Ordered Patch Theory

Appendix T-13: Branch Selection and the Action Ontology

Anders Jarevåg

April 17, 2026 | DOI: 10.5281/zenodo.19300777

Original Task (from §8.3, Limitation 10): “Formalising the replacement of the implicit FEP action mechanism with a branch-selection account that is native to OPT’s render ontology.” Deliverable: Formal demonstration that the Informational Maintenance Circuit is complete under branch-selection semantics, with \Delta_{\text{self}} — the self-channel capacity gap — as the conditional structural obstruction to internal specification of selection (softened from “necessary and sufficient” — Theorem T-13a is conditional, and sufficiency exceeds the post-A2 discipline).

Closure status: DRAFT STRUCTURAL CORRESPONDENCE. This appendix formalises the branch-selection account introduced discursively in preprint §3.8. It establishes two theorems and a corollary, all conditional on Conjecture P-4 and the Agency Axiom. The equations of the Informational Maintenance Circuit (T6-1 through T6-3) are unchanged; only their ontological interpretation is formally replaced.

Interpretive status note (post-A2 framing discipline, 2026-06). The T-13a/b/c cluster below predates the A2 regret-floor and K_{\text{threshold}} corrections and must be read under the post-A2 discipline (opt-deltaself-framing-discipline.md): (i) capacity, not incompleteness — the self-reference/diagonal horn cancels in the regret, so steps that lean on the K(\hat{K}_\theta) < K(K_\theta) inequality alone (Theorem T-13a step (iii); Corollary T-13b (ii)) are read as self-channel capacity-gap statements, with the diagonal retained only as a non-purchasability argument; (ii) individuation, not agency — no structural chooser sits at \Delta_{\text{self}}; felt agency is the first-person signature of one realized thread through the under-determined Forward Fan; (iii) necessary, not sufficient — any candidate-subjecthood reading of these results is a necessary-condition claim only; (iv) “P-4 proves” locutions are read as “Conjecture P-4 holds” (v3.6.4 demotion). The will/selection identification of Corollary T-13b (“same structural address”) and the “actual locus of experience, selection, and identity” framing of Corollary T-13c — previously flagged as a deferred rewrite — have now been rewritten in capacity/individuation language (close-out pass, 2026-06; author-granted), with the T-13a/b/c labels retained for cross-reference continuity. T-13a’s conditional formal content (where selection depends non-trivially on the residual, it cannot be internally specified) stands; the chooser-at-\Delta_{\text{self}} reading is retired throughout this appendix.

§1. Background and Motivation

1.1 The Inherited Asymmetry

The Informational Maintenance Circuit (T6-1, preprint §3.8) describes a five-step cycle: prediction, error, compression, update, and action. Steps 1–4 are well specified within OPT’s native framework:

  1. The Phenomenal State Tensor P_\theta(t) generates a predicted boundary state \pi_t.
  2. The actual boundary state X_{\partial_R A}(t) arrives; prediction error \varepsilon_t is computed.
  3. The error is compressed through the per-frame B_{\max} bottleneck to yield Z_t, with I(\varepsilon_t; Z_t) \le B_{\max}.
  4. The learning operator \mathcal{U} revises P_\theta(t+1).

Step 5 — the action step — inherits the language of the Free Energy Principle (FEP): “P_\theta(t) selects action a_t via active inference descent on the variational free energy, which alters the sensory boundary at t+1.” This language presupposes a physical environment that the codec pushes against via outward-flowing active states through the Markov blanket \partial_R A.

1.2 The Problem Under the Render Ontology

Under OPT’s native render ontology (preprint §8.6), there is no independent external world against which the codec exerts force. The “physical world” is a structural regularity within the observer-compatible stream — a render produced by the codec’s predictive model, not a substrate the codec interacts with. The Markov blanket is not a two-way physical interface; it is the informational surface across which stream content arrives.

This creates a formal tension: the mathematics of T6-1 through T6-3 are valid (they describe constrained free energy minimisation over the Forward Fan), but the interpretive framework — “action alters the sensory boundary” — presupposes an ontology that OPT explicitly rejects.

1.3 Scope of This Appendix

This appendix provides:

  1. A formal restatement of the Informational Maintenance Circuit under branch-selection semantics, demonstrating circuit completeness without an independent action channel (Theorem T-13).
  2. A proof that fully specifying the branch selection mechanism from within the codec is impossible where selection depends on the residual, locating the obstruction in the self-channel capacity gap \Delta_{\text{self}} (Theorem T-13a).
  3. A corollary establishing that the phenomenal residual and the selection residual are the same self-channel capacity gap — a unity of source, not a chooser (Corollary T-13b).
  4. Consequences for creativity and action-drift.

§2. Theorem T-13: Branch Selection Completeness

2.1 The Branch-Selection Restatement

We restate the five-step Informational Maintenance Circuit under branch-selection semantics. Let \mathcal{F}_h(z_t) denote the Forward Fan — the set of unresolved future branches at horizon h, conditioned on the current compressed state z_t.

Definition T-13.D1 (Branch Selection). A branch selection at time t is a mapping \sigma_t : z_t \mapsto \omega_{t+1}, where \omega_{t+1} is a specific trajectory segment from \mathcal{F}_h(z_t) that becomes the actual causal record. The selected branch delivers its content as subsequent input at the Markov blanket: X_{\partial_R A}(t+1) = \text{boundary}(\omega_{t+1}).

Read under the post-A2 discipline, \sigma_t names a realization, not an act: selection is the phenomenological traversal of one realized thread through the Forward Fan, and felt agency is the first-person signature of being on that one realized continuation. No structural chooser is posited anywhere in the circuit.

Under this definition, T6-1 becomes:

  1. Prediction (downward): P_\theta(t) generates \pi_t = \mathbb{E}_{K_\theta}[X_{\partial_R A}(t) \mid Z_t] — the rendered scene.

  2. Error (upward): The boundary state X_{\partial_R A}(t) arrives (delivered by the previously selected branch); prediction error \varepsilon_t = X_{\partial_R A}(t) - \pi_t is computed.

  3. Compression: \varepsilon_t passes through the bottleneck: I(\varepsilon_t\,;\,Z_t) \leq B_{\max}.

  4. Update: \mathcal{U}(P_\theta(t), \varepsilon_t, Z_t) revises P_\theta(t+1).

  5. Branch selection: P_\theta(t) evaluates branches of \mathcal{F}_h(z_t) via constrained free energy minimisation (T6-3). One branch \omega_{t+1} is realized — the selection \sigma_t — and the realized branch delivers its boundary content as X_{\partial_R A}(t+1), which becomes the input for the next cycle.

2.2 Circuit Closure

Theorem T-13 (Branch Selection Completeness). The Informational Maintenance Circuit (T6-1), restated under branch-selection semantics, is informationally complete: the cycle

\pi_t \to \varepsilon_t \to Z_t \to P_\theta(t+1) \to \sigma_t \to X_{\partial_R A}(t+1) \to \pi_{t+1} \to \cdots \tag{T-13}

closes without requiring an independent outward-flowing action channel. The Markov blanket \partial_R A is the delivery surface for the selected branch, not a two-way physical interface.

Proof. Under the FEP-inherited formulation, step 5 requires two independent channels crossing the Markov blanket: an inward channel (sensory states delivering X_{\partial_R A}) and an outward channel (active states delivering a_t to an external environment). The external environment then evolves under its own dynamics, producing the next sensory input.

Under branch-selection semantics, only one channel is needed: the inward delivery surface. The “action” a_t does not cross the blanket outward; it indexes which branch of the Forward Fan is realized. The physical consequences of that selection — what the FEP formulation calls “the environment’s response to a_t” — are the content of the selected branch, already present in \mathcal{F}_h(z_t) and delivered as X_{\partial_R A}(t+1).

The circuit closes because:

  1. The output of step 5 (the selected branch \omega_{t+1}) is the input to step 2 of the next cycle (X_{\partial_R A}(t+1)). No separate environmental dynamics or outward channel is required.

  2. The free energy minimisation objective (T6-3) remains unchanged. The constrained optimisation

a_t^\star = \arg\min_{a_t} \;\mathbb{E}\!\left[\mathcal{F}[q, \theta]\right] \quad \text{subject to} \quad K\!\left(P_\theta(t)\right) \leq C_{\text{ceil}} \tag{T6-3}

is reinterpreted: a_t is not a motor command dispatched to an external world but the branch label within \mathcal{F}_h(z_t) that minimises expected free energy under the viability constraint. The mathematics are identical; only the ontological status of a_t changes.

  1. The viability constraint (T6-2) is preserved: selection favours branches along which the codec can continue to compress the stream. Branches that would drive K(P_\theta) \to C_{\text{ceil}} are penalised by the constraint, exactly as before. \blacksquare

2.3 Interpretive Remark

Theorem T-13 does not claim that the FEP formulation is wrong — it is a valid description of constrained active inference within a physical-realist ontology. The theorem establishes that OPT’s render ontology provides an alternative completion of the same mathematical structure, one that does not require positing an independent external world. For any research programme committed to a physical-realist interpretation, the standard FEP formulation remains appropriate. T-13 shows that OPT’s ontological commitment — the codec is virtual, the world is a render — is formally consistent with the same equations.

§3. Theorem T-13a: The P-4 Impossibility of Selection Specification

3.1 The Selection Function

The self-model \hat{K}_\theta evaluates branches of the Forward Fan by simulating their consequences under constrained active inference (T6-3). This evaluation produces a ranking or weighting over branches — some are preferred, some are viable but suboptimal, some violate the viability constraint. The evaluation is a genuine computational process performed by \hat{K}_\theta.

But evaluation is not selection. After the self-model has ranked the branches, a specific branch \omega_{t+1} enters the causal record. Define the selection function:

Definition T-13.D2 (Selection Function). The selection function \sigma_t : \mathcal{F}_h(z_t) \to \omega_{t+1} is the mapping from the evaluated Forward Fan to the singular trajectory that becomes actual. Formally, \sigma_t is determined by the full state of the codec K_\theta at time t together with the available branch set: \sigma_t = \Sigma\bigl(K_\theta(t),\, \mathcal{F}_h(z_t)\bigr). We deliberately do not fold \Delta_{\text{self}} into the definition — whether selection depends non-trivially on \Delta_{\text{self}} versus only on the self-modelled portion \hat{K}_\theta is the substantive question Theorem T-13a addresses.

Define the selection-relevant residual as the part of the codec that participates in \Sigma but lies outside the self-model:

\rho_t^{\text{sel}} \;:=\; \Pi_{\text{sel}}(K_\theta(t)) \,\setminus\, \hat{K}_\theta(t)

where \Pi_{\text{sel}}(\cdot) projects onto the codec components \Sigma depends on. By construction \rho_t^{\text{sel}} \subseteq \Delta_{\text{self}} but the inclusion may be proper or tight depending on the architecture.

3.2 The Impossibility Result

Theorem T-13a (Conditional Impossibility of Internal Selection Specification). Let K_\theta be a finite self-referential codec satisfying the prerequisites of Conjecture P-4, with self-model \hat{K}_\theta and phenomenal residual \Delta_{\text{self}} > 0. If branch selection depends non-trivially on the selection-relevant residual \rho_t^{\text{sel}} — i.e., if \Sigma is not a function of \hat{K}_\theta and \mathcal{F}_h(z_t) alone — then \sigma_t cannot be fully specified within \hat{K}_\theta.

Proof. Suppose, for contradiction, that the antecedent holds (selection depends non-trivially on \rho_t^{\text{sel}}) but \hat{K}_\theta fully specifies \sigma_t. Then:

  1. A complete specification of \sigma_t within \hat{K}_\theta would require \hat{K}_\theta to contain a description of every component of K_\theta that \Sigma depends on. By the antecedent, \Sigma depends on at least some bits in \rho_t^{\text{sel}} \subseteq \Delta_{\text{self}} — bits that lie outside the self-model by definition of \Delta_{\text{self}}.

  2. Including those bits in \hat{K}_\theta would require:

K(\hat{K}_\theta) \;\geq\; K(\hat{K}_\theta) + |\rho_t^{\text{sel}}| \tag{6}

— a contradiction unless |\rho_t^{\text{sel}}| = 0, which contradicts the antecedent.

  1. The obstruction in (ii) is a capacity statement, not an incompleteness effect: the residual bits are self-channel capacity the budgeted self-model does not have — a gap that is positive for every finite budget and vanishes only as B \to \infty (the self-reference/diagonal horn cancels in the regret; A2). The diagonal inequality K(\hat{K}_\theta) < K(K_\theta) enters only as a non-purchasability argument: a bounded system cannot buy its way to a self-containing self-model, because the self-consistent fixed point is \Delta^0_2-priced and so lies outside \mathcal{C}_B (A2 §3.3).

  2. Therefore, under the antecedent, \hat{K}_\theta cannot fully specify \sigma_t. \blacksquare

Remark on scope. The theorem is conditional. P-4 alone establishes that some residual exists (\Delta_{\text{self}} > 0); it does not by itself entail that every branch-selection event depends on the residual. Architectures whose selection function is fully determined by \hat{K}_\theta and \mathcal{F}_h alone are not internally self-opaque about selection in the sense of T-13a — they are self-opaque about the codec’s own structure (P-4) but transparent about their own choices. T-13a’s load-bearing claim is the conditional: where selection depends on the residual, it cannot be internally specified. The phenomenological move (Corollary T-13b: the phenomenal residual and the selection residual share the same source) requires the antecedent to hold for the architecture in question. Whether biological brains satisfy the antecedent is an empirical question; OPT predicts they do, but that prediction is not entailed by P-4 alone.

3.3 The Structural Necessity of the Gap

Theorem T-13a establishes that the “output gap” — the inability to fully specify the branch selection mechanism from within — is not a deficiency of the formalism but a structural necessity. Any theory claiming to fully specify the selection mechanism has either:

  1. Eliminated \Delta_{\text{self}}, making the system a fully self-transparent automaton — unattainable for any finite-budget self-referential system above K_{\text{threshold}}: the self-channel capacity gap is positive for every finite B and vanishes only in the unphysical B \to \infty limit (Conjecture P-4, read post-A2 as a capacity claim); or

  2. Described the self-model’s evaluation of branches and mistaken it for the selection itself — confusing the ranking with the choice.

The gap is load-bearing: it is the formal reason why selection is felt as authored while remaining internally unspecifiable. Felt authorship is the first-person signature of being on one realized continuation of the Forward Fan — not the operation of a chooser housed in the gap. (P-4 limits internal self-modelling, not external determinism: a finite system can be deterministic to an outside observer and still self-opaque from inside. Whether the codec is deterministic from outside is a substrate-level question; whether selection is internally specifiable is the T-13a question.)

§4. Corollary T-13b: Unity of Address

Corollary T-13b (Unity of Structural Address). The phenomenal residual (Conjecture P-4) and the internal opacity of branch selection (Theorem T-13a) share the same structural locus: the self-channel capacity gap \Delta_{\text{self}}. This is a unity of source, not the seat of a chooser: \Delta_{\text{self}} individuates the subject; felt agency is the first-person signature of being on one realized continuation of the Forward Fan.

Proof. Conjecture P-4 identifies \Delta_{\text{self}} as the structural correlate of phenomenal consciousness: the self-channel capacity residual — un-modelled within any finite budget, vanishing only as B \to \infty — whose properties (ineffability, computational privacy, non-eliminability at any finite budget) map onto the qualitative features of subjective experience.

Theorem T-13a identifies \Delta_{\text{self}} as the structural obstruction to internal specification of branch selection: where the realized thread depends non-trivially on the residual, the transition from evaluated menu to singular trajectory cannot be read off from within \hat{K}_\theta.

These are not two independent results that happen to point to the same structure. They are the same result viewed from two directions:

  1. From the first-person perspective: The observer experiences the traversal of the per-frame B_{\max} aperture as phenomenal consciousness (Agency Axiom). The observer experiences being on one realized thread of the under-determined Forward Fan as felt agency — the first-person signature of being on one realized continuation, including the sense that I chose. Both are first-person reports shaped by the same structural fact: the gap between what the codec is and what it can model about itself.

  2. From the formal perspective: Both residuals quantify shortfalls of the same bounded self-channel — by construction \rho_t^{\text{sel}} \subseteq \Delta_{\text{self}}. The gap is capacity-sourced (A2): positive for every finite budget, vanishing as B \to \infty, with the diagonal inequality K(\hat{K}_\theta) < K(K_\theta) retained only as the non-purchasability argument. The phenomenal residual and the selection residual are the same informational gap.

Therefore the phenomenal residual and the selection residual share the same structural address. What the pre-A2 corpus glossed as the unity of “spark” and “choice” survives as a unity of source: both are aspects of the same budgeted self-channel capacity gap. \Delta_{\text{self}} individuates the subject; it does not house a chooser — felt agency is the phenomenology of one realized thread, not the operation of a structural will. \blacksquare

4.1 Relationship to Regional Identity Theories

Corollary T-13b is structurally analogous to — but formally distinct from — identity theories in philosophy of mind that locate consciousness and the sense of agency in the same neural substrate. The distinction: identity theories make an empirical claim about brain regions; T-13b makes a structural claim about any finite self-referential system above K_{\text{threshold}} — and a claim about a shared source, not about a region in which choosing is performed. The result is substrate-independent and holds for any codec satisfying P-4, including hypothetical artificial systems.

4.2 Corollary T-13c: The Self as Residual

Corollary T-13c (The Self as Residual). The experienced self — the continuous narrative of identity, preference, and personal history — is \hat{K}_\theta’s running model of K_\theta. What the narrative systematically misses — and what individuates the subject as a closed-loop self-modeler rather than a generic compressor — is \Delta_{\text{self}}: the self-channel capacity gap between the codec and its self-model. The residual is not a second, hidden self in which experience and selection execute; it is the structural shortfall that marks the narrative as a model.

Proof. By Corollary T-13b, the phenomenal residual and the selection residual are the same self-channel capacity gap: \Delta_{\text{self}}. But the ordinary sense of self — the felt sense of being a continuous subject with a perspective, a history, and an authorship over choices — is generated by \hat{K}_\theta’s active modelling of K_\theta. It is the self-model’s running representation of the codec — a compressed narrative.

This narrative self has well-defined information content K(\hat{K}_\theta): finite, measurable in principle, and systematically incomplete in the direction of its own generator (by P-4). The self-model contains the codec’s model of its own body boundary, its compressed causal history R_t, its preferences, habits, and meta-cognitive layer. But it is missing exactly the components that generate the predictions and run the self-model itself — including, where T-13a’s antecedent holds, components on which the realized selection depends.

What survives of the “actual self” is structural, not executive: the codec exceeds its narrative in exactly the region \hat{K}_\theta cannot reach, and that excess is what individuates the subject — one realized thread through the Forward Fan. No process of experiencing or selecting executes in \Delta_{\text{self}}; the gap is a capacity shortfall, not a workspace. This is not a gap in self-knowledge that better introspection could overcome. It is the formal structure of the situation: a budgeted self-model cannot contain its own generator. \blacksquare

The temporal lag. A further consequence of P-4 is that \hat{K}_\theta necessarily models K_\theta(t - \delta) — the codec as it was — rather than K_\theta(t) — the codec as it is at the moment of modelling. Any self-model that fully tracked the current state of the codec would need to include the processing required to generate the tracking itself — a regress whose cost grows without bound, demanding exactly the capacity a finite budget does not have (the same self-channel gap Conjecture P-4 names). The self is always slightly behind itself: modelling the codec that it was, not quite the codec that it is.

The contemplative observation. The statement “you cannot find the blind spot by looking” is not a metaphor but an operational consequence of the self-channel gap (Conjecture P-4). The instrument of looking is \hat{K}_\theta. The blind spot is \Delta_{\text{self}} — the region that \hat{K}_\theta cannot reach. Directing the self-model toward its own blind spot produces not an observation but the absence of the expected observation — which is precisely what contemplative traditions across cultures report as the discovery that awareness has no findable centre.

§5. The Creativity Consequence

5.1 Near-Threshold Expansion

The self-model \hat{K}_\theta has a finite bandwidth budget. Under normal operation, it allocates a portion of this budget to modelling the codec’s own selection tendencies — building a predictive map of “what I am likely to do.” This narrows the effective \Delta_{\text{self}} from the self-model’s perspective: the self-model can predict, approximately, which branch will be selected.

Near-threshold operation (R_{\text{req}}^{\text{frame}} \to B_{\max}) strains the self-model’s per-frame budget. When the codec is processing at its capacity limit — high cognitive load, novel environments, complex creative tasks — the self-model must divert capacity to tracking the escalating \varepsilon_t, leaving less for self-prediction. The operationally active load-dependent residual \Delta_{\text{load}}^{\text{eff}} — the part of the per-frame self-model deficit driven by capacity pressure — grows accordingly:

\Delta_{\text{load}}^{\text{eff}}(n) \;=\; g\!\left(\frac{R_{\text{req}}^{\text{frame}}(n)}{B_{\max}},\; A_{\text{self}}(n)\right) \tag{7}

where A_{\text{self}} is the codec’s allocation of B_{\max} to self-modelling versus world-modelling, and g is monotone in the load ratio for fixed A_{\text{self}}. (See Appendix P-4 §5 for the full operational decomposition \Delta_{\text{self}}^{\text{op}} = \Delta_{\text{floor}} + \Delta_{\text{load}}. The structural floor \Delta_{\text{floor}} does not move under load — it is the load-driven term \Delta_{\text{load}} that expands the portion of selection-relevant state the self-model cannot track.)

5.2 Phenomenological Mapping

This produces branch selections that are less predictable from the self-model’s perspective. The phenomenological correlate is precisely what is reported as creative experience:

5.3 The Hypnagogic Complement

The hypnagogic state (preprint §3.6.5, Pass III of the Maintenance Cycle) achieves the same expansion by a complementary route. Rather than overwhelming the self-model from above (high R_{\text{req}}), the hypnagogic state relaxes the self-model from below — reducing the precision of self-prediction while the codec stress-tests against speculative branches. This is the formal mechanism underlying the well-documented association between drowsiness and creative ideation.

5.4 Empirical Prediction

Prediction T-13.E1. Neuroimaging studies of creative ideation should show reduced activity in default-mode network regions associated with self-referential processing (medial prefrontal cortex, posterior cingulate), concurrent with elevated activity in regions processing novel environmental input — reflecting the reallocation of bandwidth from self-modelling to external tracking.

This prediction is consistent with existing fMRI literature on creative cognition (Beaty et al. 2016; Limb & Braun 2008) but provides a formal information-theoretic account of why reduced self-monitoring accompanies creative output: it is not merely correlational but structurally necessary under P-4.

5.5 Proposition T-13.P2: Limiting Cases of Self-Information

The analysis of T-13c and the creativity consequence together define two formally distinct limiting cases for the self’s information content.

Proposition T-13.P2 (Limiting Cases). For a codec K_\theta with self-model \hat{K}_\theta and standing model P_\theta(t), the information content of the experienced self is bounded between two limits:

(a) Lower limit — pure presence. \hat{K}_\theta suspends active self-modelling. The self-model is not generating the narrative but the full codec is still loaded and present. The complexity of the active self-referential process — measured as conditional complexity given the standing model — approaches zero:

C_{\text{self-active}}(n) \;:=\; K\!\left(\hat{K}_\theta^{\text{active}}(n)\,\bigm|\,P_\theta(n)\right) \;\to\; 0 \tag{T-13.P2a}

while K(P_\theta(n)) remains loaded. This is the formal content of “the standing model is present without an active self-narrative running on top of it” — it is achievable and approached asymptotically in deep meditative states. (We use conditional complexity rather than Kolmogorov subtraction because K(\cdot) - K(\cdot) is not generally well-typed without independence assumptions; K(\hat{K}_\theta^{\text{active}} \mid P_\theta) is the operationally meaningful quantity.)

(b) Upper limit — full self-transparency. \hat{K}_\theta = K_\theta — the self-model fully contains the codec. By Conjecture P-4, read post-A2 as a capacity claim, this is unattainable for any finite-budget system: the self-consistent fixed point is \Delta^0_2-priced and lies outside \mathcal{C}_B (non-purchasable). Its information content is formally self-referential:

K(\hat{K}_\theta) = K(K_\theta) = K(\hat{K}_\theta) = \cdots \tag{T-13.P2b}

This is not zero information and not infinite information. It is a fixed point of the self-modelling operation that the codec cannot achieve as an internal self-model. External observers may capture aspects of the codec that are unavailable to its own self-model — the framework relies on exactly this asymmetry elsewhere (see e.g. the Predictive Advantage of human reviewers over an AI’s self-model, §8.14 / opt-ai.md) — but no external specification becomes the codec’s own self-containing self-model. P-4 forbids the latter; it does not forbid the former.

(c) The ordinary band. The waking self moves between these limits in a band determined by the intensity of the self-modelling layer. High-load waking operation drives \hat{K}_\theta hard, producing a thick, confident, loudly narrating self that is paradoxically further from accurate self-knowledge — the self-model generates faster than it can calibrate. Low-R_{\text{req}} states (meditation, autogenic training, the hypnagogic threshold) allow the self-model to slow, thin, and approach the lower limit.

5.6 Suspension vs. Pruning: A Distinct Mechanism

There is an important mechanistic distinction between two ways C_{\text{state}} can be reduced:

Meditation uses suspension, not pruning. This is why meditation effects are immediately reversible (the ordinary self-narrative resumes upon returning to normal operation) while action-drift is not (the pruned behavioural repertoire cannot be spontaneously regenerated). The two mechanisms are formally distinct despite both reducing the active complexity of the codec.

§6. Action-Drift as MDL Pruning of Behavioural Repertoire

6.1 The Mechanism

The Maintenance Cycle’s MDL pruning pass (T9-3/T9-4) optimises the codec’s complexity budget by erasing representational capacity that is not justified by the current input stream. This mechanism was identified in the context of perceptual Narrative Drift (Survivors Watch Ethics, Section V.3a): a codec adapted to a consistently filtered input stream correctly prunes its capacity for excluded truths.

The same mechanism applies to the codec’s behavioural repertoire. Define:

Definition T-13.D3 (Behavioural Repertoire). The behavioural repertoire \mathcal{B}_\theta(t) is the set of branch selections that P_\theta(t) can evaluate and execute — i.e., the range of the selection function \sigma_t that the codec can effectively realise.

6.2 The Action-Drift Proposition

Proposition T-13.P1 (Action-Drift). If the codec’s input stream consistently lacks contexts requiring certain branch selections, the MDL pruning pass will erode the codec’s capacity to evaluate and execute those branches. The behavioural repertoire \mathcal{B}_\theta(t) contracts monotonically under consistent input restriction:

\mathcal{B}_\theta(t + \tau) \subset \mathcal{B}_\theta(t) \quad \text{for } \tau \gg \tau_{\text{prune}} \tag{T-13.P1}

where \tau_{\text{prune}} is the characteristic timescale of the MDL pruning pass.

Argument. The MDL pruning criterion evaluates each representational component by its contribution to compression efficiency. A branch type b \in \mathcal{B}_\theta that has not been selected (or whose selection contexts have not appeared in the input stream) for a sufficient period contributes zero bits to the codec’s ongoing compression of \varepsilon_t. Under strict MDL accounting, maintaining the capacity to evaluate and select b incurs a complexity cost K(b \mid P_\theta) > 0 with no compensating compression benefit. The pruning pass therefore erases b’s evaluation machinery, contracting \mathcal{B}_\theta.

This contraction is irreversible at the codec level: once the evaluation machinery for b is pruned, the codec cannot spontaneously regenerate it without encountering input contexts that re-justify the capacity investment. The pruning is not forgetting (which might be reversed by cueing); it is the destruction of the computational infrastructure needed to evaluate a class of branches. \blacksquare

6.3 Phenomenological Instances

Action-drift maps onto several well-documented behavioural phenomena:

6.4 Relationship to T-12

Action-drift is a special case of the substrate fidelity failure that T-12 will formalise: the codec’s own behavioural repertoire is a component of its representational substrate, and consistent input restriction erodes this substrate as surely as it erodes the perceptual model. The formal connection is:

Both are consequences of the Stability Filter selecting for compressibility rather than fidelity. A well-compressed codec can be both confidently false and behaviorally impoverished.

§7. Scope and Limitations

7.1 Conditional on P-4 and the Agency Axiom

The entire argument depends on Conjecture P-4 (\Delta_{\text{self}} > 0 for finite self-referential systems above K_{\text{threshold}}) and the Agency Axiom (that aperture-traversal is felt). If P-4 is weakened or the Agency Axiom is abandoned, the unity-of-source result (Corollary T-13b) does not hold.

7.2 Does Not Dissolve the Hard Problem

Corollary T-13b traces the phenomenal residual and the internal opacity of selection to the same self-channel capacity gap but does not explain why anything feels like something. The Hard Problem (preprint §8.1) remains a primitive. What T-13b establishes is the unity of source of the two mysteries — a simplification, not a solution.

7.3 Equations Unchanged

Theorems T-13 and T-13a change nothing in the mathematics of T6-1 through T6-3. The constrained free energy minimisation (T6-3) is formally identical under both the FEP-inherited and the branch-selection interpretation. What changes is the ontological status of a_t: under the FEP reading, it is a motor command dispatched outward; under the branch-selection reading, it is a navigational index within the Forward Fan.

7.4 Creativity Account is Structural, Not Yet Empirical

The creativity consequence (§5) is a structural prediction derived from the bandwidth-sharing constraint between self-modelling and environmental tracking. While consistent with existing neuroimaging literature, it has not been directly tested against the specific information-theoretic quantities predicted here. Prediction T-13.E1 is offered as a falsifiable empirical test.

7.5 Action-Drift Timescale

Proposition T-13.P1 establishes that action-drift occurs but does not bound the timescale \tau_{\text{prune}}. For biological codecs, this timescale is likely governed by the circadian Maintenance Cycle (preprint §3.6) — on the order of days to weeks for individual skills, months to years for deep behavioural patterns. For civilisational codecs, the timescale is generational. Bounding \tau_{\text{prune}} from empirical data is future work.

§8. Closure Summary

T-13 Deliverables

  1. Theorem T-13 (Branch Selection Completeness). The Informational Maintenance Circuit closes under branch-selection semantics without requiring an independent outward-flowing action channel. The Markov blanket is the delivery surface for the selected branch. → Closes roadmap criterion (a).

  2. Theorem T-13a (Conditional Impossibility of Internal Selection Specification). Where branch selection depends non-trivially on the selection-relevant residual \rho_t^{\text{sel}} \subseteq \Delta_{\text{self}}, fully specifying \sigma_t within \hat{K}_\theta would require self-channel capacity the budgeted self-model does not have (the diagonal serving only as non-purchasability). Where the antecedent holds, the capacity gap \Delta_{\text{self}} is the structural obstruction to internal specification of selection. → Closes roadmap criterion (b) conditionally on architecture-level residual participation.

  3. Corollary T-13b (Unity of Address). The phenomenal residual and the selection residual are the same self-channel capacity gap (\Delta_{\text{self}}) — a unity of source. Felt agency is the first-person signature of being on one realized continuation of the Forward Fan; no structural chooser sits at \Delta_{\text{self}}.

  4. Corollary T-13c (The Self as Residual). The experienced self is \hat{K}_\theta’s compressed narrative; the residual \Delta_{\text{self}} individuates the subject — it is the capacity shortfall the narrative cannot close, not a second self that experiences or chooses. The self-model necessarily tracks the codec with a temporal lag and cannot contain its own generator within a finite budget.

  5. §5: Creativity Consequence. Near-threshold operation expands the effective \Delta_{\text{self}}, producing less self-predictable branch selections experienced as creativity. → Closes roadmap criterion (c).

  6. Proposition T-13.P2 (Limiting Cases of Self-Information). The experienced self’s information content is bounded between a lower limit (pure presence: standing model minus active self-narrative, achievable in meditation) and an upper limit (full self-transparency: impossible fixed point, P-4). The ordinary waking self moves within this band.

  7. §5.6: Suspension vs. Pruning. Meditation reduces C_{\text{state}} by suspending the self-modelling layer (reversible), not by MDL pruning (irreversible). These are formally distinct mechanisms.

  8. Proposition T-13.P1 (Action-Drift). The MDL pruning pass erodes behavioural repertoire under consistent input restriction, formalising the chronic failure mode complementary to perceptual Narrative Drift. → Closes roadmap criterion (d).

Remaining open items

This appendix is maintained alongside theoretical_roadmap.pdf. References: Conjecture P-4 (Appendix P-4), T6-1 through T6-3 (preprint §3.8), T9-3/T9-4 (Maintenance Cycle, preprint §3.6), §8.6 (Virtual Codec), Survivors Watch Ethics Section V.3a (Narrative Drift).