Lemma 2 (parametric continuity of regularized stable points)

Apparatus §4 · the technical scaffold for Mode A

The technical scaffold Mode A (§5.1) rests on. Under the framework's regularity strengthened to global Hessian non-singularity along the parametric path, the regularized performative stable point $θ_{β}^{*'}$ is $C^{1}$ in the regularization parameter $β$ , monotone in the regularizer $B$ , and supplies a continuous path from the unregularized stable point through the regularized regime.

Mode A regularizes the platform's reward functional: $F^{'} (θ; β) = F (θ) + β B (θ)$ , where $B (θ)$ is the expected sum of a bonus function $b (τ)$ over inter-stimulus intervals — the regularizer pays for the pause exceeding the reflective threshold. Proposition 1' wants to apply the intermediate value theorem to the survival probability $P (β)$ on the parametric path indexed by $β$ . For the IVT to apply, the path needs to be continuous. Lemma 2 supplies the continuity.

What the lemma says. Three things, in increasing substantive content. First, the regularized stable point $θ_{β}^{*'}$ is well-defined and varies smoothly with $β$ . Second, the regularizer's value at the stable point grows monotonically in $β$ (and the original reward decreases). Third, there exists a smooth path from θ* at β=0 through the regularized regime, with $B$ strictly increasing on a right-neighborhood of $β = 0$ and $sup_{β} B (\tilde{θ} (β)) > 0$ .

Why the strengthened (R2).The framework's standing (R2) gives local Hessian non-singularity at the stable point. The lemma needs global non-singularity along the path of regularized stable points — a stronger condition. Under generic-bifurcation regularity, this can be replaced with Whitney-stratification machinery; the substantive results survive.

What the lemma authorizes the prose to claim. Mode A is a smooth curve through parameter space — a continuum of settings indexed by $β$ , along which the regularizer's payoff grows continuously. Any target survival probability in the achievable range corresponds to some $β_{ε}$ on the curve. The political claim “at sufficient regulation, the pause re-instantiates” depends on Lemma 2's smoothness.

The policy-design corollary follows. A regulator considering a Mode A intervention is choosing a point on a continuous curve indexed by $β$ . The binary “regulate / don't regulate” never describes the real decision. Larger $β$ yields more pause-instantiation and less platform-side reward; the curve is monotone in both, and smooth. The policy question is therefore quantitative: which target survival probability $ε$ at $θ_{ref}$ is socially acceptable, and what $β$ achieves it? Lemma 2 guarantees that the question has an answer in the framework's feasibility envelope — any $ε$ in that envelope corresponds to some $β_{ε}$ on the curve.

Lemma 2 (parametric continuity of regularized stable points)

Define $F (θ) := E_{D (θ)} [\hat{R} (θ)]$ and $B (θ) := E_{D (θ)} [\sum_{k} b (τ^{(k)})]$ , where $b : [0, \infty) \to [0, b_{m a x}]$ is bounded lower-semicontinuous with $b (τ) > 0$ for $τ > τ^{*}$ and $b (τ) = 0$ for $τ \leq τ^{*}$ . Define $F^{'} (θ; β) := F (θ) + β B (θ)$ for $β \geq 0$ . Under (R1), (R2) strengthened to global non-singularity of $H_{F} + β H_{B}$ along the relevant path, and (P-II):

(i) $θ_{β}^{*'} := ar g max_{θ} F^{'} (θ; β)$ is locally well-defined for each $β \geq 0$ ; $β \mapsto θ_{β}^{*'}$ is $C^{1}$ on $[0, \infty)$ .

(ii) Monotonicity. $B (θ_{β}^{*'})$ is non-decreasing in $β$ ; $F (θ_{β}^{*'})$ is non-increasing in $β$ .

(iii) Path existence and strict growth. There exists a $C^{1}$ path $\tilde{θ} : [0, \infty) \to Θ$ with $\tilde{θ} (0) = θ^{*}$ and $B (\tilde{θ} (β))$ strictly increasing on a right-neighborhood of $β = 0$ , with $sup_{β} B (\tilde{θ} (β)) > 0$ .

Proof — three parts

(i) Continuity via IFT. The first-order condition at $θ_{β}^{*'}$ is $G (θ, β) := \nabla F + β \nabla B = 0$ . The Jacobian in $θ$ is $H_{F} + β H_{B}$ , non-singular by the strengthened (R2). The implicit function theorem (Rudin 1976, Theorem 9.28) gives $β \mapsto θ_{β}^{*'}$ $C^{1}$ on $[0, \infty)$ .

(ii) Monotonicity via envelope theorem. Differentiating the FOC: $d θ_{β}^{*'} / d β = - (H_{F} + β H_{B})^{- 1} \nabla B$ . At a local maximum, $H_{F} + β H_{B}$ is negative definite, so $- (H_{F} + β H_{B})^{- 1}$ is positive definite, and

\frac{d B ( θ _{β}^{*'} )}{d β} = - \nabla B^{⊤} (H_{F} + β H_{B})^{- 1} \nabla B \geq 0,

in wordsAs the regulator dials up the regularization $β$ , the amount of pause the stable point delivers can only increase. The right side is a quadratic form in a positive-definite matrix, so it is never negative — which means the bonus $B$ rises monotonically along the curve. There are no dead zones where pushing harder buys nothing: the slider always responds.

strict when $\nabla B \neq = 0$ . By the envelope theorem applied to $V (β) := F^{'} (θ_{β}^{*'}; β)$ : $V^{'} (β) = B (θ_{β}^{*'}) \geq 0$ . Differentiating $V = F + β B$ directly: $d F / d β = V^{'} (β) - B - β d B / d β = - β d B / d β \leq 0$ .

(iii) Path existence and strict growth. At $β = 0$ , $\tilde{θ} (0) = θ^{*}$ . By (i), the path extends smoothly on $[0, \infty)$ . By Lemma 1 Step 2, $\overset{ˉ}{λ} (θ^{*}) = λ_{m a x}$ , so the inter-arrival distribution at $θ^{*}$ concentrates near $δ_{m i n} < τ^{*}$ , giving $B (θ^{*}) \approx 0$ (operationally). By (P-II), the $B$ -maximizer (firing at $τ^{*}$ -intervals) is in $Π$ and has $B > 0$ . At $θ^{*}$ , $\nabla B \neq = 0$ . By continuity, $\nabla B (\tilde{θ} (β)) \neq = 0$ on a right-neighborhood of $β = 0$ . Strict growth on this neighborhood follows from (ii); $sup_{β} B (\tilde{θ} (β)) > B (θ^{*}) = 0$ .

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Remark

The lemma supplies the technical scaffolding for Mode A (§5.1): a parametric path from the unregularized stable point through the regularized regime. The strengthened (R2) is the cleanest hypothesis for global IFT continuation; under generic-bifurcation regularity it can be replaced with Whitney-stratification machinery, but the substantive results survive.

Cross-references

Consumer: §5 Mode A (Proposition 1') — the lemma's payoff in the framework's central political claim
Required: Lemma 1 — Step 2's rate saturation gives $B (θ^{*}) \approx 0$
Required: §0 axioms (R1, R2, P-II)