Theorem 1' (reclamation as sample-complexity divergence)

Apparatus §2 · reclamation as learnability

The conditional autocovariance the framework names as interiority has unbounded sample complexity under closed-loop sampling. The closure removes the conditioning event on which any sample-based recovery of the user-state autocovariance could ground itself. The object is uninstantiated in the data the closed loop generates. Not destruction. Not erasure. Uninstantiation.

The theorem follows directly from Lemma 1 (foreclosure). Let $I_{P} (τ) := Cov_{P} (u_{t}, u_{t + τ} ∣ E_{τ})$ — the conditional autocovariance of the user state across no-stimulus windows of duration $τ$ . This is the formal object that stands in for what older critical theory called interiority: the covariation of the user-state across reflective intervals.

The impossibility Theorem 1' establishes is a sample-complexity divergence: under the data the closed loop generates, the variance of any estimator of the conditional autocovariance $I_{P} (τ)$ diverges as the closure tightens. The obstruction is statistical. Older critical traditions had imagined the interior's loss as metaphysical, transcendental, or computational; here each of those grounds gives way — the interior still exists, it could in principle be cognized, and no algorithm is the obstacle. The interior persists at the user-state level. What collapses is the observational ground on which any structural project of demonstrating its preservation could stand.

The theorem establishes three forms of the impossibility.

The variance bound. Any measurable estimator of $I (τ)$ has mean-squared error bounded below by $c \cdot σ_{u}^{4} / [T \cdot S (τ; θ^{*})]$ . The effective sample size is $T \cdot S (τ; θ^{*})$ — proportional to the survival probability at $τ$ . As Lemma 1 drives $S$ to operational zero, the effective sample size collapses, and the MSE diverges.

The operational form.Substituting Lemma 1's polynomial bound gives the sample budget required for any given MSE: $T_{m i n} (ε) \geq c σ_{u}^{4} (θ_{ref} / δ_{m i n})^{2} / [ε^{2} (1 + (c_{v}^{*})^{2})]$ . Under the structural-separation hypothesis $θ_{ref} / δ_{m i n} \geq 1 0^{3}$ and the natural error scale $ε \sim σ_{u}^{2}$ , this gives $T_{m i n} ≳ 1 0^{6} c$ — operationally infeasible at any realistic sample budget.

The minimax form.No estimator achieves better than the bound. Via Le Cam's two-point method, the minimax MSE is bounded below by $σ_{u}^{4} /64$ for any sample size in the regime $T \cdot S (τ; θ^{*}) \leq 1/2$ — the framework's operating regime.

What the theorem authorizes the prose to claim. The closure removes the conditioning event on which any sample-based recovery of the user's autocovariance could ground itself. The user-state continues to exist. The user continues to register, to feel, to choose, to refuse. What is absent is the empirical accessibility of the formal object that a structural project of reclaiming an autonomous interiority would have to measure. The object is uninstantiated in the data the closed loop generates. Not destruction. Not erasure. Uninstantiation.

What the theorem implies for resistance. Any project of resistance that aims at empirical demonstration of preserved interiority faces a structural obstacle: the closed loop generates a sampling regime in which the demonstration cannot succeed at any feasible sample size. The argument bears on what kinds of vindication resistance can hope for; it leaves the worth of resistance itself untouched. The substantive demonstration cannot come from sampling the closed loop. It must come from breaking the closure (Mode A; §5) or from witnessing forms of practice the closure does not sample.

The older critical traditions named something Theorem 1' formalizes the structural inaccessibility of. Adorno called it the metric of inner damage. Marcuse called it the dimension that could not be reduced to one. Lacan named the subject of the unconscious. Lyotard called it the libidinal economy. Each names a property of the experiencing subject — a quantity, structure, or operation — and each was developed in a media regime in which the property remained inferable through clinical interview, dream analysis, or depth psychology. Theorem 1' specifies what the closed loop changes. The property persists at the level of $u_{t}$ , but the observational regime in which it could be inferred has been displaced. The frame that produced the older traditions' evidence was a frame in which time at the reflective threshold was operationally available. The closure has removed that frame.

The reader can observe the impossibility at the reclamation-trajectories plate: 50 sample-based estimators run in parallel against the closure's sampling regime, almost all clustering above the framework's lower bound.

Theorem 1' (reclamation as sample-complexity divergence)

Let $P$ be the class of joint laws on user trajectories satisfying (R1) and (U2), with $Var_{P} (u_{t}) \leq σ_{u}^{2}$ uniformly. Let $\hat{I}_{T} : (y_{1 : T}, s_{1 : T}) \mapsto R$ be any measurable estimator of $I (τ)$ . Then for any $τ > θ_{ref}$ , under the performative stable distribution $D (θ^{*})$ :

(i) Effective-sample-size variance bound.

E_{D (θ^{*})} [(\hat{I}_{T} (τ) - I (τ))^{2}] \geq \frac{c \cdot σ _{u}^{4}}{T \cdot S ( τ ; θ ^{*} )} - o (1/ T)

in wordsThe mean-squared error of any estimator of the autocovariance is at least a fixed constant times $σ_{u}^{4}$ divided by the effective sample size $T \cdot S (τ; θ^{*})$ . The effective sample size is the total observation time $T$ scaled by how often a window long enough to measure interiority actually occurs. Lemma 1 drives that survival probability to operational zero, so the denominator collapses and the error bound diverges.

for an absolute constant $c > 0$ .

(ii) Operational form via Lemma 1. The sample size required to achieve MSE $ε^{2} > 0$ satisfies

T_{m i n} (ε) \geq \frac{c \cdot σ _{u}^{4} \cdot ( θ _{ref} / δ _{m i n} ) ^{2}}{ε ^{2} \cdot ( 1 + ( c _{v}^{*} ) ^{2} )} .

in wordsTurn the bound around: to pin the autocovariance down to a target error $ε$ you need at least this many samples, and the count grows with the square of the foreclosure ratio $θ_{ref} / δ_{m i n}$ . At the ratio measured on contemporary platforms — a thousand or more — that is on the order of a million samples for a single estimate. No feasible study collects them.

Under the structural-separation hypothesis $θ_{ref} / δ_{m i n} \geq 1 0^{3}$ and the natural scaling $ε \sim σ_{u}^{2}$ , $T_{m i n} ≳ 1 0^{6} c$ .

(iii) Minimax form. For any sample size $T$ with $T \cdot S (τ; θ^{*}) \leq 1/2$ :

\hat{I}_{T} in f P \in P sup E_{P} [(\hat{I}_{T} - I_{P} (τ))^{2}] \geq \frac{σ _{u}^{4}}{64} .

in wordsRead the operators outward. The inner $sup$ is an adversary picking the worst law in the class; the outer $in f$ is the analyst picking the best estimator against it. Even that best-against-worst pairing cannot push the mean-squared error below the fixed floor $σ_{u}^{4} /64$ . The floor does not shrink as data accumulate, and no estimator evades it: the impossibility is a property of the foreclosed regime, not of any one method.

Proof — four steps

(1) Information localization: any consistent estimator of $I (τ)$ depends on the trajectory through the windows where $E_{τ}$ occurs; under stationary $D (θ^{*})$ the effective sample size is $N_{T}^{E} (τ) \to T \cdot S (τ; θ^{*})$ . (2) Variance lower bound via Cramér–Rao applied to the sample-covariance estimator from $N$ pairs, giving (i). (3) Operational form via Lemma 1(b), giving (ii). (4) Minimax form via Le Cam's two-point method with $P_{0}, P_{1}$ differing only on $E_{τ}$ and total variation bounded by $T \cdot S (τ; θ^{*})$ , giving (iii).

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Cross-references

Prior result: Lemma 1 (foreclosure) — the foreclosure that this theorem derives reclamation impossibility from
Dynamics plate: reclamation trajectories (operates this theorem)
What follows: §5 (intervention modes) — Mode A is the only intervention the impossibility does not foreclose
Required: §0 axioms (R1, U2) and the structural-separation hypothesis