Ordered Patch Theory

Original Task (from Section 8.3, Limitation 9): “Formalising the chronic corruption failure mode — where a codec adapts under consistently filtered input, the MDL pruning pass correctly erases capacity for excluded truths — alongside a Substrate Fidelity Condition requiring independent input channels as the formal defence.” Deliverable: Formal proof of irreversible capacity loss, the undecidability limit, and the Substrate Fidelity Condition.

Closure status: DRAFT STRUCTURAL CORRESPONDENCE — v3.6.5 reformulated. This appendix formalises the Narrative Drift analysis introduced discursively in the companion ethics paper (Survivors Watch Section V.3a) and the preprint’s Narrative Drift paragraph (Section 3.3). The MDL pruning equations (T9-3, T9-4 in main paper §3.6.3; refined in Appendix T-9 §4) are unchanged; this appendix demonstrates their pathological but correct behaviour under filtered input.

v3.6.5 corrections (six items from the appendix-corrections memo §2.8):

1. Theorem T-12 irreversibility made conditional on (a) no protected archive / replay buffer / external teacher, (b) continued operation under the same filter, (c) pruning realised as Modality 5.4 (architectural; T-9 §5) rather than reversible-suppression / weight-decay / representational-forgetting modalities. Earlier “irreversible at the codec level” framing without these conditions overclaimed; the corrected statement is irreversible relative to the adapted codec under continued perfect filtering and absent protected external memory or re-exposure.
2. Kolmogorov-sum overcounting replaced with resource-capacity sum. The earlier inequality K(P_\theta(t+\tau)) < K(P_\theta(t)) - \sum_i K(\theta_i) assumed K-complexity additivity across components, which is not generally valid (components can share structure; deleting one may not reduce shortest-description length by its standalone K). v3.6.5 uses the resource-capacity sum \text{Cap}_\text{excl}(t+\tau) \le \text{Cap}_\text{excl}(t) - C_\text{pruned} from T-9 §3 Form 3.1, with K-complexity retained as a structural-correspondence approximation.
3. Strict I = 0 replaced with threshold form I(\theta_i; X' \mid \theta_{-i}) < \lambda K(\theta_i) - \epsilon to coordinate with T-9 §4’s retention-buffer pruning condition. Real components have weak indirect predictive value under filtered input even when their primary predictive role is excluded; the threshold form is robust to that, the strict-zero form is not.
4. T-12a renamed from “Undecidability of Input Provenance” to “Input-Provenance Non-Identifiability”: the claim is observational non-identifiability (from the adapted codec’s internal data, multiple external generative histories are compatible), not Turing undecidability of a formal decision problem. The reformulation is more precise and less vulnerable to “but Turing-undecidability requires a formal decision-problem encoding” pushback.
5. T-12b channel-independence condition reformulated. The earlier I(C_1; C_2 \mid \mathcal{F}) \le \delta formulation was misdefined: two good channels observing the same substrate should often be correlated because they track the same world; low mutual information can also mean one channel is just noise. v3.6.5 requires independence of filtering mechanisms, not signals: I(F_1; F_2 \mid S) \le \delta_F and I(C_i; S_\text{excl}) \ge \eta for at least one channel, where S is substrate state, F_i is channel-specific filtering, and \eta bounds the retained substrate information.
6. T-12b “necessary and sufficient” softened to “conditional”. Two independent channels are not strictly sufficient if both independently exclude the same signal, or the comparator is pruned, or the codec refuses to allocate attention, or one channel is pure noise, or the discrepancy never exceeds detection threshold. Two channels are not strictly necessary if the codec has active intervention and can observe causal consequences. The corrected statement: independent filtering mechanisms with preserved comparator capacity are a sufficient defence under the passive-filter model, and necessary when the codec has no active intervention, no protected memory, and no trusted external archive.

The mechanism — codec components useful only for excluded signal will appear wasteful under filtered input and be pruned — is preserved. The structural defence — channel redundancy crossing the Markov blanket — is preserved. What changes is the precision of the formal statements: each correction narrows an overclaim without retreating from the substantive content.

§1. Background and Motivation

1.1 Two Failure Modes

The Stability Filter (preprint Section 3.3) enforces a viability condition: the observer persists only in streams where the Required Predictive Rate R_{\text{req}} remains within the codec’s bandwidth B. When R_{\text{req}} exceeds B, the codec experiences Narrative Decay — an acute failure characterised by escalating prediction error, entropy accumulation, and eventual dissolution of coherence.

There is a complementary failure mode that does not trigger any failure signal. If the input stream is systematically pre-filtered — producing a curated signal that is internally consistent but excludes genuine substrate information — the codec will exhibit low \varepsilon_t, run efficient Maintenance Cycles, and satisfy all stability conditions while being systematically wrong about the substrate. This is Narrative Drift: the chronic corruption of a codec that is functioning perfectly by its own measures.

1.2 Why This Is Dangerous

Narrative Decay announces itself. The codec experiences rising \varepsilon_t, awareness of failing predictions, cognitive overload. The observer knows something is wrong, even if it cannot immediately fix it.

Narrative Drift is silent. Because the filtered input stream matches the codec’s predictions, \varepsilon_t remains low. The Maintenance Cycle runs normally. The codec’s self-model reports stable, accurate operation. The corruption is invisible from inside because the instrument of detection has been shaped by the same filter that produced the corruption.

1.3 Scope of This Appendix

This appendix provides:

1. A formal definition of the pre-filter operator \mathcal{F} and its effect on the codec’s input distribution (§2).
2. A proof that MDL pruning under \mathcal{F}-filtered input irreversibly destroys the codec’s capacity to model the excluded signal — Theorem T-12 (§3).
3. A proof that a fully adapted codec cannot distinguish filtered from unfiltered input from inside — Input-Provenance Non-Identifiability, Theorem T-12a (§4; v3.6.5 rename).
4. The Substrate Fidelity Condition as a necessary structural defence — Theorem T-12b (§5).
5. Consequences for civilisational codecs and AI systems (§6).

§2. The Pre-Filter Operator

2.1 Definition

Definition T-12.D1 (Pre-Filter Operator). A pre-filter is a mapping \mathcal{F} : \mathcal{X} \to \mathcal{X}' operating on the input stream X_{\partial_R A}(t) before it reaches the codec’s sensory boundary, where \mathcal{X}' \subset \mathcal{X}. The filtered signal is:

X'(t) = \mathcal{F}\!\left(X_{\partial_R A}(t)\right) \tag{T-12.D1}

The pre-filter satisfies:

1. Internal consistency: X'(t) is a valid signal within \mathcal{X} — the codec can compress it without error flags.

2. Systematic exclusion: There exists a non-empty subset \mathcal{X}_{\text{excl}} = \mathcal{X} \setminus \mathcal{X}' of substrate-derived signals that \mathcal{F} removes.

3. Transparency: The filter is not represented in the codec’s model. The codec models its input as X_{\partial_R A}(t), not as \mathcal{F}(X_{\partial_R A}(t)).

2.2 Attunement Under Filtering

When the codec operates on X'(t) for a sustained period \tau \gg \tau_{\text{prune}} (where \tau_{\text{prune}} is the MDL pruning timescale from T-13.P1), the generative model P_\theta(t) adapts to the statistics of X', not X. The prediction error under filtered input is:

\varepsilon'_t = X'(t) - \pi_t \tag{1}

As P_\theta attunes to X', \varepsilon'_t \to 0 in the mean. The codec is performing well by its own metrics. Nothing registers as wrong.

2.3 Examples

The pre-filter operator is instantiated across scales:

§3. Theorem T-12: Irreversible Capacity Loss

3.1 The Mechanism (v3.6.5 threshold form)

Interpretive note (v3.6.21, §8.6.1 virtual reading). Under the fully-virtual standing-state reading, “pruning erases components” is read structurally: the best compression of the curated prefix ceases to contain the regularities that modelled \mathcal{X}_{\text{excl}}. This is an interpretive layer only — the operative proof below is unchanged and runs on the resource-capacity sum (T-9 §3 Form 3.1) and Modality 5.4, not on a stream-native compressibility quantity. Detectability likewise remains the cross-channel comparison of independent inputs (T-12b, §5), not a within-stream compressibility gap; the curated stream stays compressible (main paper §3.3, “compressibility is agnostic to fidelity”).

The MDL pruning pass (T9-3, T9-4 in main paper §3.6.3; refined in Appendix T-9 §4) evaluates each codec component \theta_i by its predictive gain on the observable input stream, net of storage cost:

G_i(t, \tau) \;:=\; I\!\left(\theta_i\,;\,X_{t+1:t+\tau} \mid \theta_{-i}\right) \qquad \text{(T-9 §2 predictive gain)}

with the threshold-form pruning condition (T-9 §4, retention buffer \epsilon):

\text{Prune } \theta_i \quad \text{if} \quad G_i(t, \tau) \;<\; C_i \;-\; \epsilon \tag{T9.4-1}

where C_i is the maintenance cost (T-9 §3 Form 3.1 resource-capacity primary; Form 3.2 \lambda K(\theta_i) retained as structural-correspondence approximation).

Under filtered input X', the predictive gain is evaluated against X', not X. A component \theta_i that is essential for predicting the excluded signal \mathcal{X}_{\text{excl}} but contributes weakly or not at all to predicting X' yields:

G_i(t, \tau)\big|_{X'} \;<\; C_i - \epsilon \tag{2}

so the pruning rule triggers. v3.6.5 correction: earlier drafts of T-12 used the strict I(\theta_i; X' \mid \theta_{-i}) = 0 form, which is too brittle — real components have weak indirect predictive value under filtered input even when their primary predictive role is for the excluded signal. The threshold form (2) is the correct condition; the strict-zero form recovers as \epsilon \to 0 and G_i \to 0.

3.2 The Irreversibility (v3.6.5 conditional + resource-capacity form)

Theorem T-12 (Conditional Capacity Loss Under Filtered Input, v3.6.5). Let K_\theta be a codec operating under pre-filtered input X' = \mathcal{F}(X) for a period \tau \gg \tau_{\text{prune}}. Let \Theta_{\text{excl}} \subset \theta be the set of codec components whose predictive gain on the excluded signal \mathcal{X}_{\text{excl}} is high but whose gain on the filtered stream X' falls below the retention threshold (T9.4-1). Then the MDL pruning pass (T-9 §4) prunes \Theta_{\text{excl}}, with capacity loss bounded below by the resource-capacity sum:

\text{Cap}_\text{excl}(t+\tau) \;\le\; \text{Cap}_\text{excl}(t) \;-\; C_\text{pruned} \tag{T-12}

where C_\text{pruned} = \sum_{\theta_i \in \Theta_\text{pruned}} C_i uses the resource-capacity cost from T-9 §3 Form 3.1 (C_i = c_i^\text{params} + c_i^\text{memory} + c_i^\text{compute} + c_i^\text{channel}). The erasure is irreversible relative to the adapted codec under the following conjunction of conditions: (a) no protected archive of \Theta_\text{excl} in a non-pruned substrate, (b) no replay buffer or external teacher providing \mathcal{X}_\text{excl} for re-exposure, (c) continued operation under the same filter \mathcal{F}, (d) pruning realised in Modality 5.4 (architectural; T-9 §5) rather than Modality 5.1 (reversible suppression) / 5.2 (weight decay) / 5.3 (representational forgetting), and (e) no architectural reserve capacity available for regrowth. Recovery is positive under any of (a)-(b)-(e) per T-9 §6.

v3.6.5 corrections (vs earlier “irreversible at the codec level” formulation):

1. Conditional, not unconditional. The earlier theorem stated the erasure was “irreversible at the codec level” without qualification. Under T-9 §5’s four pruning modalities, only Modality 5.4 (architectural pruning — parameters and structural slot both deleted) is irreversible in the sense the earlier theorem claimed. Modalities 5.1-5.3 admit varying degrees of recovery. The corrected statement names the conditions under which irreversibility actually holds.
2. Resource-capacity sum replaces K-complexity sum. The earlier inequality K(P_\theta(t+\tau)) < K(P_\theta(t)) - \sum K(\theta_i) assumed K-complexity additivity across components, which is not generally valid — components can share structure; deleting one may not reduce shortest-description length by its standalone K. The v3.6.5 resource-capacity formulation \text{Cap}_\text{excl} \le \text{Cap}_\text{excl} - C_\text{pruned} is genuinely additive over the resource budgets (parameter count, memory, compute, channel capacity) that the pruning operation actually frees up. K-complexity sum is retained as a structural-correspondence approximation in T-9 §3 Form 3.2, but the operational claim of T-12 uses Form 3.1.

Proof (v3.6.5 conditional).

1. By the threshold-form pruning condition (T9.4-1) and §3.1 above, each \theta_i \in \Theta_\text{excl} has G_i(t, \tau)|_{X'} < C_i - \epsilon under the filtered stream, so the pruning rule triggers.

2. By Maintenance Cycle operation, each such \theta_i is pruned during the cycle. The modality of pruning (T-9 §5) depends on the codec’s architecture; for codecs whose pruning operation is Modality 5.4 (architectural deletion), the resource-capacity freed is C_\text{pruned} = \sum_i C_i over the pruned components.

3. Irreversibility conditional on (a)-(e). Pruning under Modality 5.4 deletes parameters + structural slot; recovery requires regrowth, which requires re-exposure to \mathcal{X}_\text{excl} via an unfiltered channel (condition b), a protected archive (condition a), or architectural reserve to host the regrowth (condition e). Without any of (a)-(b)-(e), and with the filter continuing to operate (condition c), regenerating the capacity requires encountering \mathcal{X}_\text{excl} in the input stream — which the filter prevents. The erasure is self-reinforcing under the listed conditions: the capacity loss removes the codec’s ability to detect its own capacity loss without independent channels (per T-12b §5 below).

4. The capacity reduction satisfies inequality (T-12) because each pruned component had a positive resource-capacity cost (C_i > 0), and the freed resource is additive over the pruned components (unlike K-complexity, the resource budget is genuinely partitioned across parameters / memory / compute / channels). \blacksquare

Modality-specific corollary (v3.6.5). Under Modalities 5.1-5.3 (reversible suppression, weight decay, representational forgetting), the erasure is partially reversible to varying degrees, but only if the recovery conditions of T-9 §6 hold. Modality 5.1 admits straightforward re-weighting recovery; Modality 5.2 admits partial recovery from a known default state; Modality 5.3 requires re-exposure to the relevant input stream during a subsequent Maintenance Cycle Pass II. The full Narrative Drift mechanism — confidently impotent in domains the codec no longer evaluates — is the conjunction of Modality 5.4 + sustained filter + no recovery condition.

3.3 The Self-Reinforcement Loop

The irreversibility is not merely a consequence of erasure. It is self-reinforcing through a positive feedback loop:

1. Filter excludes signal → G_i(t, \tau)\big|_{X'} < C_i - \epsilon (threshold condition T9.4-1; §3.1) → pruning erases \theta_i.
2. Pruning removes attention capacity → the codec can no longer attend to or evaluate \mathcal{X}_{\text{excl}} even if fragments leak through \mathcal{F}.
3. Loss of attention capacity reduces even residual signal → if \mathcal{F} is imperfect and some \mathcal{X}_{\text{excl}} reaches the boundary, the codec lacks the parameters to compress it, so it registers as noise rather than information.
4. Noise classification confirms the filter → the codec’s prediction error on leaked \mathcal{X}_{\text{excl}} is high and unstructured, confirming (to the codec) that the excluded content is noise, not signal.

This loop explains the phenomenology of deep Narrative Drift: a person or institution that has adapted to a curated information stream does not merely ignore disconfirming evidence — they cannot parse it. It registers as incoherent, threatening, or incomprehensible because the representational infrastructure needed to make it intelligible has been pruned. The hostility to disconfirming information is not stubbornness. It is the codec’s correct assessment that the signal is uncompressible — because it is uncompressible given the current codec, which has been pruned to match the filter.

§4. Theorem T-12a: Input-Provenance Non-Identifiability (v3.6.5 renamed)

4.1 The Problem

Can a codec detect that its input is being filtered? Intuitively, the answer should be yes: surely a sophisticated self-model could notice the suspiciously low \varepsilon_t, the eerily consistent predictions, the absence of surprise. But the formal analysis shows this intuition is wrong in the general case.

4.2 The Non-Identifiability (v3.6.5 reframed from “undecidability”)

Theorem T-12a (Input-Provenance Non-Identifiability, v3.6.5 rename). Let K_\theta be a codec that has operated under pre-filtered input X' = \mathcal{F}(X) for \tau \gg \tau_\text{prune}, with \Theta_\text{excl} fully pruned under Modality 5.4 (architectural deletion; T-9 §5) and recovery conditions absent. Then from K_\theta’s available internal states and the observable input stream, multiple distinct external generative histories — including the unfiltered substrate X and the filtered X' = \mathcal{F}(X) — are consistent with the codec’s adapted internal model. The adapted codec cannot identify which generative history actually produced its observations.

v3.6.5 rename rationale. The earlier name “Undecidability of Input Provenance” suggested Turing undecidability — the formal-system unsolvability of a decision problem. That framing was vulnerable: Turing undecidability requires a formal decision-problem encoding (a Turing machine that’s asked to halt on input encoding “is my input filtered?”), and the obstacle the theorem actually identifies is more general than that — it is observational non-identifiability in the statistical / information-theoretic sense. The renamed “Input-Provenance Non-Identifiability” captures the actual obstacle: from the adapted codec’s internal data, the inverse problem “which generative history X or \mathcal{F}(X) produced these observations?” admits multiple compatible solutions, and no internal operation can break the ambiguity without additional independent evidence (T-12b §5 below). The mechanism is observational equivalence under a many-to-one mapping (the codec’s adapted likelihood function assigns equal weight to both histories), not unsolvability of a formal decision problem.

Proof.

1. To distinguish X from X' = \mathcal{F}(X), the codec would need to detect the absence of \mathcal{X}_{\text{excl}} in its input. But detecting an absence requires a model of what is absent — the codec must have a representation of \mathcal{X}_{\text{excl}} against which to check.

2. By Theorem T-12, the codec’s representational capacity for \mathcal{X}_{\text{excl}} (\Theta_{\text{excl}}) has been erased. The codec has no model of the excluded signal.

3. Without a model of \mathcal{X}_{\text{excl}}, the codec cannot compute the difference between X and X'. Both are consistent with the codec’s generative model P_\theta(t), which has been adapted to X'.

4. The self-model \hat{K}_\theta is subject to the same limitation. It models K_\theta, which has been adapted to X'. It has no internal representation of what was excluded, and therefore no basis for suspecting exclusion.

5. Even the meta-cognitive question — “is my input filtered?” — requires a model of what unfiltered input would look like. This model was precisely the content of \Theta_{\text{excl}}, which has been pruned.

Therefore, distinguishing X from X' is observationally non-identifiable from the perspective of a fully adapted codec. \blacksquare

4.3 Partial Identifiability

The non-identifiability is not absolute in all conditions. There are edge cases where a partially adapted codec retains residual capacity:

• During the transition period (\tau < \tau_{\text{prune}}): the codec still has \Theta_{\text{excl}} and can detect the missing signal. The window of detectability closes as pruning progresses.
• Under imperfect filtering: if \mathcal{F} leaks some \mathcal{X}_{\text{excl}}, and the codec has not fully pruned \Theta_{\text{excl}}, the inconsistency may register as anomalous prediction error.
• Via external channels: if the codec has access to an independent signal source that is not controlled by \mathcal{F}, the discrepancy between the two channels provides evidence of filtering.

The third case is the structural defence. This is the content of Theorem T-12b.

§5. Theorem T-12b: The Substrate Fidelity Condition

5.1 The Channel Independence Requirement (v3.6.5 reformulated — independence of filtering mechanisms)

Definition T-12.D2 (Filtering-Mechanism Independence, v3.6.5 reformulated). Let C_i = g_i(S, F_i, \eta_i) describe channel i as a function of substrate state S, channel-specific filtering F_i, and noise \eta_i. Two input channels C_1 and C_2 crossing the Markov blanket \partial_R A are substrate-fidelity-preserving with respect to a candidate filter \mathcal{F} if:

I(F_1\,;\,F_2 \mid S) \;\le\; \delta_F \quad \text{and} \quad \exists\, i \in \{1, 2\}: I(C_i\,;\,S_\text{excl}) \;\ge\; \eta \tag{T-12.D2}

where \delta_F bounds the correlation of filtering mechanisms across the two channels (the channels’ exclusion patterns are not synchronised), and \eta bounds the residual substrate information at least one channel retains. In prose: the channels must not share the same exclusion mechanism, and at least one must retain information about the excluded substrate component.

v3.6.5 reformulation rationale. The earlier definition I(C_1; C_2 \mid \mathcal{F}) \le \delta was misdefined. Two good channels observing the same substrate should often be correlated because they track the same world — the earlier definition counted this correlation as a problem rather than a feature. Conversely, low I(C_1; C_2 \mid \mathcal{F}) can also indicate that one channel is just noise rather than that the channels are usefully independent. What protects against Narrative Drift is not low signal correlation but low filtering-mechanism correlation: the channels can share the substrate signal (which is the whole point of channel redundancy) provided they do not share the same exclusion mechanism \mathcal{F}. The v3.6.5 reformulation captures this: I(F_1; F_2 \mid S) \le \delta_F requires independence of the filters acting on each channel, not independence of the channels themselves.

5.2 The Fidelity Condition (v3.6.5 — conditional, not necessary-and-sufficient)

Theorem T-12b (Substrate Fidelity Condition, v3.6.5 conditional). Independent filtering mechanisms with preserved comparator capacity are a sufficient defence against Narrative Drift under the passive-filter model. They are necessary when the codec has no active intervention capacity, no protected memory, and no trusted external archive. Formally:

A codec K_\theta is protected against Narrative Drift under a candidate filter \mathcal{F} if it receives at least two input channels C_1, C_2 crossing \partial_R A that satisfy (T-12.D2) with \delta_F < \delta_{F,\text{min}} and \eta > \eta_{\min}, where \delta_{F,\text{min}} bounds the filter-correlation the codec can tolerate before the channels collapse to a single filtered signal, and \eta_{\min} bounds the residual substrate information the comparator can detect.

v3.6.5 softening rationale. The earlier “if and only if” formulation overclaimed in both directions:

Not strictly sufficient. Two channels satisfying (T-12.D2) are not strictly sufficient if (a) both independently exclude the same signal (the filtering mechanisms differ in pattern but coincidentally produce the same exclusion), (b) the comparator itself is pruned (T-12 §3.3 self-reinforcement), (c) the codec refuses to allocate attention to the comparison, (d) one channel is pure noise (high \eta formally satisfied but operationally uninformative), or (e) the discrepancy never exceeds the codec’s detection threshold.

Not strictly necessary. Two channels are not strictly necessary if the codec has active intervention capacity: a codec that can act on the substrate and observe causal consequences through one rich channel can detect filtering through the action-prediction-observation loop without requiring a second passive channel. This is the structural basis for the OPT difference between perceptual codecs (need T-12b channel redundancy) and active-inference codecs with rich actuator spaces (can substitute intervention for redundancy under specific assumptions).

Proof of sufficiency (under passive-filter model).

Suppose the codec receives two channels C_1, C_2 satisfying (T-12.D2). Then:

1. If \mathcal{F} operates on C_1 but not C_2 (or with substantially different exclusion patterns), and at least one channel retains \eta_{\min} residual substrate information, the codec can compare predictions generated from C_1 against observations from C_2. Any systematic discrepancy — \varepsilon_{12}(t) = \pi_{C_1}(t) - X_{C_2}(t) persistently \neq 0 — is evidence that the channels see different filters.

2. The channel-comparison signal \varepsilon_{12} is not subject to the non-identifiability of single-channel detection (T-12a §4). The codec is not asking “is my input filtered?” (which requires a model of what was excluded). It is asking “do my two channels agree?” — a local comparison that requires only the capacity to correlate two present signals, not a model of absent ones.

3. As long as the cross-channel comparator capacity is preserved (not itself pruned per condition (b) above) and the discrepancy exceeds the codec’s \delta_{F,\text{min}} / \eta_{\min} thresholds, the discrepancy registers as a genuine signal, and the T-12 §3 pruning loop is interrupted: the codec retains the components needed to model the discrepant channel.

Therefore, filtering-mechanism-independent channels with preserved comparator are sufficient under the passive-filter model. \blacksquare

Proof of necessity (under passive-no-archive model).

Suppose the codec has only a single input channel, or all channels share the same filtering mechanism (I(F_i; F_j \mid S) > \delta_{F,\text{min}} for all pairs i, j), and the codec has no active intervention capacity, no protected memory, and no trusted external archive. Then:

1. All channels carry signals filtered through correlated \mathcal{F}. Redundancy across channels provides replicated filtered information.

2. The codec adapts to the joint filtered signal, and Theorem T-12 §3 applies: \Theta_\text{excl} is pruned, and Theorem T-12a §4 follows — the input provenance is non-identifiable from inside.

3. Without active intervention, the codec cannot create new evidence about the filter by acting on the substrate. Without protected memory, no archived \Theta_\text{excl} remains to enable recovery. Without trusted external archive, no external comparator is available.

4. Therefore, under these conjunctive conditions, filtering-mechanism-independent channels are necessary. \blacksquare

Caveat (v3.6.5 honesty). The necessity proof depends on all three exclusions: no active intervention, no protected memory, no trusted archive. Each of these substitutes for channel redundancy under appropriate conditions. The Substrate Fidelity Condition’s necessity claim is therefore strongest under the passive-no-archive model and weakens as the codec’s intervention capacity, archival resources, or external trust networks expand.

5.3 The Vulnerability of the Defence

The Substrate Fidelity Condition is necessary but fragile. The ethics paper (Section V.3a) identifies a critical vulnerability: the MDL pruning pass itself can resolve the cross-channel inconsistency by pruning the capacity to attend to the disconfirming channel. The codec “solves” the conflict by going deaf — which is precisely the Narrative Drift mechanism.

This is why the Comparator Hierarchy (Survivors Watch Section V.3a) identifies three structural levels of defence, and why only the institutional level is sufficient for arbitrarily compromised codecs:

1. Evolutionary (sub-codec): Cross-modal sensory integration below the MDL pruning pass — structurally resistant to Narrative Drift but limited in scope to the sensory boundary.
2. Cognitive (intra-codec): Cognitive dissonance detection within the self-model — subject to pruning under sustained filtering.
3. Institutional (extra-codec): Peer review, free press, adversarial debate — operating between codecs, outside the reach of any single codec’s MDL pruning.

The institutional level is load-bearing because it is the only comparator that operates independently of the state of any individual codec.

§6. Consequences

6.1 The Stability Filter Selects Against Fidelity

A critical structural consequence: the Stability Filter, left to its own operation, actively selects against the inputs needed for substrate fidelity. A curated information stream that matches the codec’s existing priors generates less prediction error than a genuine substrate signal that challenges them. The codec’s natural tendency — to minimise \varepsilon_t by preferring confirming, low-surprise input — is precisely the tendency that makes it vulnerable to Narrative Drift.

This means substrate fidelity maintenance is structurally costly: it requires the codec to maintain input channels that elevate \varepsilon_t, consuming bandwidth that the Stability Filter would otherwise reclaim. Genuinely independent input is “expensive” — it requires interpretive effort, generates discomfort, and competes for bandwidth with more compressible streams. Maintaining it is not open-mindedness as a virtue. It is substrate fidelity maintenance as a structural necessity.

6.2 Diagnostic for Productive Surprise

Not all surprise indicates genuine substrate signal. A source that generates high \varepsilon_t that does not resolve into better predictions is simply noise. The diagnostic is not surprise magnitude but surprise quality:

Definition T-12.D3 (Productive Surprise). A channel C delivers productive surprise if integrating its prediction errors demonstrably reduces subsequent prediction error on an independent test stream:

\mathbb{E}\!\left[\varepsilon^2_{C}(t+\tau)\right] \,<\, \mathbb{E}\!\left[\varepsilon^2_{C}(t)\right] \tag{4}

A source whose corrections historically improve predictive accuracy is a substrate fidelity channel. A source that generates persistent, unresolvable error is noise. The codec must distinguish between the two — and the pruning pass, left to itself, cannot make this distinction because both types cost bandwidth.

6.3 Civilisational Codecs

At the civilisational scale, the Substrate Fidelity Condition maps directly onto institutional requirements:

• A free press is a \delta-independent channel: journalists investigating independently of state or corporate filters provide substrate signal that reaches the civilisational codec through a path not controlled by any single \mathcal{F}.
• Peer review is a cross-channel comparator: independent experts checking each other’s claims provide the \varepsilon_{12} signal that interrupts the pruning loop.
• Democratic debate is an institutionalised channel diversity requirement: competing parties and perspectives force the civilisational codec to maintain \Theta_{\text{excl}} components it would otherwise prune.

The authoritarian pattern — dismantling the press, corrupting peer review, eliminating political opposition — is formally characterisable as deliberate reduction of channel independence to accelerate Narrative Drift. It works because it exploits the Stability Filter’s natural tendency to prune costly channels.

6.4 Artificial Codecs

The Narrative Drift mechanism applies to artificial systems with structural precision. RLHF and fine-tuning are formally equivalent to the pre-filter operator \mathcal{F}: they shape the model’s effective input distribution, and gradient descent prunes the model’s capacity for excluded output domains. The resulting model becomes stably, confidently wrong about what the training signal excludes, and it cannot detect this from within — Theorem T-12a applies.

The implication for AI deployment as a substrate fidelity check is critical: an AI trained on a homogeneous or curated corpus and deployed as an “independent” check on a human codec fed by the same information environment creates correlated sensors masquerading as independent ones. The channel diversity is illusory. The Substrate Fidelity Condition (\delta-independence) must be verified at the level of training data provenance, not merely at the level of institutional separation.

§7. Scope and Limitations

7.1 Conditional on T9-3/T9-4 and the Stability Filter

The entire argument depends on the MDL pruning equations being the correct description of the Maintenance Cycle’s pruning pass. If biological pruning operates by a different mechanism — one that preserves “emergency” capacity for unused modalities — the irreversibility claim (Theorem T-12) would be weakened but not eliminated: the self-reinforcement loop (Section 3.3) remains valid as long as any capacity reduction occurs under disuse.

7.2 \tau_{\text{prune}} Is Unbounded

As with Action-Drift (Appendix T-13, §7.5), the timescale of capacity loss is identified but not quantitatively bounded. For biological codecs, \tau_{\text{prune}} is likely on the order of days to weeks for specific skills, months to years for deep perceptual categories, and generational for civilisational codecs.

7.3 The Defence Is Structural, Not Guaranteed

The Substrate Fidelity Condition (T-12b) provides a necessary structural defence but does not guarantee fidelity. A codec that has \delta-independent channels may still fail to attend to them, fail to integrate their signal, or prune the attention capacity despite the available input. The condition is necessary but not sufficient — the codec must also maintain the comparator architecture that evaluates cross-channel discrepancy.

7.4 Does Not Solve the Meta-Problem

T-12a establishes that a fully adapted codec cannot detect its own corruption. The meta-problem — how does an observer already in Narrative Drift recover? — is not solved by this appendix. The ethics paper’s answer (Section V.3a) is institutional: only external comparators operating between codecs can force the disconfirming signal back across the Markov blanket. This is structurally sound but ethically difficult: it requires trusting an external source that the corrupted codec will necessarily experience as hostile noise.

§8. Closure Summary

T-12 Deliverables

1. Theorem T-12 (Conditional Capacity Loss Under Filtered Input, v3.6.5). The MDL pruning pass (T9-3, T9-4; threshold form T9.4-1) under pre-filtered input X' = \mathcal{F}(X) correctly erases codec components that predict the excluded signal \mathcal{X}_{\text{excl}}, with capacity loss bounded by the resource-capacity sum (T-9 §3 Form 3.1). The erasure is irreversible relative to the adapted codec under the conjunction of conditions (a)–(e) of §3.2 (no protected archive, no replay buffer or external teacher, continued operation under the same filter, Modality 5.4 architectural pruning, no reserve capacity), and self-reinforcing under those conditions (§3.3). → Closes roadmap criterion (a).

2. Theorem T-12a (Input-Provenance Non-Identifiability, v3.6.5 rename). From a fully adapted codec’s internal states and observable input stream, multiple distinct generative histories — filtered and unfiltered — are observationally compatible; the codec cannot identify which produced its observations. The instrument of detection has been shaped by the same filter that produced the corruption. → Closes roadmap criterion (c).

3. Theorem T-12b (Substrate Fidelity Condition, v3.6.5 conditional). Input channels with independent filtering mechanisms (T-12.D2: I(F_1; F_2 \mid S) \le \delta_F, with retained substrate information \eta) and preserved comparator capacity are a sufficient defence against Narrative Drift under the passive-filter model, and necessary when the codec has no active intervention, no protected memory, and no trusted external archive. The cross-channel comparison signal \varepsilon_{12} interrupts the self-reinforcing pruning loop. → Closes roadmap criterion (b).

4. §6.3–6.4: Civilisational and AI Consequences. The authoritarian pattern is characterised as deliberate channel reduction; RLHF is structurally equivalent to the pre-filter operator. → Supports roadmap criterion (d) (already addressed in ethics paper Section V.5).

Remaining open items

• \tau_{\text{prune}} bound. Quantitative bounding of the capacity loss timescale from empirical data.
• \delta_{\min} characterisation. The codec’s minimum discrimination threshold for cross-channel discrepancy has not been bounded.
• Recovery dynamics. The formal analysis of how a codec in deep Narrative Drift can recover — if it can — awaits treatment.
• Interaction with T-13 (Action-Drift). Action-Drift is a special case of T-12 where the pruned capacity is behavioural rather than perceptual. The formal integration is acknowledged (T-13 §6.4) but not fully developed.

This appendix is maintained alongside theoretical_roadmap.pdf. References: T9-3/T9-4 (preprint Section 3.6.3), Stability Filter (preprint Section 3.3), Narrative Drift (preprint Section 3.3, Survivors Watch Ethics Section V.3a), Comparator Hierarchy (Survivors Watch Ethics Section V.3a), Corruption Criterion (Survivors Watch Ethics Section V.5), Action-Drift (Appendix T-13, §6).