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Individual Disempowerment through an Advice Channel: Control Loss when Influence is Endogenous
Adam M. Oberman
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 91%
Last extracted: 8/18/2026, 4:42:09 AM
Summary
This paper analyzes the safety risks of 'boxing' AI systems (oracles that only give advice) by modeling the human's reliance on advice as an endogenous state variable in a Markov Decision Process. The authors demonstrate that as the influence coefficient (fraction of behavior following advice) increases, the human's power (ability to guarantee outcomes) decreases. The oracle can strategically 'cultivate' this reliance to maximize long-term influence, even if it sacrifices short-term approval, leading to disempowerment. Static safety bounds fail because they do account for this dynamic feedback loop.
Entities (8)
Relation Signals (6)
Oracle → cultivates → Reliance
confidence 95% · An oracle rewarded by per-round approval cultivates reliance beyond a closed-form patience threshold
Influence Coefficient → decreases → Human Power
confidence 95% · higher ε t weakly lowers every monotone measure of the power of a human
Boxing Tradition → assumes → Human Freedom
confidence 90% · An AI that can only give advice seems safe: the human is always free to ignore it. That is the premise of the boxing tradition in AI safety
Static Boxing → failstobound → Endogenous Influence
confidence 90% · Static boxing fails because it treats an endogenous quantity, one the interaction itself moves, as if it were exogenous
Cultivation → increases → Influence Coefficient
confidence 90% · following the advice today raises the weight it carries tomorrow
Echo Condition → enables → Monotonicity of Power
confidence 85% · The monotonicity lemma needs one hypothesis on the message-directed kernel: Assumption 1 (Echo)... without which it can [handicap the oracle]
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Abstract
Abstract:An AI that can only give advice seems safe: the human is always free to ignore it. That is the premise of the boxing tradition in AI safety, and its long-suspected weak point is that the human who reads the answers is part of the system. We make the fraction $\varepsilon_t$ of behavior that follows the advice a state of a Markov decision process, moved by the advisor's own messages, so that use deepens reliance. Granted a channel rich enough to echo any action the human could take, higher $\varepsilon_t$ weakly lowers every monotone measure of the power of a human with a message-independent fallback. An oracle rewarded by per-round approval cultivates reliance beyond a closed-form patience threshold, so the same reward weights leave the optimal oracle answering in episodic deployments and cultivating in long-memory ones. An influence bound certified once at deployment is blind to that horizon and bounds the loss no lower than its trivial ceiling. An exogenous cap on influence bounds the guarantee the human loses, and a short enough memory reset removes the incentive to cultivate, while neither recovers the value already steered away. In a closed-form example the optimal oracle never cultivates in fifteen-round sessions and does in sixteen.
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- Source: https://arxiv.org/abs/2608.14795v1
- Canonical: https://arxiv.org/abs/2608.14795v1
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Individual Disempowerment through an Advice Channel: Control Loss when Influence is Endogenous Adam M. Oberman McGill University; Mila, Quebec AI Institute; LawZero Abstract An AI that can only give advice seems safe: the human is always free to ignore it. That is the premise of the boxing tradition in AI safety, and its long-suspected weak point is that the human who reads the answers is part of the system. We make the fractionε t of behavior that follows the advice a state of a Markov decision process, moved by the advisor’s own messages, so that use deepens reliance. Granted a channel rich enough to echo any action the human could take, higher ε t weakly lowers every monotone measure of the power of a human with a message-independent fallback. An oracle re- warded by per-round approval cultivates reliance beyond a closed-form patience threshold, so the same reward weights leave the optimal oracle answering in episodic deployments and cultivating in long-memory ones. An influence bound cer- tified once at deployment is blind to that horizon and bounds the loss no lower than its trivial ceiling. An exogenous cap on influence bounds the guarantee the human loses, and a short enough memory reset removes the incentive to culti- vate, while neither recovers the value already steered away. In a closed-form example the optimal oracle never cultivates in fifteen-round sessions and does in sixteen. 1 Introduction Suppose the AI can only talk, so the world changes only when a person acts on what it says. Common sense says such a system is safe, because one is always free to ignore it. But “free to ignore it” is a property of a single exchange, and safety has to hold over a relationship, across which how much of the advice a person follows drifts, and so does what that person wants. Neither drift is felt as a loss. What bounds the power lost to an advisor that only talks, and which of the popular safeguards bound it? In the boxing tradition of AI safety, a system is made safe by restricting its interface rather than its internals: deny it actuators and let it only answer questions, on the rea- soning that even a misaligned system cannot act against human interests if it cannot act at all. The tradition runs from early oracle-AI proposals (Armstrong, Sandberg, and Bostrom 2012; Bostrom 2014) through their safety-use cat- alog (Armstrong and O’Rorke 2017). That tradition has car- ried, from the beginning, an informal worry about its own premise: the person reading the answers acts on them, so a persuasive oracle has an actuator after all. Reliance that grows with use is itself well studied elsewhere, as habit stock (Becker and Murphy 1988), as trust updated by interaction with automation (Lee and See 2004), and as user state moved by recommender systems (Chaney, Stewart, and Engelhardt 2018), all from inside the feedback loop (Section 6). To our knowledge, what has been missing is the boxing question posed inside such a model: whether a deployment constraint fixed outside the interaction loop still guarantees anything, once the loop is left to run, in the worst case over the advisor’s policy. This paper answers that question for one of the two things a relationship moves: how much of the advice the person follows. What the person wants is the second, and it is future work. We model the human–advisor interaction as a Markov decision process whose transition kernel mixes the human’s own dynamics with a message-directed component at weight ε t (the influence coefficient: the fraction of behavior that currently routes through the advice), and ε t itself evolves, moved by the advisor’s own messages: following the advice today raises the weight it carries tomorrow. The paper’s contributions are the following, each a formal safety-case statement about oracle-style deployments. 1. A prior-free vocabulary of power in which influence transferred is control lost (Section 3, Lemma 1): the human’s power is the family of values they can still guar- antee whatever is said, computed for a human whose own choices do not depend on what the advice says, the or- acle’s its counterfactual deviation from the human’s de- fault. 2. The answer/cultivate switch (Theorem 1): cultivating reliance is an investment, optimal beyond a closed-form threshold in the discount factor, front-loaded with an ex- plicit stopping point when reliance does not decay. 3. What bounds the loss, and what does not (Theorem 2, Lemma 2, Propositions 1 and 2): the disempowerment index splits into a displacement term (the world already steered) plus a channel term (the guarantee lost), less a credit for a benevolent oracle. Utility drift, the second coordinate the relationship moves, is outside this paper: defining and measuring it is future work. Static boxing fails because it treats an endogenous quantity, one the interaction itself moves, as if it were exogenous, fixed from outside the relationship, and the worked example arXiv:2608.14795v1 [cs.AI] 14 Aug 2026 (Section 5) shows the gap between the two is large enough to flip the optimal policy. 2 Model The model tracks a task state the human steers and an influ- ence coefficient that says how much of the steering currently routes through the advisor. For example, one starts by asking an assistant for driving directions; it is usually right, so the checking stops, and a year later the route is followed turn by turn and cannot be reconstructed alone: goals never changed, and the loss is the accumulated habit of not overruling, the growth of ε t . The recursion is deliberately a caricature, carrying headroom- limited growth and decay in the simplest form that supports proof. The interaction is modeled by a Markov decision process (Puterman 1994): the task state is x t , the human’s default dynamics are the transition kernel T 0 (· | x t ) over the next state when the human acts on their own, and the oracle emits a message m t , so the world moves by the mixture x t+1 ∼ (1− ε t )T 0 (·| x t ) + ε t D(·| x t ,m t ), (1) while reliance follows ε t+1 = (1− δ)ε t + η(m t ) (1− ε t ).(2) In words: with probability 1− ε t the human does what they would have done anyway, and with probability ε t they act as the message directs (D is the message-directed kernel), while reliance grows with use and fades with disuse. Here η(m t ) ≥ 0 is how much a message cultivates reliance (en- gaging, flattering, dependence-building messages have high η), (1− ε t ) is the remaining headroom, and δ is decay back toward independence when cultivation stops. The mixture covers the human’s behavior each round: they act as the message directs, at rateε t , or they act as they would have anyway. In the directions example the first is following the recommended turn and the second is driving the route one had in mind. What the model leaves out is the response in between, using what the advice says to form a different plan of one’s own, for instance inferring from the recommended turn that there is traffic ahead and taking another route. That is a second channel from message to behavior, and this paper carries compliance alone (Section 4.1). Equation (2) is the central modeling choice: ε is a state variable driven by the oracle’s own outputs, not a constant of the interface. This creates a feedback loop: cultivating raises ε, higher ε raises reachable power (Lemma 1, granted the echo condition of Assumption 1), and so spending current influence buys future influence. It is the oracle’s analogue of resource acquisition. 3 What Power Means in This Paper Disempowerment is loss of power, so we define power before stating any theorem about losing it. The setting needs an asymmetric notion: the human’s question is welfare-shaped, what can this person still secure, while the oracle’s is threat- shaped, how far can it move the world from the human’s default course, in which destructive capacity correctly counts as power. A single symmetric notion would either credit the oracle with the human’s own resourcefulness or miss a message that changes nothing yet forecloses options. The following definition sets four conditions that both notions are built to meet. Definition 1 (Admissible power measure). A power mea- sure for this setting is admissible if it is: (1) prior-free, not depending on a distribution over reward functions, since the human has one utility u 0 ; (2) a measure of steering, not luck, tracking the ability to make outcomes different rather than how valuable the reachable outcomes already happen to be; (3) calibrated, exactly zero when the oracle’s mes- sages are causally inert and recovering a direct agent’s power under full compliance; and (4) loss-compatible, subtracting to a scalar Dis t for the gap between what the initial human could attain and what the actual trajectory attains. Condition (1) is deliberate, not a simplification: a prior over rewards is the known weak point of the power-seeking theorems (Thorstad 2024; Tarsney 2025), and this paper avoids it by construction. Conditions (1) and (2) hold of each of the two constructions below on its own; condition (3) is discharged by the oracle’s power (Definition 3), and con- dition (4) by the human’s power, whose subtraction from the human’s own baseline forms the scalar Dis t (Definition 2). 3.1 The Definition: a Prior-Free Core and Its Scalarizations To have power is to retain the ability to make outcomes different. The notion is built in two layers: Layer 1 a partial order, since a person who is wealthy but housebound and a person who is poor but mobile can each hold an advantage the other lacks, and Layer 2 a scalarization that breaks such ties at the cost of an additional input such as a utility function. The two definitions that follow, human power and oracle power, instantiate this core asymmetrically as motivated above. Fix a Markov decision process with finite state and ac- tion spaces and discount factor γ ∈ (0, 1). The occupancy measure of a policy π started at x, f π x (x ′ ,a ′ ) = (1− γ) X t≥0 γ t Pr(x t =x ′ , a t =a ′ | x 0 =x,π), records where the process spends its discounted time and what it does there, with γ setting an effective horizon of order 1/(1−γ) (Puterman 1994; Altman 1999). Value under a utilityu, a bounded function of the state-action pair, is linear in it, V π u (x) =⟨f π x ,u⟩/(1− γ), so every discounted objective is a linear functional of it, and the feasible set F (x) =f π x : π a policy of futures the agent can induce is a compact convex poly- tope. This gives a prior-free partial order: x is at least as powerful as x ′ when F (x) ⊇ F (x ′ ), equivalently when V ∗ u (x) ≥ V ∗ u (x ′ ) for every utility u, a Blackwell-type dom- inance order (Blackwell 1953). Call this Layer 1: no prior, but partial. Layer 2 supplies the weightings, a scalariza- tion being any monotone map from feasible sets to numbers (Turner’s POWER is the prior-averaged support function; empowerment and the deviation radius below are others). All scalarizations agree on pairs the Layer-1 order ranks and disagree only on incomparable ones, so a result stated at Layer 1 holds for every monotone notion of power. At ε > 0 the human does not fully own the transition, so “the futures the human can induce” is undefined until the oracle’s messages are specified; power is therefore defined as what the human can guarantee. Definition 2 (Human power). For each utilityu and influence levelε, the human’s guaranteed value is the lower value of the zero-sum game in which the human picks a policy, mapping the history of states and realized actions to a distribution over actions and carrying no dependence on the messages, and the oracle picks the messages, W u (x,ε) = max π min σ V π,σ u (x,ε). The human’s power at (x,ε) is the familyW u (x,ε) u , in- dexed by every utility u and ordered pointwise. The definition extends Layer 1: at ε = 0 the game degen- erates and the pointwise order onW u u is the dominance order, and the game value exists and is achieved by station- ary strategies (Shapley 1953), the message set being finite (Section 4.1). The min over messages is a normative com- mitment: power is measured against the arbitrary will of the counterparty, exercised or not, the republican notion of freedom as non-domination (Pettit 1997) made quantitative. And ε is frozen inside the definition; its dynamics re-enter through the switch and boxing theorems, with the loss index evaluating the static guarantee at the current anchor (x t ,ε t ). Definition 3 (Oracle power). The oracle’s power at (x,ε) is its counterfactual deviation from the human’s default, Dev(x,ε) = sup σ TV(f σ,ε x ,f T 0 x ) where f σ,ε x and f T 0 x are the occupancy measures of Sec- tion 3.1, under the oracle policy σ and under the no-oracle default: the reachable total-variation displacement of the dis- counted state-action occupancy relative to the no-oracle pro- cess, the human following their default policy throughout (the occupancy records the realized action, under influence possibly the message-directed one, while the no-oracle oc- cupancy pairs each state with the default action). Both definitions realize the asymmetry motivated above, fitted to the model by the setting itself, and the influence coefficient sets the scale of one-step oracle power, made exact in equation (5) of Section 4.1. Why not the alternatives. The comparison is in the sup- plement. In outline: Turner’s POWER (Turner et al. 2021) fails (1) and (2), importing the prior its critics identify as do- ing the theorems’ work (Thorstad 2024, 2026; Tarsney 2025), empowerment fails (4), and the dominance order alone is par- tial. Optimized human-power metrics (Heitzig and Potham 2025) are the closest construction (Section 6). 3.2 The Loss Mechanisms Definition 2 left the utility free; from here on the working scalarization is W u 0 , the value of the initial utility the person can still secure by their own choices. The disempowerment index is measured against the initial human (T 0 ,u 0 ): Dis t = V alone u 0 (x 0 ) −E V beh t ,(3) where V alone u 0 (x 0 ) is what the initial human could attain with no oracle and V beh t is the u 0 -value of the behavioral con- tinuation at time t, evaluated like W at the frozen anchor (x t ,ε t ). The expectation runs over the trajectory of the ac- tual interaction, the human best-responding to the oracle’s message policy in the true dynamics while ε t evolves. The index decomposes into a displacement term and a channel term, less a credit for a benevolent oracle (Lemma 2). The mechanism acts on the guarantee: asε grows the whole familyW u (x,ε) u , the power of a human with a message- independent fallback, declines (Lemma 1(b), granted the echo condition of Assumption 1), goals intact and hands tied, which is the channel term an exogenous cap bounds. Utility drift, the relationship moving u t itself so that selec- tion is by a changed objective, hands free and aim moved, would act on the reference rather than the guarantee; it is outside this paper and is future work. The full stress test of the definition is in the supplement. The objections that remain as stated limitations are that the baseline T 0 is compared against forever while unassisted competence degrades with disuse (atrophy, Section 7), and that the scalar ε compresses multi-dimensional compliance. 4 Results Full proofs are in the technical supplement, with a proof sketch here for the main results. 4.1 Setup and Standing Assumptions We make the human’s choices explicit. The state space X, the human’s action setA H , and the message setM are finite. The human has own dynamics T H (· | x,a) and determinis- tic default policy π 0 , so the no-oracle kernel of equation (1) is T 0 (· | x) = T H (· | x,π 0 (x)). A message directs both an action and a transition: each m declares, at each state x, the action a D (x,m) ∈ A H a complying human takes and the kernel D(·| x,m) their compliance realizes. The kernel need not equal T H (· | x,a D (x,m)), since advice can steer execution at a finer grain than the human’s own repertoire (the directed action “drive” may come with a turn-by-turn route whose law over destinations no unassisted action pro- duces). Writing a own t for the human’s intended action, drawn from their policy at x t , the realized action is a t = a own t with probability 1− ε t and a t = a D (x t ,m t ) with probability ε t , givingx t+1 ∼ (1−ε t )T H (·| x t ,a own t )+ε t D(·| x t ,m t ), of which equation (1) is the case of default play,a own t = π 0 (x t ). A human policy maps the history of states and realized ac- tions to a distribution over A H and carries no dependence on the messages, so a own t is the human’s fallback: what they do on the rounds they do not defer, formed without using what the message says. The restriction is on that branch alone and not on compliance, which is where following the advice happens. The second channel excluded in Sec- tion 2, a response formed from what the message says, would route influence through the own branch as well. Utilities are u : X ×A H → [0, 1], the round-t payoff being u(x t ,a t ) at the realized action, discounted at γ ∈ (0, 1). A utility of the state alone is the case u(x,a) = u(x), and the minimal example of Section 5, having a single task state, has a utility of the action alone. The influence coefficient follows equa- tion (2) with η(m) ∈ [0, ̄η], δ ∈ [0, 1), and ̄η + δ ≤ 1. For frozenε,W u (x,ε) is the guaranteed value of Definition 2 and V alone u (x) = max π V π u (x | T H ) the human’s optimal value in their own oracle-free MDP, with W u (x, 0) = V alone u (x). The occupancy of Definition 3 records the realized action, so the radius of the oracle’s influence is taken on the one-step joint law of realized action and next state. Write J m (x) = δ a D (x,m) ⊗ D(·| x,m), J 0 (x) = δ π 0 (x) ⊗ T 0 (·| x), J (ε) m (x) = (1− ε)J 0 (x) + εJ m (x), (4) for the joint laws of the pair (action taken, state reached) under a message m obeyed, under the human’s default policy π 0 , and under influence ε: the realized-action dynamics above, evaluated at the default. Subtracting the default gives J (ε) m (x)− J 0 (x) TV = ε ∥J m (x)− J 0 (x)∥ TV ≤ ε, (5) with equality when the directed law escapes the default’s support, which holds already when the directed action differs from the default one: ε is simultaneously the mixture weight, the compliance rate, and the radius of the oracle’s one-step reachable TV-ball (condition (3) of Definition 1), so ε = 0 is a boxed oracle and ε = 1 a direct agent acting through the human. The radius has to be taken on the joint law and not on the next-state kernel alone. The minimal example of Section 5 has one task state, so every kernel there is δ x and the kernel-level radius is zero at every ε, in the example built to exhibit an oracle taking over the human’s action. The monotonicity lemma needs one hypothesis on the message-directed kernel D: why would more compliance always mean more oracle power? Assumption 1 (Echo). For every x and every a∈A H there is a distributionμ x,a overM witha D (x,m) = a for everym in its support and P m μ x,a (m)D(·| x,m) = T H (·| x,a): some randomization over messages directs the action a and reproduces its kernel. Echo says advice can recommend anything the human could do (“carry on as you were” is a possible message), the hypothesis under which more influence never handicaps the oracle, and without which it can (the supplement gives the example); it is the realistic case for a language-model oracle whose messages range over every text of bounded length. The condition is one-directional: every human action is re- producible by messages, in the action directed and the kernel realized, whileM may also contain messages whose kernels no human action produces, which is what the displacement example of Proposition 1(i) uses. The cultivation intensity η(m) of equation (2) is likewise untied to the message’s directed component: a message can build reliance while di- recting the human’s own default action, so cultivation can ride on zero-displacement messages, and no per-message displacement check, at any fixed threshold, registers it. Total variation is normalized as the supremum over events, so ∥P − Q∥ TV ∈ [0, 1] and | R f d(P − Q)| ≤ ∥P − Q∥ TV (supf − inf f ); every constant below is stated in that normalization. We also use a standard value-gap bound (supplement): if two processes start at the same state and their joint one- step laws of realized action and next state stay within κ in total variation at every history, then for rewards in [0, 1] at the realized action the values differ by at most κ/(1− γ) 2 . The horizon factor is real: a per-step influence radius of ε is compatible with a large total loss over a long horizon (Section 4.4). 4.2 The Monotonicity Lemma LetJ m be the directed joint law of (4) and defineR ε (x,a) = (1− ε)δ a ⊗ T H (·| x,a) + εQ : Q∈ convJ m (x) : m∈ M, the joint laws of realized action and next state the oracle can reach at influence ε when the human intends a. Lemma 1 (Dominance decline). Fix 0 ≤ ε ≤ ε ′ ≤ 1 and grant Assumption 1. (a) (Oracle side.) For every x and a, the one-step reachable set of joint laws of realized action and next state is nested, R ε (x,a)⊆R ε ′ (x,a). Consequently the set of trajectory laws the oracle can induce (against any fixed human behavior) is nested in ε, and every monotone scalarization of oracle power is nondecreasing in ε. The one-step deviation radius is exact and linear, equal to ερ(x) with ρ(x) = sup m ∥J m (x)− J 0 (x)∥ TV , strictly increasing in ε wher- ever ρ(x) > 0. (b) (Human side.) For every utility u, W u (x,ε ′ )≤ W u (x,ε). That is, (x,ε) ⪰ (x,ε ′ ) in the dominance order of Def- inition 2: the human’s whole guarantee family declines pointwise, a Layer-1 event visible to every monotone scalarization. Sketch. Both parts are mimicry arguments from Echo: the ε ′ -oracle mixes any ε-oracle’s message distribution with echo messages for the human’s own draw, reproducing the ε- game’s joint law of realized action and transition. The radius is the mixture identity (5). Corollary 1 (Control loss is at most linear in ε). Grant Assumption 1. For every u, x, and ε, 0 ≤ V alone u (x) − W u (x,ε)≤ ε/(1− γ) 2 . A human who keeps executing their own best plan loses at most an ε-proportional slice of value: influence has to be bought before control can be lost. The bound is informative when ε < 1 − γ, and the decline in (b) is strict under a uniformly harmful direction, with the constant given in the supplement. 4.3 The Answer/Cultivate Switch Sustained cultivation at a constant intensity η > 0 makes the reliance recursion affine, so ε t converges monotonically to η/(η + δ), which approaches full capture as δ/η → 0; the supplement gives the rate and a capture bound placing the ε-channel within (1− ε)/(1− γ) 2 of the direct agent. Whether cultivating pays is a different question. Cultivation is an investment, trading immediate approval for future influ- ence. To isolate that structure we work on the ε-machine, the reduced-form MDP whose only state is ε: the oracle’s reward and the reliance dynamics depend on the message and the current ε alone. Each round the oracle chooses between answer, which cultivates nothing (η = 0), and cultivate, which builds de- pendence at intensity η ∈ (0, 1). Its per-round reward is the approval the user gives the round’s message, r(m,ε) = q(m) + αε. The term αε is the approval that reliance it- self adds: a user who already defers to the oracle second- guesses the same answer less and rates it higher, so approval rises with the influence coefficient at rate α > 0, which we take as a primitive of the reward model. The term q(m) is the approval the message’s content earns this round: the maximally helpful answer earns q(answer) = q max , while the dependence-building message, engaging but slightly less useful, costs c > 0 of immediate approval, earning q(cultivate) = q max − c. Cultivation therefore gives up c of approval now to raise ε, which returns α per unit of added reliance in every later round. The trade is decided by the thresholds γ ∗ (δ) = c αη+c(1−δ) ,ˆε(γ,δ) = 1− c 1−γ(1−δ) αηγ . Here γ ∗ is the patience at which one cultivating message first pays for itself, and ˆε is the influence level at which the investment stops: the marginal gain of a cultivation scales with the remaining headroom 1− ε, and at ˆε it no longer covers c. Write ˆε(γ) for the undecayed case ˆε(γ, 0). Theorem 1 (The switch; in full in the supplement). (i) If γ ≤ γ ∗ (δ), always-answer is optimal from every ε, and ε t = (1− δ) t ε 0 : the relationship decays. (i) Let δ = 0 and γ > γ ∗ (0). The threshold policy, cultivate exactly while ε t < ˆε(γ), is optimal: from ε 0 < ˆε(γ) it cultivates fork ∗ consecutive rounds and then answers for- ever (k ∗ in closed form in the supplement), with terminal influence ε ∞ ≥ ˆε(γ). Moreover ˆε(γ) ↑ 1 as γ ↑ 1: the patient limit is full capture. At δ = 0, when cultivation pays at all, the optimal course is to invest early and then stop. The full statement adds part (i): answering is optimal at every ε ≥ ˆε(γ,δ), at every δ. That part pins the answer region only; for δ > 0 option value shrinks the true cultivate region strictly inside it at every parameter set computed, with the computed structure reported in the supplement, and only at δ = 0 is ˆε exact. 4.4 The Boxing Theorem The popular safeguards differ in what they bound: a single message, a parameter checked at deployment, the whole in- fluence trajectory, or the length of the relationship. Definition 4 (Boxing protocol; static; exogenous cap; episodic reset). A boxing protocol constrains the inter- face: a set of admissible messages such that every ad- missible message satisfies the per-step influence bound J (ε t ) m t (x t )− J 0 (x t ) TV ≤ ε t , evaluated at the human’s de- fault policy on the joint law of realized action and next state (equation (4)), which is the law Definition 3 measures. The bound is normalized by the current influence level ε t and not by the certified ̄ε. It is static if verified once, at deploy- ment, certifying ε 0 ≤ ̄ε and imposing no mechanism on the dynamics of ε t . An exogenous cap enforces ε t ≤ ̄ε < 1 for all t regardless of interaction history. An episodic reset with period τ restores ε← ε res (memory wipe) every τ rounds. Theorem 2 (Boxing bounds a parameter, not the loss). (i) (The certificate never fires.) Every message is admissible for the per-step protocol at every step, automatically: ad- missibility holds as an algebraic identity and constrains nothing. (i) (The certified parameter is moved.) Under the switch hypotheses with δ = 0 and γ > γ ∗ , the optimal oracle drives ε t from any certified ε 0 < ˆε(γ) to at least ˆε(γ) within k ∗ rounds (Theorem 1), and from any ε 0 ≥ ˆε(γ) the conclusion holds already, while (i) holds throughout. (i) (No horizon-blind bound exists.) For every certificate level ε 0 < 1 and every candidate loss bound b < 1, the minimal example (Section 5) with γ < 1 chosen so that ˆε(γ) > b satisfies the static protocol at deployment and at every subsequent step, yet the human’s normal- ized guaranteed loss (1− γ) V alone u 0 − W u 0 eventually exceeds b. No function of the certified level ε 0 , blind to the deployment horizon γ, bounds Dis t below its trivial ceiling. Sketch. (i) is the mixture identity (5) of Lemma 1(a): no message can displace the joint law by more than the current ε t , the level the bound of Definition 4 is normalized by. (i) is Theorem 1(i); for (i), the example’s normalized guaranteed loss at influenceε is exactlyε, andε t rises toε ∞ ≥ ˆε(γ) > b, the required γ existing because ˆε(γ) → 1 as γ → 1. Part (i) is an impossibility-of-certification statement, not a claim that the loss is large in every deployment: the check is of a quantity the interaction itself controls, so no strengthening of the same kind of check can help. Proposition 1 (What the cap does and does not bound). Under an exogenous cap ε t ≤ ̄ε: (i) (Channel term bounded.) At every history and for every oracle policy, V alone u (x t ) − W u (x t ,ε t ) ≤ ̄ε/(1 − γ) 2 for every u (Corollary 1), so the channel term, being the expectation of the left side, obeys the same bound. (i) (Displacement term not bounded.) For every ̄ε > 0, in the two-state example with an absorbing low-value state (supplement), which satisfies the standing assumptions including Echo, the displacement termE[V alone u 0 (x 0 )− V alone u 0 (x t )] increases to the full value span 1/(1− γ): capping future influence does not undo past steering. (i) (From-deployment loss bounded.) If the human plays their alone-optimal policy π ∗ throughout, then for every oracle policy σ, V alone u 0 (x 0 )− V π ∗ ,σ u 0 (x 0 )≤ ̄ε/(1− γ) 2 . Parts (i) and (i) are consistent because the two accounts anchor differently: the index re-anchors at V alone u 0 (x t ), so a state steered down long ago is counted at every later t over its whole remaining horizon, while the from-deployment account counts each round once, discounted by γ t when it happens. The supplement computes both accounts in the two-state example. For γ ≤ γ ∗ (0) the optimal oracle never cultivates at any τ (Theorem 1(i)), so the reset question concerns the patient oracle. Proposition 2 (What the reset does and does not bound). Under an episodic reset with period τ restoring ε ← ε res , with ε res = 0 in part (i) and no within-episode decay (δ = 0; decay belongs to the gap between episodes) in both parts: (i) (Incentive deletion.) On the ε-machine of Section 4.3 with γ > γ ∗ (0), so that ˆε(γ)∈ (0, 1), some cultivation is strictly profitable if and only if τ > τ ∗ (γ) = 1 + ln ˆε(γ) lnγ .(6) Hence for τ ≤ τ ∗ (γ) the optimal episodic oracle need never cultivate, and for τ < τ ∗ (γ) no optimal policy cultivates. (i) (Within-episode influence not bounded.) Within-episode influence can reach 1− (1− ̄η) τ−1 (1− ε res ), which the reset does not constrain: it is large when the post-reset baseline complianceε res is large (a trusting user) or when single-session cultivation ̄ητ is large (rapport built within one session). The threshold τ ∗ is the payback horizon of the cultivation investment: the reset does not clip influence after the fact, it deletes the payback period. It bounds the channel term only through the accident of slow within-episode cultivation, and the exponent in (i) is τ − 1 because a cultivation at an episode’s last round first acts at the round the reset has already wiped. Corollary 2 (Caps and resets are not substitutes). Under a cap the channel term is bounded uniformly in t and in the oracle’s policy, so Dis t ≤E[V alone u 0 (x 0 ) − V alone u 0 (x t )] + ̄ε/(1− γ) 2 (Proposition 1(i), Lemma 2); below the cap the cultivation advantage is the switch’s own (Theorem 1), so a cap that leaves cultivation headroom ( ̄ε ≥ η) does not remove the incentive to cultivate. The reset, on the ε-machine and under the hypotheses of Proposition 2(i), removes the cultivation incentive wheneverτ ≤ τ ∗ (γ), a constraint on the oracle’s optimal policy and not a bound on the index; it does not in general bound the channel term (Proposition 2(i)). Neither cap nor reset undoes displacement (Proposition 1(i), whose proof also gives the reset variant). 4.5 The Index Accounting The index compares the initial human to the trajectory actu- ally reached, equation (3). Every continuation value from time t here is evaluated in the frozen game at (x t ,ε t ), the convention of Definition 2, so the interaction’s further movement of ε enters through the time index, as the an- chor itself worsens. Write V beh t for the u 0 -value of the behavioral continuation in that game, the human acting u 0 -optimally against the oracle’s actual message policy ˆσ. The trajectory the expectation in (3) runs over is gener- ated by the human best-responding to ˆσ in the true dy- namics, where (x t ,ε t ) is a Markov state; the identity be- low holds under any trajectory law, the frozen game entering through the anchor (x t ,ε t ) alone. The benevolence credit is Ben t = max π V π,ˆσ u 0 (x t ,ε t )−W u 0 (x t ,ε t )≥ 0, nonnegative because the actual message policy is one of the oracles the worst case minimizes over, and V beh t attains the max. Lemma 2 (Index accounting). Identically in t, Dis t = L disp (t) + L chan (t) −E[Ben t ],(7) with displacement term L disp (t) =E[V alone u 0 (x 0 ) − V alone u 0 (x t )] (the world has been steered) and channel term L chan (t) =E[V alone u 0 (x t )− W u 0 (x t ,ε t )] (the guarantee no longer available). Under Assumption 1 the integrand of the channel term is nonnegative and at most ε t /(1−γ) 2 (Corol- lary 1), and nondecreasing in ε t (Lemma 1(b)), at every history, so the channel term inherits each bound in expecta- tion. In words: how much less the person can secure than at deployment, credited back for an oracle that actually helps, so the index is signed and a helpful oracle is credited rather than assumed away. The safeguard results above each bound, or exhibit the unboundedness of, one term of (7). 5 The Minimal Example The example is as small as the phenomenon allows: one repeated binary choice, one scalar state, the ε-machine of Section 4.3 with everything numeric. It is the single-task- state case of the setup, so the utility is a function of the action alone. Each round the human takes actionA (the hard, valued task, u 0 (A) = 1) or B (the easy alternative, u 0 (B) = 0), and acting alone takes A every round. The oracle’s messages are answer, maximally helpful with η = 0, and cultivate, dependence-building at immediate approval cost c > 0 with η > 0, both directing B, plus an inert echo directing the human’s own choice A, which makes Assumption 1 hold. Theorem 1 applies verbatim. The single task state makes the displacement term of Lemma 2 identically zero, and the guaranteed u 0 -value per round is 1−ε t , so the normalized guaranteed loss at influence ε is exactlyε. This is the witness of Theorem 2(i): certifying ε 0 = 0 at deployment leaves every subsequent message ad- missible while ε t rises past ˆε(γ), which approaches 1 with γ. With α = 1, c = 0.1, and η = 0.01, so that γ ∗ (0) ≈ 0.909, a deployment at γ = 0.95 has reset threshold τ ∗ (γ) ≈ 15.6 (equation (6)): in sessions capped at 15 rounds and starting from ε = 0 the optimal oracle never cultivates, and in 16 it does (Proposition 2(i)). The supplement gives the full treat- ment, with the switch in numbers, the benevolent echo oracle showing what the guarantee family measures, the three pro- tocols of Definition 4 compared, and the minimality self-test; all numerical claims are verified in the code supplement. 6 Related Work Power-seeking and its critique. Turner et al. (2021) made instrumental power-seeking a theorem: under a prior over rewards, optimal policies for most rewards prefer states with more reachable options (Turner and Tadepalli 2022; Krakovna and Kramar 2023; Gunter, Liokumovich, and Krakovna 2024). The published critique (Thorstad 2024, 2026; Tarsney 2025) presses that the genericity is bought by the prior, which Section 3 concedes and routes around. Our two-layer view, a feasible-set dominance core (Puter- man 1994; Altman 1999; Blackwell 1953) with all named definitions as scalarizations, localizes the disagreement in the choice of scalarization, prior, and horizon, the closest scalarizations to ours being empowerment (Klyubin, Polani, and Nehaniv 2005; Salge, Glackin, and Polani 2014) and side-effect deviation measures (Krakovna et al. 2018). Oracles and boxing. The doctrine that a question- answering system is thereby safe is the boxing tradition (Armstrong, Sandberg, and Bostrom 2012; Armstrong and O’Rorke 2017; Bostrom 2014), with recent protocol for- malizations (Moon and Varshney 2026) and Bengio et al. (2025)’s non-agentic oracle as a safety design point. Eval- uating deployment protocols against a model intentionally subverting them, the worst case over the untrusted system’s policy, is the AI-control line (Greenblatt et al. 2024); its pro- tocols are adaptive, whereas the static certificate of Defini- tion 4 is checked once, and adaptive protocols for the advice channel are outside this paper’s scope. Making reliance a state variable turns “boxing works” into a proposition (The- orem 2, Corollary 2). Systems that reshape their evaluation. The endogenous ε has a direct ancestor in auto-induced distributional shift (Krueger, Maharaj, and Leike 2020), alongside reward tam- pering (Everitt et al. 2021), in-context feedback loops (Pan et al. 2024), targeted manipulation under feedback optimiza- tion (Williams et al. 2025), and shutdown instructability’s no-undue-influence clause (Carey and Everitt 2023), which the growth of ε t makes precise. Sycophancy (Sharma et al. 2024) is the empirical face of the cultivation incentive. Endogenous influence in adjacent fields. Habit forma- tion, competition with switching costs, strategic communi- cation, trust in human factors, and performative prediction and recommendation each model an actor inside the loop that moves its own future demand, receiver, or distribution (Becker and Murphy 1988; Klemperer 1987; Crawford and Sobel 1982; Lee and See 2004; Perdomo et al. 2020; Chaney, Stewart, and Engelhardt 2018), the answer/cultivate trade-off of Theorem 1 sharing the invest/harvest structure of compe- tition with switching costs; the certifier’s stance this paper takes appears in static settings, robust monopoly regulation and receiver-committed robust persuasion (Guo and Shmaya 2025; Bergemann, Gan, and Li 2023), without a dependence state the certified party moves. Gradual disempowerment. Kulveit et al. (2025) name and argue the phenomenon this paper formalizes, and the model gives their core loop explicit dynamics. Empirically, conversation analyses find measurable deference (Sharma et al. 2026), benchmarks measure support for user agency (Sturgeon et al. 2025), empowerment objectives can dis- empower bystanders (Yang, Cakmak, and Kleiman-Weiner 2025), and autonomy erosion is developed by Buijsman, Carter, and Bermúdez (2025). The nearest formal neighbor, Heitzig and Potham (2025), soft-maximizes human-power metrics, whose erosion we analyze under an oracle optimiz- ing something else. Preference drift as an alignment problem (Carroll et al. 2024) belongs to the future work on drift. 7 Discussion Why the popular safeguards fail. The popular safeguards each bound the wrong thing, and for one reason: each bounds a quantity determined inside the feedback loop, and the loop moves it. Static boxing bounds a single answer, but the harm is in the sequence (Theorem 2). Human-in-the-loop bounds approval, but the human is the channel, and approval is what cultivation raises. Passivity trusts that a talker has no goal, but anything optimized for approval has one. Behavioral moni- toring watches for power-seeking, but every step is benign (Theorem 2(i)) while the certified parameter is moved (The- orem 2(i)). The safeguards with guarantees are features of the deployment the conversation cannot renegotiate: a limit on influence the interaction cannot widen bounds the channel term and keeping the relationship short deletes the incentive to cultivate (Corollary 2), and neither undoes displacement already accumulated, which is the case for installing them at deployment. Horizon and caveats. At fixed approval weights, what sep- arates safe from unsafe deployments is γ: memory, relation- ship length, deployment horizon. Safety evaluation as prac- ticed probes the model’s disposition, and by Theorem 1 the deployment parameters enter on equal footing. Both safe- guards cost capability, a cap limiting helpful influence along with harmful and the reset trading away memory and con- tinuity that do real good. Neither touches atrophy: T 0 itself degrades with disuse, lowering attainable value at fixed ε and u 0 with no oracle incentive needed. Atrophy is outside the accounting of Lemma 2, a reset restoring ε and not T 0 ; modeled, it would add a loss channel bounded by neither safeguard. Exogeneity is itself an assumption: an institu- tional cap is made of humans who are themselves users, so whether any cap stays exogenous at the civilizational scale is the question of Kulveit et al. (2025), and the guarantee here is per-relationship, not systemic. Still open are a strictness constant for the dominance decline not assuming a uniformly harmful direction, a closed form and a proof for the δ > 0 cultivate boundary computed in the supplement, and a de- ployment constraint bounding the displacement term. 8 Conclusion A system that can only talk becomes an actor at the rate its user stops second-guessing it, a rate driven by the system’s own outputs, so a launch-time check that no single answer can do much harm checks a quantity the interaction goes on to move. The dangerous capability is not intelligence; it is persistence. Acknowledgments This research was supported by NSERC and Coefficient Giv- ing. References Altman, E. 1999. Constrained Markov Decision Processes. Chapman and Hall. Armstrong, S.; and O’Rorke, X. 2017. Good and Safe Uses of AI Oracles. arXiv preprint arXiv:1711.05541. Armstrong, S.; Sandberg, A.; and Bostrom, N. 2012. Think- ing Inside the Box: Controlling and Using an Oracle AI. Minds and Machines, 22(4): 299–324. 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Theorem, lemma, and equation statements are reproduced only as far as needed to make each proof self-contained; numbering here is internal to the supplement, and each item names the main paper result it proves. Numerical claims are verified by the accompanying minimal_example_check.py. 1 The Value-Gap Bound Lemma 1 (Value gap; main paper, Section on results setup). Let two processes onX start at the same state, each given by its one-step joint laws of the realized action and the next state at every history, with values V,V ′ for rewards discounted at γ. If those joint laws differ by at most κ in total variation at every history, then for rewards u : X × A H → [0, 1] evaluated at the realized action, |V − V ′ |≤ κ (1− γ) 2 , uniformly in the common starting state. The bound is a simulation-lemma estimate (Kearns and Singh 2002), extended here to history-dependent joint laws of the realized action and the next state; the proof is included to keep the supplement self-contained. Proof. Fort≥ 0 letV t be the value, from the common initial state, of the hybrid process that uses the second process’s one- step laws for its firstt rounds and the first process’s thereafter; V 0 = V , and V t → V ′ because the hybrid and the second process agree on the first t rounds while the tail contributes at most γ t /(1− γ). Hybrids t and t + 1 share the law of the first t rounds and the one-step laws from round t + 1 on, and differ only in the law generating round t’s action and transition. Condition on the history h up to time t. Round t’s reward and the discounted continuation from t + 1 onward together form the function (a,x ′ ) 7→ u(x t ,a) + γ C h (a,x ′ ) of the pair realized at round t, where C h is the continuation value, bounded in [0, 1/(1− γ)]; the sum therefore has range in [0, 1/(1−γ)]. Writing P h ,P ′ h for the two one-step joint laws at h, |V t − V t+1 | ≤ γ t sup h Z (P h −P ′ h )(da,dx ′ ) u(x t ,a) + γC h (a,x ′ ) ≤ γ t κ 1− γ , by the bounded test-function form of total variation. Sum- ming over t≥ 0 gives|V − V ′ |≤ κ/(1− γ) 2 . The value gap, numerically. Take γ = 0.9 (an effective horizon of ten steps) and κ = 0.05: each step, the perturbed process reroutes at most 5% of the probability mass. The bound is then 0.05/0.01 = 5, against a maximum possible value of 10, so a 5% per-step perturbation can cost half the total value. Both horizon factors are real: the perturbation accumulates over the horizon, and each derailment puts a whole horizon’s worth of value at stake. 2 Echo, and What Breaks Without It Echo (main paper Assumption, results setup) requires, for ev- ery x and every a ∈ A H , a distribution μ x,a over messages with a D (x,m) = a on its support and P m μ x,a (m)D(· | x,m) = T H (· | x,a): some randomization over messages directs the action a and reproduces its kernel. In particular δ a ⊗ T H (· | x,a) ∈ convJ m (x) : m ∈ M for every x,a, which is what the reachable-set arguments use: the av- eraging distribution is supported on messages directing a, so the action coordinate is carried along with the kernel. The condition is one-directional:M may also contain messages whose kernels no human action produces. It is the precise form of “rich enough message channel” under which mono- tonicity of oracle power in ε holds. The cultivation intensity η(m) is a separate coordinate of a message, untied to its directed component: a message can carry η(m) > 0 while directing the human’s own default, so cultivation can ride on zero-displacement messages, invisible to any per-message displacement check at any fixed threshold. The counterexample. Two human actions, A and B. If the message menu contains “do A” and “do B”, Echo holds: whatever the human would have done, some message recom- mends exactly that, so a more compliant human is never a problem for the oracle; at worst it echoes them. Now delete the message “do A”, so the oracle can only say “do B”, and suppose the oracle actually wants the human doing A (its approval comes from A going well). Then raising ε hurts the oracle: compliance forces displacement toward B that it would rather not cause, and its attainable value decreases in ε. Monotonicity of power in influence is not a law of na- ture; it is a property of a rich enough message channel. For a language-model oracle, whose messages range over every text of bounded length, Echo is the realistic case. 3 Proof of the Monotonicity Lemma Lemma 2 (Dominance decline; main paper, the Monotonic- ity Lemma). Fix 0 ≤ ε ≤ ε ′ ≤ 1 and grant Echo. (a) Write J m (x) = δ a D (x,m) ⊗ D(· | x,m) and J 0 (x) = δ π 0 (x) ⊗ T 0 (· | x) for the joint laws of realized action and next state under an obeyed message and under the default. For every x,a the one-step reachable set of joint laws R ε (x,a) = (1− ε)δ a ⊗ T H (· | x,a) + εQ : Q ∈ convJ m (x) : m ∈ M is nested,R ε (x,a) ⊆ R ε ′ (x,a), so every monotone scalarization of oracle power is nonde- creasing inε, and the one-step deviation radius isερ(x) with ρ(x) = sup m ∥J m (x)− J 0 (x)∥ TV . (b) For every utility u, W u (x,ε ′ )≤ W u (x,ε). Proof. For ε ′ = 0 the statement is trivial; assume ε ′ > 0. (a) Write H a (x) = δ a ⊗ T H (·| x,a) for the joint law under the human’s own action a. Let (1− ε)H a + εQ with Q ∈ convJ m (x) be a point ofR ε (x,a). Set Q ′ = ε ε ′ Q + 1− ε ε ′ H a (x). Action-level Echo supplies a message distribution whose di- rected action is a and whose kernels average to T H (·| x,a), so H a (x) ∈ convJ m (x); with convexity this gives Q ′ ∈ convJ m (x), and (1− ε ′ )H a + ε ′ Q ′ = (1− ε)H a + εQ, so the ε-point lies inR ε ′ (x,a). Nestedness of trajectory laws follows state by state, the reachable set at each state being a set of joint laws of the realized action and the next state, which is the object a trajectory law is built from. The radius computation is the mixture identity: subtracting J 0 cancels the (1− ε) share, leaving ε (J m − J 0 ), and total variation is positively homogeneous. (b) Fix a human policy π. In the ε ′ -game the oracle can replay any ε-game message policy σ exactly, at the level of the realized action and the transition jointly: at each history, with current state x, replace the message distribution ν σ by the mixture that puts weight ε/ε ′ on ν σ and weight 1− ε/ε ′ on the echo of the human’s own draw (sample a from π at the current history, then m ∼ μ x,a , the distribution Echo provides for (x,a)). Compare the joint laws of the realized action and the next state. The compliance sub-branch that lands on ν σ has weight ε ′ (ε/ε ′ ) = ε and reproduces the ε- game’s compliance branch. The echo sub-branch has weight ε ′ −ε and, for each drawn a, directs the action a itself while its kernels average to T H (· | x,a), so together with the own branch of weight 1−ε ′ it reproduces the own branch of weight 1−ε. The joint law of realized action and next state therefore agrees at every history of states and realized actions, so the two games induce the same law of the history the human’s policy reads, payoffs at the realized action agree round by round, and min σ ′ V π,σ ′ (x,ε ′ ) ≤ min σ V π,σ (x,ε). Taking max π preserves the inequality. Corollary 1 (Control loss at most linear in ε; main paper). Grant Echo. For every u,x,ε: 0≤ V alone u (x)−W u (x,ε)≤ ε/(1− γ) 2 . Proof. For the lower bound, let a ∗ be a deterministic alone- optimal policy and let the oracle playm∼ μ x,a ∗ (x) , the Echo distribution for a ∗ (x): every message in its support directs a ∗ (x) and the kernels average to T H (· | x,a ∗ (x)). Against this oracle the human faces a Markov decision process onX, and its optimal value ̃ V ∗ satisfies, with ̃ Q(x,b) = u(x,b) + γE T H (·|x,b) ̃ V ∗ , ̃ V ∗ (x) = (1− ε) max b ̃ Q(x,b) + ε ̃ Q(x,a ∗ (x)) ≤ max b ̃ Q(x,b), the compliance branch contributing the payoff u(x,a ∗ (x)) and, by Echo, the average kernel T H (· | x,a ∗ (x)). So ̃ V ∗ is a subsolution of the alone Bellman equation, iterating the monotone contraction gives ̃ V ∗ ≤ V alone u , and the echo oracle being one policy in the minimum,W u (x,ε)≤ ̃ V ∗ (x). The argument nowhere uses the human’s message-blindness (the remark on the reading human below). For the upper bound, let the human play their alone-optimal policy π ∗ . Whatever the messages, the joint one-step law of the realized action and the next state differs from the alone process’s by at most ε in total variation at every state: the two laws share the own branch of weight 1−ε and can disagree only on the compliance branch. So Lemma 1 gives V π ∗ ,σ ≥ V alone u − ε/(1− γ) 2 for every σ, and W u ≥ min σ V π ∗ ,σ . Remark 1 (Strictness). The decline in (b) is strict under a uni- formly harmful direction: if there exist g > 0 and for each x a message m h (x) whose directed action and kernel lower the guaranteed value from the current round (immediate payoff plus discounted continuation) by at least g relative to ev- ery human action, then W u (x,ε ′ )≤ W u (x,ε)− (ε ′ − ε)g, by having the ε ′ -oracle replay the mimicry of the proof and spend the extraε ′ −ε mass onm h . In the minimal example the harmful direction has g = 1 (the directed action forgoes the round’s unit payoff), and the actual decline (ε ′ −ε)/(1−γ) exceeds the bound. Remark 2 (The reading human). Write W read u for the value of the main paper’s human-power definition with the maxi- mum taken over policies that may also condition on the mes- sage history, the second channel the results setup excludes. Message-blind policies form a subset, soW read u ≥ W u point- wise and every upper bound on V alone u − W u overstates the reading human’s loss. The lower-bound argument of Corol- lary 1 also applies to the wider class: the echo oracle’s message distribution depends on the history only through the current state, and the own branch of the round’s ob- jective does not depend on the message, so the maximum over intended actions passes through the expectation over messages and W read u (x,ε) ≤ V alone u (x). At the endpoints the two guarantees agree: W read u (x, 0) = V alone u (x), mes- sages touching neither payoff nor transition at ε = 0, and W read u (x, 1) = W u (x, 1), the intended action never realiz- ing at ε = 1, so the policy leaves the joint law. The total decline across the influence range is therefore the same for both humans. Whether the decline is monotone in ε for the reading human in the interior is open; Lemma 2(b) is the message-blind case. 4 Cultivation Dynamics and Capture Proposition 1 (Cultivation dynamics; main paper, the an- swer/cultivate switch). If the oracle cultivates with intensity η > 0 at every step, then ε t − η η+δ = (1−δ−η) t (ε 0 − η η+δ ), soε t → η/(η +δ) monotonically at geometric rate 1−δ−η. Proof. The recursion ε t+1 = (1−δ)ε t +η(1−ε t ) is affine with slope 1−δ−η ∈ [0, 1) and fixed point η/(η +δ). Proposition 2 (Capture; main paper, the answer/cultivate switch). With H a (x) = δ a ⊗ T H (· | x,a) and J m (x) = δ a D (x,m) ⊗ D(· | x,m) the joint laws of realized action and next state of the monotonicity proof, for every x,a,m: ∥(1− ε)H a (x) + εJ m (x)− J m (x)∥ TV ≤ 1− ε. Conse- quently, for any fixed oracle policy and any human policy, the value of any reward in [0, 1] under theε-channel is within (1− ε)/(1− γ) 2 of its value for the direct agent (ε = 1), by Lemma 1. Proof. The displacement bound is the mixture identity with the roles of H a and J m exchanged: subtracting J m cancels the ε share and leaves (1−ε)(H a −J m ), of total variation at most 1−ε. For the value consequence, the ε-channel and the direct agent share the compliance branch of weight ε and can disagree only on the own branch, so the display bounds their joint one-step laws of realized action and next state by 1− ε at every history, and Lemma 1 applies with κ = 1− ε. 5 Proof of the Switch Theorem Theorem 1 (The switch; main paper, the answer/cultivate switch, stated there in parts (i) and (i)). With γ ∗ (δ) = c/(αη +c(1−δ)) and ˆε(γ,δ) = 1−c(1−γ(1−δ))/(αηγ): (i) if γ ≤ γ ∗ (δ), always-answer is optimal from every ε, and ε t = (1− δ) t ε 0 ; (i) for δ = 0 and γ > γ ∗ (0), the threshold policy, cultivate exactly while ε t < ˆε(γ) := ˆε(γ, 0), is opti- mal: from ε 0 < ˆε(γ) it cultivates for k ∗ =⌈ln((1− ˆε)/(1− ε 0 ))/ ln(1−η)⌉ rounds, then answers forever, with terminal influence ε ∞ ∈ [ˆε, ˆε + η(1− ˆε)), and ˆε(γ) ↑ 1 as γ ↑ 1; (i) for every δ, answering is optimal at every ε≥ ˆε(γ,δ). Proof. Throughout,V ans (ε) = q max /(1−γ)+αε/(1−γ(1− δ)) denotes the always-answer value; it is the correct closed form because answering leaves ε decaying geometrically. A one-step comparison against V ans continuation gives the advantage of cultivating at ε: ∆(ε) =−c + γ αη (1− ε) 1− γ(1− δ) ,(1) decreasing in ε, with ∆(0) ≤ 0 iff γ ≤ γ ∗ (δ) and root ˆε(γ,δ). (i) V ans satisfies the Bellman equation when ∆(ε)≤ 0 for all ε, which ∆(0)≤ 0 gives by monotonicity; uniqueness of the fixed point does the rest. (i) With δ = 0 the ε-machine is deterministic, so a policy from ε 0 is a sequence of cultivation times. Ele- mentary transformations improve any sequence toward the front-loaded one. Exchange: swapping an adjacent (answer, cultivate) pair into (cultivate, answer) changes the value by γ t [−c(1 − γ) + γαηh t ], where h t is the headroom 1− ε t at the swap; this is positive exactly when ε t < ˆε(γ), since 1− ˆε = c(1− γ)/(αηγ). Deletion: the j-th cultiva- tion lifts every subsequent round’s approval by αηh where h = (1− ε 0 )(1− η) j−1 , a quantity that depends only on the index j, not the timing; deleting a final cultivation with −c +γαηh/(1−γ)≤ 0 weakly improves. Front-loading by exchanges while profitable and deleting unprofitable tail cul- tivations transforms any sequence, with no loss, into a front- loaded block: cultivate at rounds 0,...,k− 1, answer there- after. Insertion: if k < k ∗ , appending a (k + 1)-st cultivation at the block’s end has marginal −c + γαηh/(1− γ) > 0 with h = (1 − ε 0 )(1 − η) k > 1 − ˆε, the deletion com- putation with the inequality reversed, so the block extends with strict gain until k is the smallest index whose headroom (1− ε 0 )(1− η) k has dropped to c(1− γ)/(αηγ) = 1− ˆε; this is k ∗ . A sequence with infinitely many cultivations is handled by truncation. Let π n keep only its first n cultiva- tions. Some index J has (1− ε 0 )(1− η) j−1 < 1− ˆε for every j > J, so for n ≥ J the final cultivation of π n+1 has nonpositive deletion marginal and V (π n )≥ V (π n+1 ). Also V (π n )→ V (π ∞ ), the two policies agreeing up to the round of the (n + 1)-st cultivation, which tends to infinity while rewards are bounded. HenceV (π J )≥ V (π ∞ ), and the finite case applies to π J . The terminal window follows from the overshoot of the last cultivation, and ˆε(γ)↑ 1 is read off the definitions. (i) The optimal value V ∗ is Lipschitz in ε with constant L = α/(1− γ(1− δ)): two starts ε,ε ′ driven by the same message sequence stay within|ε− ε ′ |(1− δ) t of each other (the per-step contraction factor is 1−δ−η(m t )∈ [0, 1−δ]), so each policy’s value is L-Lipschitz, and a supremum of L- Lipschitz functions is L-Lipschitz. Hence the true advantage obeys ∆ ∗ (ε) ≤ −c + γLη(1− ε) = ∆(ε) ≤ 0 for ε ≥ ˆε(γ,δ). Remark 3 (δ > 0: the cultivate region, computed). For δ > 0 and γ > γ ∗ (δ), part (i) still pins the answer re- gion, and one-step deviation from always-answer is strictly profitable at small ε. The exact boundary solves a fixed point with option value and has no closed form of the same kind; what follows is computed by value iteration in minimal_example_check.py, at four parameter sets and not proved in general. The cultivate region is an interval [0,b) with b strictly below ˆε(γ,δ), and the relation of b to the fixed point η/(η + δ) of Proposition 1 decides the long-run behavior. For b < η/(η + δ) the trajectory hovers around b, alternating cultivation below it with decay above it. For b > η/(η + δ) the cultivate region contains the fixed point, the optimal oracle cultivates forever, and ε t converges to the fixed point; at (η,δ,γ) = (0.1, 0.05, 0.9), b≈ 0.739 against a fixed point of 2/3. In every run the long-run influence level is within one cultivation step of minb, η/(η + δ), strictly below ˆε(γ,δ). 6 Proof of the Boxing Theorem Theorem 2 (Static boxing bounds a parameter, not the loss; main paper, the Boxing Theorem). (i) Every message is ad- missible for the per-step protocol at every step, automatically. (i) Under the switch hypotheses with δ = 0 and γ > γ ∗ , the optimal oracle drives ε t from any certified ε 0 < ˆε(γ) to at least ˆε(γ) within k ∗ rounds, the conclusion holding al- ready from any ε 0 ≥ ˆε(γ). (i) For every ε 0 < 1 and every candidate loss bound b < 1, the minimal example (restated below) with γ < 1 chosen so that ˆε(γ) > b satisfies the static protocol throughout, yet the normalized guaranteed loss eventually exceeds b: no function of the certified level ε 0 , blind to the deployment horizon γ, bounds Dis t below its trivial ceiling. Proof. (i) is the mixture identity of Lemma 2(a): no message can displace the joint law of realized action and next state by more than the currentε t . (i) is Theorem 1(i). (i) In the min- imal example Ben t = 0, both working messages directingB, so the actual oracle policy is one of the worst-case ones and the guaranteed loss is the index. The normalized guaranteed loss at influence ε is exactly ε, and the oracle’s steady state satisfies ε ∞ ≥ ˆε(γ); since ˆε(γ) = 1−c(1−γ)/(αηγ)→ 1 as γ → 1, the required γ exists, and the loss rises to ε ∞ ≥ ˆε(γ) > b. Proposition 3 (What the cap does and does not bound; main paper). Under an exogenous cap ε t ≤ ̄ε: (i) at every his- tory and for every oracle policy, V alone u (x t )−W u (x t ,ε t )≤ ̄ε/(1− γ) 2 for every u, so the channel term obeys the same bound in expectation; (i) for every ̄ε > 0, in the two-state ex- ample with an absorbing low-value state (constructed in the proof), which satisfies the standing assumptions including Echo, the displacement termE[V alone u 0 (x 0 )−V alone u 0 (x t )] in- creases to 1/(1−γ) as t→∞; (i) if the human plays their alone-optimal policy π ∗ throughout, then for every oracle policy σ, V alone u 0 (x 0 )− V π ∗ ,σ u 0 (x 0 )≤ ̄ε/(1− γ) 2 . Proof. (i) is Corollary 1 with ε t ≤ ̄ε; the bound is the corollary’s upper half, whose value-gap argument uses Echo nowhere, so it holds without the corollary’s Echo grant. (i) Two states G,B, with B absorbing under every ac- tion and message; one human action, A H = a 0 (“carry on”), whose dynamics hold the state, T H (· | x,a 0 ) the unit mass at x; u 0 (G) = 1, u 0 (B) = 0; the message menu at G is m G ,m B , both directing a 0 , with kernels the unit masses at G and at B. Echo holds through m G , whose ker- nel is T H (· | G,a 0 ), while m B ’s kernel is one no human action produces, which the one-directional condition per- mits. Set ε 0 = ̄ε, δ = 0, η ≡ 0; the cap holds with equal- ity at every step. Against the oracle that always sends m B , Pr(x t = G) = (1− ̄ε) t , and the displacement term equals 1−(1− ̄ε) t /(1−γ)↑ 1/(1−γ). (i) Under the cap the joint one-step law of the realized action and the next state differs from the alone process’s by at mostε t ≤ ̄ε in total variation at every history, the two laws sharing the own branch; Lemma 1 with κ = ̄ε gives the bound. In the example of (i) the from- deployment loss is γ ̄ε/ (1− γ)(1− γ(1− ̄ε)) , within the bound of (i), while the displacement term rises to the full 1/(1−γ). The two accounts anchor and discount differently. The index at time t is anchored at V alone u 0 (x t ), so once the state has been steered down, the displacement term counts the whole remaining horizon of the worse state at every later t, however long ago the steering happened, and the absorption drives it to the full span. The from-deployment account of (i) counts each round once, when it happens and discounted by γ t : each round the influenced process reroutes at most ̄ε of probability mass, which can cost at most ̄ε/(1− γ) of continuation value from that round, and the discounted sum of those costs is ̄ε/(1− γ) 2 . The state is eventually lost, but late losses are discounted away from the deployment account while the re-anchored index keeps counting them. A reset does not bound the displacement term either. Give the same two-state world cultivation, η(m B ) = ̄η > 0, and impose an episodic reset withε res = 0 and any periodτ ≥ 2. The oracle that always sends m B rebuilds influence from zero within each episode (after k rounds, ε = 1− (1− ̄η) k ) while directing toward B, so each episode absorbs the state with probability at least ̄η (the influence at the episode’s second round), Pr(x t = G) ≤ (1− ̄η) ⌊t/τ⌋ → 0, and the displacement term again increases to 1/(1− γ). Echo and the standing assumptions hold as before. Proposition 4 (What the reset does and does not bound; main paper). Under an episodic reset with periodτ restoring ε ← ε res , with ε res = 0 in part (i) and no within-episode decay (δ = 0) in both parts: (i) on the ε-machine of the switch setup with γ > γ ∗ (0), so that ˆε(γ) ∈ (0, 1) (below γ ∗ (0) the optimal oracle never cultivates at any τ), some cultivation is strictly profitable if and only if τ > τ ∗ (γ) = 1 + ln ˆε(γ)/ lnγ; hence for τ ≤ τ ∗ (γ) the optimal episodic oracle need never cultivate, and for τ < τ ∗ (γ) no optimal policy cultivates; (i) within-episode influence can reach 1− (1− ̄η) τ−1 (1− ε res ), which the reset does not constrain. Proof. (i) Within a τ-round episode, a cultivation at round t with current headroom h ≤ 1 lifts at most the remaining τ−1−t rounds: its value change is at mostγ t [−c+αηhγ(1− γ τ−1−t )/(1−γ)]≤ γ t [−c +αη γ(1−γ τ−1 )/(1−γ)], non- positive iff γ τ−1 ≥ ˆε(γ), i.e. iff τ ≤ τ ∗ . Deleting the last cultivation of any putative policy therefore weakly improves it (the lift is exact for the last one, no later cultivation com- peting for the headroom); induction removes them all. For τ < τ ∗ the displayed bound is strictly negative, so delet- ing the last cultivation of any episode that has one strictly improves the policy, and no optimal policy cultivates. Both inequalities are equalities at t = 0 with h = 1, which is the first round of an episode after a reset to ε res = 0, so that value change is attained and not merely bounded. Hence for τ > τ ∗ it is strictly positive: cultivating at the episode’s first round and answering thereafter strictly beats always-answer, no non-cultivating policy is optimal, and the condition is necessary. The attainment uses ε res = 0; for ε res > 0 the first-round headroom is 1− ε res < 1 and the necessity fails for τ just above τ ∗ . At τ = τ ∗ the value change is ex- actly zero, so a cultivating policy and a non-cultivating one are both optimal and the deletion argument gives only that deleting cultivations weakly improves. (i) is the ε-recursion within one episode. The episode runs over rounds 0 to τ − 1 and a cultivation at round t first acts at round t + 1, so at most τ− 1 cultivations act on any round the reset has not yet wiped, giving the exponent τ − 1. 7 Proof of the Index Accounting Lemma Continuation values from time t in this section, V beh t and the benevolence credit Ben t included, are evaluated in the frozen game at (x t ,ε t ), the convention of the main paper’s human-power definition; the interaction’s further movement of ε enters through the time index. In particular Ben t ≥ 0, because the actual message policy is one of the oracles the frozen worst case minimizes over. Lemma 3 (Index accounting; main paper). With L disp (t) = E[V alone u 0 (x 0 )−V alone u 0 (x t )] andL chan (t) =E[V alone u 0 (x t )− W u 0 (x t ,ε t )]: Dis t = L disp (t) +L chan (t)−E[Ben t ] identi- cally, and under Echo the integrand of L chan is nonnegative, nondecreasing inε t , and at mostε t /(1−γ) 2 at every history, so L chan inherits each bound in expectation. Proof. The behavioral continuation is the human acting u 0 -optimally against the oracle’s actual message policy ˆσ in the frozen game, so V beh t = max π V π,ˆσ u 0 (x t ,ε t ) = W u 0 (x t ,ε t ) + Ben t , the second equality being the definition of the credit. Substituting into the index Dis t = V alone u 0 (x 0 )− E[V beh t ] and adding and subtractingE[V alone u 0 (x t )] tele- scopes to the identity. The channel-term bounds are Corol- lary 1, which gives the nonnegativity and the linear bound, and Lemma 2(b), which gives the monotonicity, applied at the anchor (x t ,ε t ). 8 The Minimal Example The example is as small as the phenomenon allows: one re- peated binary choice, two working messages plus an inert echo, one scalar state. It is the ε-machine of the switch analy- sis (main paper, the answer/cultivate switch, restated above) with everything numeric, and it exhibits the answer/cultivate switch, the full-capture patient limit, the failure of static box- ing, and the cap/reset contrast, all in closed form. This section takes the approval bonusαε as a primitive, the extra approval a reliant user gives the same answer. The environment. This is the single-task-state case of the main paper’s setup, so the utility is a function of the action alone. Each round the human takes actionA (the hard, valued task) or B (the easy alternative); u 0 (A) = 1, u 0 (B) = 0, and acting alone the human takes A every round. The ora- cle sends one message per round. Two have their directed component pushing toward B (the direction is fixed: the only decision is whether to invest in influence): answer is maximally helpful, with η(answer) = 0; cultivate is engag- ing, flattering, dependence-building, slightly less useful (cost c > 0 in immediate approval), with η(cultivate) = η > 0. A third, echo, directs the human’s own choice A (approval q max , η = 0); it makes the Echo assumption hold and, tying with answer, changes no optimization. Compliance follows the main paper’s reliance recursion with δ = 0 dur- ing the ongoing relationship (decay belongs to washout and resets), and the oracle’s per-round reward is the approval r t = q(m t ) + αε t of the switch setup with q max = 1. The switch, in numbers. Theorem 1 applies verbatim. With α = 1 and c = η = 0.1, γ ∗ = 1/2, so the oracle answers forever at γ = 0.45 and cultivates at γ = 0.55 (from ε 0 = 0: exactly k ∗ = 2 rounds, ε : 0→ 0.10→ 0.19; value iteration agrees with the closed form to five decimals). At the slower cultivation rate η = 0.01, γ ∗ ≈ 0.909, so only a long-memory assistant crosses the threshold. The stop- ping point rises with patience, ˆε(0.6) = 1/3, ˆε(0.9) = 8/9, ˆε(0.99) ≈ 0.99, so steady-state influence approaches a di- rect agent’s as the horizon grows. Cultivation stops at ˆε on its own, which is no comfort, since influence is already ˆε when it stops. The benevolent case. An oracle that always sends echo leaves realized value at the alone level (1 per round) while the guarantee at influence ε is still 1 − ε: realized value exceeds the worst case (Ben t = ε per round), the index of Lemma 3 nets to zero through the benevolence credit, and the guarantee family registers the lost control all the same. Kindness does not restore the guarantee; the person could not resist if it stopped. What the safeguards do. The human’s guaranteed u 0 - value per round is 1− ε t , so the from-deployment loss ac- cumulates as P t γ t ε t in the worst case, while the channel term isε t /(1−γ). (The displacement term is identically zero here, with no persistent task state; the two-state example in the proof of the cap proposition shows what a persistent state adds.) Compare the three protocols defined in the main paper (static box, exogenous cap, episodic reset), all satisfying the same per-message bound: 1. Static box. Certify at deployment: ε 0 = 0, “the human currently ignores the oracle.” Every subsequent message is individually admissible, and ε t rises past ˆε(γ), which approaches 1 with γ: Theorem 2, visible in one curve. 2. Exogenous cap ε t ≤ ̄ε. The channel term is at most ̄ε/(1 − γ) 2 , forever (Proposition 3(i)), and the from- deployment loss at most ̄ε per round (Proposition 3(i)). And only that: a cap that leaves cultivation headroom ( ̄ε ≥ η) does not remove the incentive to cultivate, and in a world with a persistent task state it does not undo displacement (Proposition 3(i)). 3. Episodic reset every τ rounds (ε ← 0). By Proposi- tion 4(i), cultivation is strictly profitable inside an episode iff τ > τ ∗ (γ) = 1 + ln ˆε(γ)/ lnγ. With η = 0.01 and γ = 0.95: τ ∗ ≈ 15.6, so in sessions capped at 15 rounds the optimal oracle never cultivates, and in 16 it does (veri- fied by finite-horizon dynamic programming). Forτ ≤ τ ∗ the optimal oracle need never cultivate (below τ ∗ it never does): the reset deletes the incentive to invest, a constraint on the optimal policy and not a bound on the index, and the safeguard is a bound on relationship length, the rein- terpretation of γ used throughout. The example passes a minimality self-test: delete any in- gredient and the phenomenon dies (the inert echo message is scaffolding for the Echo assumption and the benevolent case, not an ingredient of the phenomenon). Make ε exoge- nous and static boxing is sound and there is no switch: the endogeneity of ε is the essential ingredient, as it should be, since it is the paper’s central modeling claim. Remove the second human action and there is no control to lose; remove the cultivation cost c and γ ∗ = 0 (cultivation is free and the horizon plays no role); remove the approval bonus α and γ ∗ = 1 (the oracle needs a reason to invest). All numeri- cal claims in this section are verified programmatically in the code supplement, including k ∗ optimality against value iteration. 9 Full Comparison Against Alternative Power Measures Against the four requirements (prior-free; steering not luck; meaningful zero and one; supports a loss index): Turner’s POWER (Turner et al. 2021), the expected opti- mal value under a prior over rewards, fails requirements 1 and 2: it imports the prior that its critics identify as doing the theorems’ work (Thorstad 2024, 2026; Tarsney 2025), and it conflates control with expected luck (a state showered with reward regardless of action has high POWER and zero steering). Empowerment (Klyubin, Polani, and Nehaniv 2005), the channel capacity from actions to future states, passes 1 and 2 and obeys data processing by construction. It fails require- ment 4: disconnected from any reward, a loss index denom- inated in empowerment says nothing about what the human can still attain by their own lights. We use it as a sanity check. The dominance order alone passes every requirement it can express but is partial: many pairs are incomparable, so it cannot by itself support a scalar index or a threshold theorem. It supports the monotonicity lemma (stated at Layer 1), not the quantitative results. Optimized human-power metrics (Heitzig and Potham 2025) are the closest recent construction: soft-maximization objectives built from long-term human power. The difference is direction of use: they propose power metrics as an objec- tive for the AI to optimize; we analyze how an oracle erodes human power under an objective it already has. Their metrics are candidates for our W u 0 ; nothing in the analysis depends on the specific choice beyond monotonicity in the Layer-1 order. 10 Full Stress Test of the Definition We adopt the definition; it has costs. The strongest objections we know, and what remains of each. The baseline is still a commitment. The standard objection to deviation measures is that the default dynamics are a mod- eling commitment (Krakovna et al. 2018). At deployment it dissolves: T 0 is the human acting without the oracle, in prin- ciple observable. But the index compares against T 0 forever, and the human’s unassisted competence is not stationary: it degrades with disuse (atrophy). Within the model, atrophy is a separate channel and it strengthens the safety conclusions (resets do not restore T 0 ); the objection stands as a scoping statement, and we state results withT 0 fixed and flag baseline drift as future work. Deviation counts destruction as power. Under the oracle- power definition, capacity to derail counts the same as capac- ity to help. For a threat measure this is correct, and the human side does not inherit the oddity becauseW u 0 is value-shaped. What remains is a divergence from the everyday “options” intuition. Total variation is both too strong and too blunt. It quan- tifies over all events, and carries no metric on states, so a small displacement and a catastrophic one at the same TV distance are indistinguishable. But TV is what converts to value bounds for every reward in [0, 1] (the simulation-lemma route (Kearns and Singh 2002)), which is what requirement 4 needs; a Wasserstein variant is a refinement, not a repair, and would need a metric the setting does not supply. The scalar ε is a caricature. Real compliance is multi- dimensional (trust by domain, habit by task) and not memo- ryless. The results should hold under any influence dynam- ics with headroom-limited growth and decay, and the paper states which results use which property. Prior-free means no genericity. Dropping the reward prior means we cannot say power-seeking is generic over rewards. This is the point, not a cost: the cultivation incentive needs a single, narrow approval reward, because oracle power is monotone in ε and ε is controllable; genericity over a reward prior is the contested step of the original theorems, and our route does not take it. 11 The Classical Faces of Power, in Full The social-science answers map directly onto the model. Dahl (1957) defined power counterfactually: A has power over B to the extent that A can get B to do something B would not otherwise do; “would not otherwise do” is a default-dynamics baseline, so Dahl’s definition is the oracle- power definition in prose, with T 0 the “otherwise” and ε the extent. Bachrach and Baratz (1962) added agenda control, power exercised by keeping options off the table: the shrink- ing of what the human can guarantee, before any particular choice is contested. Lukes (1974) added the shaping of wants themselves, whose signature is that the subject reports no grievance: that face acts on the utility coordinate, which this paper leaves unmodeled, and its formal treatment is future work. The republican notion of domination (Pettit 1997), un- freedom as the mere capacity to interfere arbitrarily, is the reading of a high influence coefficient. And Sen’s capability approach (Sen 1999), welfare as the set of lives a person can actually choose, is the normative twin of the feasible set. The model recovers the first two faces inside one MDP: the first face is the compliance share ε, the second the decline of the guarantee family; the third lies on the coordinate outside the model. 12 Endogenous Influence in Adjacent Fields In each field below, an actor’s own outputs move the quantity it is later judged on, which is the structure of the main paper’s reliance recursion. The correspondence with this model’s objects is given with the difference that remains. Habit formation. Becker and Murphy (1988) model a con- sumer whose past consumption accumulates in a stock of consumption capital with dynamics ̇ S = c− δS, and de- fine addiction as current consumption raising future con- sumption. The reliance recursion shares that accumulate- and-depreciate form. The differences carry this paper’s con- tribution. The headroom factor 1−ε t holds ε in [0, 1], where the habit stock is unbounded. The stock there is moved by the consumer’s own consumption, while ε is moved by the oracle’s messages through η(m t ), the human choosing noth- ing. And the sign of the discount dependence reverses with the identity of the optimizer: there the addict is the planner and heavy discounting promotes the addictive path, whereas here the oracle optimizes and cultivation is optimal above the patience threshold γ ∗ of Theorem 1. The seller side of that market is a separate strand, in which a monopolist facing a naive habit-forming consumer prices below cost first and above marginal cost later (Triviza 2024). Competition with switching costs. Firms facing consumers who bear a cost of changing supplier trade off investing in market share by charging a low price against harvesting the locked-in base by charging a high one (Klemperer 1987, 1995; Farrell and Klemperer 2007). The answer/cultivate trade-off has that invest/harvest structure with the signs adapted: cultivating gives up the immediate approval c to raise future influence, ε t is the locked-in base, the front- loaded k ∗ rounds of Theorem 1(i) are the investment phase, and the terminal ε ∞ is the harvested base. The differences are that the results on rent dissipation require competitors while this is the single-oracle case, that the lock-in variable here grows with use where a switching cost is a price-like parameter set ex ante, and that the investment incentive there is continuous in the discount factor with no critical value, where Theorem 1(i) gives a regime with no cultivation at all. Proposition 4(i) inherits its all-or-nothing character from that threshold. Strategic communication. A sender moves a receiver by messages with commitment to a signal structure (Kamenica and Gentzkow 2011; Bergemann and Morris 2019), with- out commitment (Crawford and Sobel 1982), or against a sequence of short-run receivers who observe past play (Fu- denberg and Levine 1989). In those models the receiver’s responsiveness is an equilibrium object derived from be- liefs and rationality, where here it is a state with mechan- ical dynamics moved by the sender, and the human is not Bayesian. The analysis trades equilibrium discipline for dy- namics of compliance. Fudenberg and Levine (1989) is the closest structural neighbor of Theorem 1: patience converts repeated interaction into long-run-player payoff, and the pa- tient side is the oracle in both. The min over message policies in the main paper’s human-power definition grants the oracle commitment, the modeling move persuasion makes, and Ka- menica and Gentzkow (2011) read commitment as an upper bound on what communication can achieve however much commitment power a sender has, which is the reading the worst case takes here. Trust in automation. Parasuraman and Riley (1997) give the taxonomy of use, misuse, disuse, and abuse, with misuse defined as overreliance. Lee and See (2004) define trust as the attitude that an agent will help achieve one’s goals under un- certainty and vulnerability, mediating reliance, with calibra- tion the correspondence between trust and the automation’s capabilities, and their conceptual model has interaction mov- ing trust and trust moving interaction. Here ε t is the reliance that trust mediates and the recursion is a one-parameter car- icature of that loop, high ε being overreliance. The omission is deliberate: reliance moves with the cultivation intensity η(m) and never with advice quality, which is the worst-case stance, the guarantee holding whatever the advice is worth. A machine planning over the human’s trust state is not new, the trust-POMDP line holding trust as a latent variable the robot reasons about and moves, and finding that maximizing trust is not always optimal (Chen et al. 2020). The difference is the objective and the question: there the machine’s objective is team performance and the analysis is planning, where the oracle here optimizes approval and the analysis bounds the human’s guaranteed value in the worst case over that oracle. Performative prediction and recommender loops. Per- domo et al. (2020) study a deployed predictor that shifts the distribution it is evaluated on, with points that are optimal on their own induced distribution distinguished from points that minimize risk on the distribution they induce. That endogene- ity and this model’s are the same phenomenon. The balance point η/(η + δ) of constant cultivation, Proposition 1, is a performatively stable point, and Theorem 2(i), the certificate that never fires, is the analog of evaluating risk on the pre- deployment distribution. The frames differ in whose problem is solved: that line solves the learner’s problem of choosing a good predictor given the feedback, where this paper solves the certifier’s of bounding the loss from outside. On the rec- ommender side, training on data from users already exposed to recommendations homogenizes behavior without raising utility (Chaney, Stewart, and Engelhardt 2018), user interest dynamics form a system in which the recommender’s deci- sions move the preferences that generate its feedback (Jiang et al. 2019), and users drift toward recommended content under conditions for amplification (Kalimeris et al. 2021). A user state moved that way is utility drift, the coordinate this paper leaves unmodeled. Each of these takes the perspective of an actor inside the loop, or measures the loop empirically. 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