Proof of Perron's Method on Unitary Laws: the Bump Construction

Each result cited below is universally quantified over the data in its own statement.

Conventions. The conventions of Unitary Laws with Free Unitary Noise: Standing Data §spaces and Unitary Laws with Free Unitary Noise: Standing Data §test-functions are in force. Viscosity subsolutions are those of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with the admissible Hamiltonian $H$ of the statement. By The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §inner-product and The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §hilbert, $E_{d}$ is a real Hilbert space with inner product $\langle\cdot,\cdot\rangle_{d}$ and norm $\lVert\cdot\rVert_{d}$, and its distance is $(y,y')\mapsto\lVert y-y'\rVert_{d}$ (Real Hilbert Spaces: Standing Notation and Background §space); Elementary Identities in a Real Inner Product Space, the symmetry axiom of Real Inner Product Space §inner-product, The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity, Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space and Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space are applied with $E_{d}$ as the space and $U=E_{d}$, which is open in itself (Real Hilbert Spaces: Standing Notation and Background §topology); every member of $C^{2}(E_{d})$ belongs to $C^{1}(E_{d})$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c2, and every member of $C^{1}(E_{d})$ is differentiable at every point of $E_{d}$ by The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1. For $\mu,\nu\in\mathcal{L}_{d}$, $d_{\mathcal{L}}(\mu,\nu)=\lVert\iota_{d}(\mu)-\iota_{d}(\nu)\rVert_{d}$ (The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance §distance); $(\mathcal{L}_{d},d_{\mathcal{L}})$ is a metric space (The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §metric), nonempty (Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words §nonempty) and compact (The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It §compact). Convergence of real sequences is convergence in $(\mathbb{R},d_{\mathbb{R}})$ (The Real Numbers: Standing Notation and Background §sequences). For $G\in C^{1}(E_{d})$ and $\lambda\in\mathcal{L}_{d}$ put

$$T_{G}(\lambda)=H\bigl(\lambda,DG(\iota_{d}(\lambda))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DG(\iota_{d}(\lambda)),\widehat{\Theta}\lambda\bigr\rangle_{d}-f(\lambda).$$

Preliminary facts. Let $G\in C^{1}(E_{d})$.

(C1) $G\circ\iota_{d}$ is continuous and bounded on $\mathcal{L}_{d}$. $G$ is continuous on $E_{d}$ by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space §continuous. Let $\lambda\in\mathcal{L}_{d}$ and $\varepsilon>0$; choose $\eta>0$ with $|G(y)-G(\iota_{d}(\lambda))|<\varepsilon$ whenever $\lVert y-\iota_{d}(\lambda)\rVert_{d}<\eta$ (Continuous Map Between Metric Spaces). If $d_{\mathcal{L}}(\lambda,\lambda')<\eta$ then $\lVert\iota_{d}(\lambda')-\iota_{d}(\lambda)\rVert_{d}=d_{\mathcal{L}}(\lambda',\lambda)<\eta$, so $|G(\iota_{d}(\lambda'))-G(\iota_{d}(\lambda))|<\varepsilon$. Thus $G\circ\iota_{d}$ is continuous on $(\mathcal{L}_{d},d_{\mathcal{L}})$, hence upper semicontinuous (Upper Semicontinuous Function on a Subset of a Metric Space), and Extreme Value Theorem on a Compact Subset of a Metric Space (with $X=K=\mathcal{L}_{d}$) gives $\lambda_{\min},\lambda_{\max}\in\mathcal{L}_{d}$ with $G(\iota_{d}(\lambda_{\min}))\le G(\iota_{d}(\lambda))\le G(\iota_{d}(\lambda_{\max}))$ for all $\lambda$; so $C_{G}=\max\{|G(\iota_{d}(\lambda_{\min}))|,|G(\iota_{d}(\lambda_{\max}))|\}$ satisfies $|G(\iota_{d}(\lambda))|\le C_{G}$ for every $\lambda\in\mathcal{L}_{d}$.

(C2) Sequential continuity. Let $(\lambda_{n})_{n\in\mathbb{N}}$ converge to $\lambda$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$. Then $(G(\iota_{d}(\lambda_{n})))_{n}$ converges to $G(\iota_{d}(\lambda))$ and $(T_{G}(\lambda_{n}))_{n}$ converges to $T_{G}(\lambda)$. Indeed, $\lVert\iota_{d}(\lambda_{n})-\iota_{d}(\lambda)\rVert_{d}=d_{\mathcal{L}}(\lambda_{n},\lambda)$, so $(\iota_{d}(\lambda_{n}))_{n}$ converges to $\iota_{d}(\lambda)$ in $E_{d}$ (Convergent Sequence in a Metric Space). The first assertion follows from (C1) and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. The gradient map $DG$ is continuous on $E_{d}$ (The Classes $C^1$ and $C^2$ on an Open Subset of a Real Inner Product Space §c1), so by the same lemma $p_{n}=DG(\iota_{d}(\lambda_{n}))$ converges to $p=DG(\iota_{d}(\lambda))$ in $E_{d}$. By claim 1 of Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space, $((\lambda_{n},p_{n}))_{n}$ converges to $(\lambda,p)$ in $\mathcal{L}_{d}\times E_{d}$ with the product metric of Admissible Hamiltonians on Unitary Laws §continuous, on which $H$ is continuous by that property, $H$ being admissible; so $(H(\lambda_{n},p_{n}))_{n}$ converges to $H(\lambda,p)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. By The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge §continuity and the same lemma, $(\widehat{\Theta}\lambda_{n})_{n}$ converges to $\widehat{\Theta}\lambda$ in $E_{d}$, so $(\langle p_{n},\widehat{\Theta}\lambda_{n}\rangle_{d})_{n}$ converges to $\langle p,\widehat{\Theta}\lambda\rangle_{d}$ by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §continuity. As $f$ is continuous (Unitary Laws with Free Unitary Noise: Standing Data §source), $(f(\lambda_{n}))_{n}$ converges to $f(\lambda)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential. Claims 1 and 3 of Arithmetic of Limits of Real Sequences give the second assertion.

The data. By Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution, $w$ is bounded, with a bound $C_{w}\ge0$ (Bounded Real-Valued Function on a Set), and upper semicontinuous. For every $\lambda\in\mathcal{L}_{d}$, $-C_{w}\le w(y)$ for all $y\in\mathcal{L}_{d}$, so $-C_{w}\in B_{w}(\lambda)$ and $w$ is bounded below near each point (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function §near-bounds); we apply Properties of the Lower Semicontinuous Envelope, by Duality with the metric space $(\mathcal{L}_{d},d_{\mathcal{L}})$ and $S=\mathcal{L}_{d}$, and Properties of the Lower Semicontinuous Envelope, by Duality §bounds gives $w_{*}\le w$. By Local Minimum of a Function Relative to a Subset of a Metric Space, fix a real $r_{1}>0$ with

$$w_{*}(\lambda)-F(\iota_{d}(\lambda))\ge w_{*}(\lambda_{0})-F(\iota_{d}(\lambda_{0}))\qquad(\lambda\in\mathcal{L}_{d},\ d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{1}),\tag{Lm}$$

and put $\gamma=g(\lambda_{0})-w_{*}(\lambda_{0})>0$. Let $r>0$ be real. The objects below are chosen in this order: $F_{2}$ (Step 1); $\eta_{0}$ and $r_{2}$ (Step 2); $r_{g}$, $r_{F}$, $r_{0}$ and $\delta$ (Step 3); $\widetilde{w}$ (Step 4); $\lambda'$ (Step 6).

Step 1 (A strict minimum). Put $c=w_{*}(\lambda_{0})-F(\iota_{d}(\lambda_{0}))$ and let $q:E_{d}\to\mathbb{R}$, $q(y)=\lVert y-\iota_{d}(\lambda_{0})\rVert_{d}^{2}=\tfrac22\lVert y-\iota_{d}(\lambda_{0})\rVert_{d}^{2}$; by Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$ §quadratic (with $2$ and $\iota_{d}(\lambda_{0})$ in the roles of $\alpha$ and $y_{0}$), $q\in C^{1}(E_{d})$ and $Dq(y)=2(y-\iota_{d}(\lambda_{0}))$. Let $F_{2}=(F-q)+c$, where $c$ denotes the constant function. By Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §difference, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §constant and Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §sum, $F_{2}\in C^{1}(E_{d})$ and $DF_{2}(y)=DF(y)-2(y-\iota_{d}(\lambda_{0}))$; in particular $DF_{2}(\iota_{d}(\lambda_{0}))=DF(\iota_{d}(\lambda_{0}))$, as $\iota_{d}(\lambda_{0})-\iota_{d}(\lambda_{0})$ is the zero vector. For $\lambda\in\mathcal{L}_{d}$, $F_{2}(\iota_{d}(\lambda))=F(\iota_{d}(\lambda))-d_{\mathcal{L}}(\lambda,\lambda_{0})^{2}+c$, so $F_{2}(\iota_{d}(\lambda_{0}))=w_{*}(\lambda_{0})$, and by (Lm)

$$w_{*}(\lambda)-F_{2}(\iota_{d}(\lambda))=\bigl(w_{*}(\lambda)-F(\iota_{d}(\lambda))\bigr)-c+d_{\mathcal{L}}(\lambda,\lambda_{0})^{2}\ge d_{\mathcal{L}}(\lambda,\lambda_{0})^{2}\qquad(d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{1}).\tag{Q1}$$

Step 2 (The strict inequality near $\lambda_{0}$). Put $Q(\lambda)=\rho\,F_{2}(\iota_{d}(\lambda))+T_{F_{2}}(\lambda)$ for $\lambda\in\mathcal{L}_{d}$. By Step 1 and the hypothesis of the lemma,

$$Q(\lambda_{0})=\rho\,w_{*}(\lambda_{0})+H\bigl(\lambda_{0},DF(\iota_{d}(\lambda_{0}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF(\iota_{d}(\lambda_{0})),\widehat{\Theta}\lambda_{0}\bigr\rangle_{d}-f(\lambda_{0})<0 .$$

Put $\eta_{0}=-Q(\lambda_{0})/2>0$. By (C2) with $G=F_{2}$ and claims 1 and 3 of Arithmetic of Limits of Real Sequences, $(Q(\lambda_{n}))_{n}$ converges to $Q(\lambda_{0})$ for every sequence $(\lambda_{n})_{n}$ converging to $\lambda_{0}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$; so $Q$ is continuous at $\lambda_{0}$ relative to $\mathcal{L}_{d}$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, and there is a real $r_{2}>0$ with $|Q(\lambda)-Q(\lambda_{0})|<\eta_{0}$, hence

$$Q(\lambda)<Q(\lambda_{0})+\eta_{0}=-\eta_{0}\qquad(\lambda\in\mathcal{L}_{d},\ d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{2}).\tag{Q2}$$

Step 3 (Radii and the height $\delta$). As $g$ is lower semicontinuous at $\lambda_{0}$ (Lower Semicontinuous Function on a Subset of a Metric Space), choose a real $r_{g}>0$ with $g(\lambda)>g(\lambda_{0})-\gamma/3$ for $d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{g}$; by (C1) for $G=F_{2}$, choose a real $r_{F}>0$ with $|F_{2}(\iota_{d}(\lambda))-F_{2}(\iota_{d}(\lambda_{0}))|<\gamma/3$ for $d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{F}$. Put

$$r_{0}=\min\{r,r_{1},r_{2},r_{g},r_{F}\}>0,\qquad\delta=\min\Bigl\{\frac{r_{0}^{2}}{8},\ \frac{\gamma}{3},\ \frac{\eta_{0}}{2\rho}\Bigr\}>0,$$

using $\rho>0$ (Unitary Laws with Free Unitary Noise: Standing Data §constants). Then for $\lambda\in\mathcal{L}_{d}$:

(B1) if $d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0}$, then $F_{2}(\iota_{d}(\lambda))+\delta<g(\lambda)$, since $F_{2}(\iota_{d}(\lambda))+\delta<w_{*}(\lambda_{0})+\tfrac{\gamma}{3}+\tfrac{\gamma}{3}=g(\lambda_{0})-\tfrac{\gamma}{3}<g(\lambda)$;

(B2) if $r_{0}/2\le d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0}$, then $F_{2}(\iota_{d}(\lambda))+\delta<w(\lambda)$, since by $w_{*}\le w$, (Q1) (as $r_{0}\le r_{1}$) and Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, $w(\lambda)\ge w_{*}(\lambda)\ge F_{2}(\iota_{d}(\lambda))+r_{0}^{2}/4$, and $\delta\le r_{0}^{2}/8<r_{0}^{2}/4$;

(B3) if $d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0}$, then $Q(\lambda)<-\eta_{0}$ by (Q2), as $r_{0}\le r_{2}$; moreover $\rho\,\delta\le\eta_{0}/2$.

Step 4 (The bump). Define $\widetilde{w}:\mathcal{L}_{d}\to\mathbb{R}$ by

$$\widetilde{w}(\lambda)=\max\bigl\{w(\lambda),\ F_{2}(\iota_{d}(\lambda))+\delta\bigr\}\ \text{ if }d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0},\qquad\widetilde{w}(\lambda)=w(\lambda)\ \text{ otherwise},$$

with the maximum of two real numbers, which is one of them and is at least each of them. Then:

(W1) $w\le\widetilde{w}\le g$ on $\mathcal{L}_{d}$, by the definition, $w\le g$ and (B1).

(W2) $\widetilde{w}(\lambda)=w(\lambda)$ whenever $d_{\mathcal{L}}(\lambda,\lambda_{0})\ge r_{0}/2$: for $d_{\mathcal{L}}(\lambda,\lambda_{0})\ge r_{0}$ by definition, and for $r_{0}/2\le d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0}$ by (B2). In particular $\widetilde{w}(\lambda)=w(\lambda)$ whenever $r\le d_{\mathcal{L}}(\lambda,\lambda_{0})$, as $r_{0}\le r$.

(W3) $\widetilde{w}$ is bounded: with $C_{F_{2}}$ as in (C1), $-C_{w}\le w\le\widetilde{w}\le\max\{C_{w},C_{F_{2}}+\delta\}$.

(W4) $\widetilde{w}$ is upper semicontinuous on $\mathcal{L}_{d}$. Let $\lambda\in\mathcal{L}_{d}$ and $\varepsilon>0$, and let $t_{1}>0$ be a radius for the upper semicontinuity of $w$ at $\lambda$ with $\varepsilon$. If $d_{\mathcal{L}}(\lambda,\lambda_{0})<r_{0}$, let $t_{2}>0$ be such a radius for the continuous (hence upper semicontinuous, by (C1)) function $\varphi=F_{2}\circ\iota_{d}+\delta$, and $t=\min\{t_{1},t_{2},r_{0}-d_{\mathcal{L}}(\lambda,\lambda_{0})\}>0$; for $d_{\mathcal{L}}(\lambda,\lambda')<t$ the triangle inequality gives $d_{\mathcal{L}}(\lambda',\lambda_{0})<r_{0}$, and $w(\lambda')<w(\lambda)+\varepsilon\le\widetilde{w}(\lambda)+\varepsilon$ and $\varphi(\lambda')<\varphi(\lambda)+\varepsilon\le\widetilde{w}(\lambda)+\varepsilon$, so $\widetilde{w}(\lambda')<\widetilde{w}(\lambda)+\varepsilon$. If $d_{\mathcal{L}}(\lambda,\lambda_{0})\ge r_{0}$, let $t=\min\{t_{1},r_{0}/2\}$; for $d_{\mathcal{L}}(\lambda,\lambda')<t$ the triangle inequality gives $d_{\mathcal{L}}(\lambda',\lambda_{0})\ge d_{\mathcal{L}}(\lambda,\lambda_{0})-d_{\mathcal{L}}(\lambda,\lambda')>r_{0}/2$, so by (W2) $\widetilde{w}(\lambda')=w(\lambda')<w(\lambda)+\varepsilon=\widetilde{w}(\lambda)+\varepsilon$. This is Upper Semicontinuous Function on a Subset of a Metric Space.

Step 5 ($\widetilde{w}$ is a viscosity subsolution). By (W3), (W4) and Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution, it remains to prove the inequality. Let $\psi\in C^{1}(E_{d})$ and $\widehat{\lambda}\in\mathcal{L}_{d}$ be such that $\widetilde{w}-\psi\circ\iota_{d}$ has a local maximum at $\widehat{\lambda}$ relative to $\mathcal{L}_{d}$, with a radius $\widehat{r}>0$ as in Local Maximum of a Function Relative to a Subset of a Metric Space. We must show

$$\rho\,\widetilde{w}(\widehat{\lambda})+H\bigl(\widehat{\lambda},D\psi(\iota_{d}(\widehat{\lambda}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle D\psi(\iota_{d}(\widehat{\lambda})),\widehat{\Theta}\widehat{\lambda}\bigr\rangle_{d}\le f(\widehat{\lambda}).\tag{Goal}$$

Case A: $\widetilde{w}(\widehat{\lambda})=w(\widehat{\lambda})$. For $\lambda'\in\mathcal{L}_{d}$ with $d_{\mathcal{L}}(\widehat{\lambda},\lambda')<\widehat{r}$, by (W1), $w(\lambda')-\psi(\iota_{d}(\lambda'))\le\widetilde{w}(\lambda')-\psi(\iota_{d}(\lambda'))\le\widetilde{w}(\widehat{\lambda})-\psi(\iota_{d}(\widehat{\lambda}))=w(\widehat{\lambda})-\psi(\iota_{d}(\widehat{\lambda}))$. So $w-\psi\circ\iota_{d}$ has a local maximum at $\widehat{\lambda}$, and since $w$ is a viscosity subsolution, Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws §subsolution gives (Goal) with $w(\widehat{\lambda})=\widetilde{w}(\widehat{\lambda})$ in place of $\widetilde{w}(\widehat{\lambda})$, which is (Goal).

Case B: $\widetilde{w}(\widehat{\lambda})\ne w(\widehat{\lambda})$. By (W2), $d_{\mathcal{L}}(\widehat{\lambda},\lambda_{0})<r_{0}/2$, so $\widetilde{w}(\widehat{\lambda})$ is the maximum of $w(\widehat{\lambda})$ and $F_{2}(\iota_{d}(\widehat{\lambda}))+\delta$ and, being different from the first, equals the second. Let $\widehat{t}=\min\{\widehat{r},r_{0}/2\}$ and $\lambda'\in\mathcal{L}_{d}$ with $d_{\mathcal{L}}(\widehat{\lambda},\lambda')<\widehat{t}$. Then $d_{\mathcal{L}}(\lambda',\lambda_{0})<r_{0}$ by the triangle inequality, so $\widetilde{w}(\lambda')\ge F_{2}(\iota_{d}(\lambda'))+\delta$, and

$$F_{2}(\iota_{d}(\lambda'))+\delta-\psi(\iota_{d}(\lambda'))\le\widetilde{w}(\lambda')-\psi(\iota_{d}(\lambda'))\le\widetilde{w}(\widehat{\lambda})-\psi(\iota_{d}(\widehat{\lambda}))=F_{2}(\iota_{d}(\widehat{\lambda}))+\delta-\psi(\iota_{d}(\widehat{\lambda})).$$

Let $\Xi=\psi-F_{2}:E_{d}\to\mathbb{R}$. The display says $\Xi(\iota_{d}(\widehat{\lambda}))\le\Xi(\iota_{d}(\lambda'))$, so $\Xi\circ\iota_{d}$ has a local minimum at $\widehat{\lambda}$ relative to $\mathcal{L}_{d}$ (Local Minimum of a Function Relative to a Subset of a Metric Space). As $\psi$ and $F_{2}$ are differentiable at $\iota_{d}(\widehat{\lambda})$, Constants, Sums, Scalar Multiples and Differences of Differentiable Functions on an Open Subset of a Real Inner Product Space §difference shows that $\Xi$ is differentiable there with gradient $\widehat{q}=D\psi(\iota_{d}(\widehat{\lambda}))-DF_{2}(\iota_{d}(\widehat{\lambda}))$, and Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones §minimum gives $0\le\langle\widehat{q},v\rangle_{d}$ for $v\in T^{+}_{\widehat{\lambda}}$ and $\langle\widehat{q},v\rangle_{d}=0$ for $v\in T^{\pm}_{\widehat{\lambda}}$. Since $D\psi(\iota_{d}(\widehat{\lambda}))=DF_{2}(\iota_{d}(\widehat{\lambda}))+\widehat{q}$, Elementary Identities in a Real Inner Product Space §bilinear and symmetry give $\langle D\psi(\iota_{d}(\widehat{\lambda})),y\rangle_{d}=\langle DF_{2}(\iota_{d}(\widehat{\lambda})),y\rangle_{d}+\langle\widehat{q},y\rangle_{d}$ for $y\in E_{d}$. The difference $D\psi(\iota_{d}(\widehat{\lambda}))-DF_{2}(\iota_{d}(\widehat{\lambda}))=\widehat{q}$ satisfies $\langle\widehat{q},v\rangle_{d}=0$ for every $v\in T^{\pm}_{\widehat{\lambda}}$, so Admissible Hamiltonians on Unitary Laws §tangential, applied with $\widehat{\lambda}$, $p=D\psi(\iota_{d}(\widehat{\lambda}))$ and $p'=DF_{2}(\iota_{d}(\widehat{\lambda}))$, gives $H(\widehat{\lambda},D\psi(\iota_{d}(\widehat{\lambda})))=H(\widehat{\lambda},DF_{2}(\iota_{d}(\widehat{\lambda})))$. Also $\widehat{\Theta}\widehat{\lambda}\in T^{+}_{\widehat{\lambda}}$ by The Free Unitary Heat Generator Points into the Space of Unitary Laws §inward, so $\langle D\psi(\iota_{d}(\widehat{\lambda})),\widehat{\Theta}\widehat{\lambda}\rangle_{d}\ge\langle DF_{2}(\iota_{d}(\widehat{\lambda})),\widehat{\Theta}\widehat{\lambda}\rangle_{d}$, and as $\beta^{2}/2\ge0$ the left side of (Goal) is at most

$$\rho\bigl(F_{2}(\iota_{d}(\widehat{\lambda}))+\delta\bigr)+H\bigl(\widehat{\lambda},DF_{2}(\iota_{d}(\widehat{\lambda}))\bigr)-\frac{\beta^{2}}{2}\bigl\langle DF_{2}(\iota_{d}(\widehat{\lambda})),\widehat{\Theta}\widehat{\lambda}\bigr\rangle_{d}=Q(\widehat{\lambda})+\rho\,\delta+f(\widehat{\lambda})<-\eta_{0}+\frac{\eta_{0}}{2}+f(\widehat{\lambda})<f(\widehat{\lambda}),$$

by (B3), as $d_{\mathcal{L}}(\widehat{\lambda},\lambda_{0})<r_{0}$. This is (Goal).

Step 6 (A point where the bump is strict). By Properties of the Lower Semicontinuous Envelope, by Duality §approximation, choose a sequence $(\lambda_{k})_{k\in\mathbb{N}}$ converging to $\lambda_{0}$ in $(\mathcal{L}_{d},d_{\mathcal{L}})$ with $(w(\lambda_{k}))_{k}$ converging to $w_{*}(\lambda_{0})$. By (C2), $(F_{2}(\iota_{d}(\lambda_{k})))_{k}$ converges to $F_{2}(\iota_{d}(\lambda_{0}))=w_{*}(\lambda_{0})$, so $(w(\lambda_{k})-F_{2}(\iota_{d}(\lambda_{k})))_{k}$ converges to $0$ by claim 3 of Arithmetic of Limits of Real Sequences. Choose $N_{1}\in\mathbb{N}$ with $w(\lambda_{k})-F_{2}(\iota_{d}(\lambda_{k}))<\delta$ for $k\ge N_{1}$ and $N_{2}\in\mathbb{N}$ with $d_{\mathcal{L}}(\lambda_{k},\lambda_{0})<r_{0}/2$ for $k\ge N_{2}$, and let $\lambda'=\lambda_{k}$ with $k=\max\{N_{1},N_{2}\}$. Then $d_{\mathcal{L}}(\lambda',\lambda_{0})<r_{0}/2<r$, and $\widetilde{w}(\lambda')\ge F_{2}(\iota_{d}(\lambda'))+\delta>w(\lambda')$.

Conclusion. By Steps 4 to 6, $\widetilde{w}$ is a viscosity subsolution of The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise §equation with Hamiltonian $H$, with $w\le\widetilde{w}\le g$ on $\mathcal{L}_{d}$ (W1), $\widetilde{w}(\lambda)=w(\lambda)$ whenever $r\le d_{\mathcal{L}}(\lambda,\lambda_{0})$ (W2), and $w(\lambda')<\widetilde{w}(\lambda')$ for some $\lambda'$ with $d_{\mathcal{L}}(\lambda',\lambda_{0})<r$. This proves the clause (Bump).