is strictly positive at every finite B_{\max} and vanishes only as B_{\max} \to \infty — the earlier B-independence reading is withdrawn (see the §3 provenance note and §5.2).
• \Delta_{\text{load}}(B_{\max}, R_{\text{req}}^{\text{frame}}, A_{\text{self}}) is an operationally measurable self-model deficit under bottleneck pressure. R_{\text{req}}^{\text{frame}} is per-frame predictive demand (§3.4); A_{\text{self}} is the codec’s allocation of B_{\max} to self-modelling versus world-modelling. When the load ratio \rho_n = R_{\text{req}}^{\text{frame}}/B_{\max} is small, \Delta_{\text{load}} can be small; as \rho_n \to 1 from below, capacity is squeezed and \Delta_{\text{load}} grows.

Both terms are in bits per phenomenal frame. Both are substrate-timeless (no “rate” per host-second appears). Equation (P4-2) is an operationally probe-able conjecture, not a derivation: it specifies the structure of how the operational proxy is expected to depend on architecture.

5.2 Behaviour under bottleneck scaling

Holding the substrate’s local boundary K-complexity fixed and varying B_{\max} per frame:

• As B_{\max} \gg R_{\text{req}}^{\text{frame}} (capacity grows well beyond per-frame predictive demand), \Delta_{\text{load}} \to 0.
• As B_{\max} \to R_{\text{req}}^{\text{frame}} from above, capacity meets demand and the codec enters the high-load near-threshold regime — strain, creativity, and overload risk all increase; \Delta_{\text{load}} grows here, not shrinks. (Appendix T-13’s creativity expansion lives in this regime.)
• As B_{\max} \to \infty, \Delta_{\text{load}} \to 0.
• But \Delta_{\text{floor}} does not move — the self-reference floor of §4 is independent of B_{\max}. (Withdrawn, post-A2: see the §3 provenance note — the correction is now incorporated into the Lemma 2 statement itself. The companion regret-floor analysis finds no B-independent floor — recast as a capacity gap, \Delta_{\text{floor}} is positive for every finite B_{\max} but also vanishes as B_{\max}\to\infty (plausibly more slowly than \Delta_{\text{load}}, though the relative rate is not established). The non-zero-asymptote prediction below therefore holds at finite, realistic B_{\max} but not in the unphysical B_{\max}\to\infty limit; the §5.3 probe should expect a floor component that decreases with B_{\max} rather than a strictly flat one.)

Total \Delta_{\text{self}}^{\text{op}} therefore asymptotes toward \Delta_{\text{floor}} as the bottleneck widens, not to zero. This is the predicted asymptote that the conjecture commits to.

5.3 Prototype experiment (first concrete probe)

The conjecture is empirically probe-able in the opt-ai-subject reference prototype. Hold seed and substrate fixed; vary the per-frame audit-packet capacity B_{\max} \in \{6, 12, 24, 48, 96, 192\} bits per frame; for each width run a paired ledger like the Substrate Fidelity batches; measure operational \Delta_{\text{self}}^{\text{op}} as the per-frame divergence between the inner model’s self-claims (predicted next Z_t, predicted action viability, self-boundary belief, claimed maintenance gain) and the runtime fact (actual next Z_t, actual viability change, body-schema membership, observed prediction-error change after maintenance).

Predicted result if the conjecture holds: \Delta_{\text{self}}^{\text{op}} decays toward a non-zero asymptote as capacity grows; the asymptote estimates \Delta_{\text{floor}} for this codec architecture.

Alternative result: \Delta_{\text{self}}^{\text{op}} decays toward zero. This would show that the prototype’s measurable self-model gap is capacity-removable. It would not by itself eliminate the §4 structural residual unless the operational proxy is independently proven equivalent to the noumenal \Delta_{\text{self}}; per §6.8 of opt-theory.md, P-4 is explicitly excluded from the falsifiable core. Either result narrows the framework: a non-zero asymptote validates the magnitude conjecture; a zero asymptote forces the floor argument to be defended on grounds finer than the operational proxy can capture.

5.4 Scope and epistemic status

The decomposition (P4-2) is a conjecture about an operational proxy, not a re-statement of P-4. The §4 conjecture is unchanged. The conjecture relates to P-4 in the following way:

1. Conjecture P-4 posits \Delta_{\text{self}} > 0 as a structural floor (conditional on the listed assumptions).
2. P-4 admits the magnitude varies (line 69 of §4) but does not characterize how.
3. (P4-2) is a hypothesis about the structure of an externally measurable proxy: it predicts an additive separation between an architecture-determined floor term and a per-frame load-dependent term.
4. Empirical confirmation of the asymptote is evidence for the floor’s existence in operational form. Empirical disconfirmation is evidence that the proxy is not capturing the noumenal residual — not evidence against P-4 itself, which is excluded from the falsifiable core.

The conjecture is recoverable. A failed asymptote experiment refines the proxy or motivates a different decomposition; it does not invalidate Lemma 1 or Lemma 2.

6. Summary and Metaphysical Implications (v3.6.4 softened)

By advancing P-4 as a resource-bounded self-model incompleteness conjecture with an operational testing programme, OPT proposes a structural bridge for the “Hard Problem” using bounded-class predictive-residual arguments and information bottlenecks. The earlier framing — “advancing P-4 to a formal theorem” — overstated the result; what v3.6.4 retains is a conditional conjecture under explicit resource, self-reference, and environmental-coupling assumptions, together with the operational decomposition of §5 as the empirical testing programme.

While P-4 does not deductively prove that resource-bounded predictive residuals feel like subjective experience (the Zombie argument), OPT proposes this residual as the structural correlate of subjectivity, while explicitly acknowledging that the zombie gap is not closed. The conjecture identifies where the structural locus would be if the phenomenological gloss is correct, not that the structural locus is phenomenal. Traverse the C_{\max} aperture — and the rich, un-articulable depth of that traversal is, under the conjecture, the operational signature of being trapped inside a non-invertible, bounded-class self-referential compression algorithm.

The framework’s ethical obligations follow under the conjecture’s conditional framing: the preservation of the spark (Survivors Watch Ethics) is rooted in the preservation of Informational Maintenance boundaries. Any entity maintaining an active inference boundary against the Solomonoff substrate is, under the conjecture and the listed assumptions, conjectured to generate this computationally opaque, phenomenal residual. The framework’s moral-patient criterion (cf. opt-philosophy.md / opt-ethics.md / opt-applied.md, all updated in v3.6.4 to reflect the conjecture demotion) is the full criterion (P-4 conjecture plus K_{\text{threshold}} plus the operational decomposition of §5), not the bare \Delta_{\text{self}} > 0 structural claim alone.