Proof of Information Bound on the Synthetic Copy with Mean-Field Data: Prior Energy of the Profile, Observation Information Along the Profile Response, and the Non-Close Record Mass

Throughout, $\mathbb{E}$ is the integral with respect to $P$ on $(\Omega,\mathcal{F},P)$ (Expectation, Variance, and Moments), and on either of the measure spaces $(\Omega,\mathcal{F},P)$ and $(\mathbf{R},\mathcal{R},\rho)$ linearity refers to claim 2 of Linearity and Monotonicity of the Lebesgue Integral (finite linear combinations of integrable functions are integrable, by induction on the number of terms, the integral is linear, $|\int f|\le\int|f|$, and $\int f\le\int g$ when $f\le g$ pointwise), monotonicity to claim 1 of that theorem (for nonnegative measurable functions), and domination to the following consequence of claims 1 and 2 there and of Integrable Function and the Lebesgue Integral (a measurable $f$ is integrable if and only if $\int|f|<\infty$): if $f$ is measurable and $|f|\le g$ pointwise with $g$ integrable, then $f$ is integrable and $|\int f|\le\int g$. Sums, products, absolute values and indicators of measurable real functions are measurable by claims 1--4 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; monotonicity of the square root means claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to nonnegative square roots ($0\le u\le v$ implies $u^{1/2}\le v^{1/2}$, and for $u\ge0$, $u\le v^{1/2}$ if and only if $u^{2}\le v$); and the triangle inequality is claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers together with claim 1 there. We abbreviate $\varrho_q=\varrho_q(\mathsf{K}_q)$ for the random variables of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass.

Step 0 (constants and the integrand of $\mathsf{Q}^{\mathrm{cl}}$). No $\omega$ is fixed in this step. (a) Positivity of $\varepsilon_0$. By claim 1 of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (valid for every $\omega$), $N\underline{b}\le\lambda^{\sharp,\omega,\upsilon}_t(r)\le N\tilde{B}$ at any point $(t,r)\in[0,T]\times\mathbf{R}$ and channel $\upsilon$ (there are such points: $T>0$, $\tilde{l}\ge1$, and $\mathbf{R}$ contains the empty record, the cell $C_\emptyset$ having $\rho(C_\emptyset)=1$ in The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), the upper bound because $N\tilde{B}$ is a bound of the causal intensity $\lambda^{\sharp,\omega}$; hence $\tilde{B}\ge\underline{b}>0$ by (OC). The derivative bounds $K$ and $\tilde{K}$ are nonnegative by Twice Continuously Differentiable Extension of a Transition-Rate Family and Twice Continuously Differentiable Extension of an Observation-Rate Family, so $\Gamma=\sqrt{l}(\tilde{B}+\tilde{K})>0$; $A_0=\sqrt{2}(\mathsf{m}+l(l-1)(D+\mathsf{m}))\exp(\sqrt{2}l(l-1)\Lambda_1T)>0$ by (W), as $\mathsf{m}\ge1$, $l\ge2$, $D\ge0$ and $\exp>0$ (claim 2 of Basic Properties of the Exponential Function); therefore $\varepsilon_0=\Gamma A_0/(N\underline{b})>0$, and $\varepsilon_0\le\tfrac12\le1$ by (P). Consequently $\mathsf{E}^{\star}_N\ge0$, $\exp(9\mathsf{E}^{\star}_N)\ge1$ (claim 4 of Basic Properties of the Exponential Function), so $\mathsf{e}^{\star}_N$ is defined and $\mathsf{e}^{\star}_N\ge0$; likewise $\bar{E}_N\ge0$ and $\mathsf{c}_N=\exp(\bar{E}_N)-1\ge0$; and $\kappa_0\ge1$, its formula being $1$ plus a nonnegative term. (b) The integrand of $\mathsf{Q}^{\mathrm{cl}}$. By the setting of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound, $z\cdot(\tilde{D}(x)z)=\sum_{\upsilon=1}^{\tilde{l}}(g_\upsilon(x)\cdot z)^{2}/\tilde{b}^\upsilon(x)$ for $x\in\Delta^l$ and $z\in\mathbb{R}^l$, where $g_\upsilon(x)=(\partial_1\bar{\tilde{b}}^\upsilon(x),\dots,\partial_l\bar{\tilde{b}}^\upsilon(x))$ and $\tilde{b}^\upsilon(x)=\bar{\tilde{b}}^\upsilon(x)\ge\underline{b}>0$ on $\Delta^l$ (claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift and (OC)). By claims (i) and (ii) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift, $\bar{\tilde{b}}^\upsilon$ and each $\partial_\gamma\bar{\tilde{b}}^\upsilon$ are $C^1$ maps on the open set $\tilde{U}\supseteq\Delta^l$ in the sense used there, whose first clause is that each such map is continuous at every point of $\tilde{U}$; and $|\partial_\gamma\bar{\tilde{b}}^\upsilon|\le\tilde{B}+\tilde{K}$ on $\Delta^l$. A real function $g$ on $\tilde{U}$ continuous at every point is sequentially continuous on the nonempty set $\Delta^l$ (nonempty by Probability Simplex) in the sense of Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable: if $x_n,x\in\Delta^l$ with Euclidean distance $d(x_n,x)\to0$, then given $\varepsilon>0$ the continuity of $g$ at $x$ provides $\delta>0$ with $|g(y)-g(x)|<\varepsilon$ whenever $y\in\tilde{U}$ and $d(y,x)<\delta$, and by Limit of a Sequence of Real Numbers there is $n_0$ with $d(x_n,x)<\delta$ for $n\ge n_0$, so $|g(x_n)-g(x)|<\varepsilon$ for $n\ge n_0$, i.e. $g(x_n)\to g(x)$. Moreover $|g_\upsilon(x)|\le\Gamma$ on $\Delta^l$: by claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ and claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, $|g_\upsilon(x)|^{2}=\sum_{\gamma=1}^{l}(\partial_\gamma\bar{\tilde{b}}^\upsilon(x))^{2}\le l(\tilde{B}+\tilde{K})^{2}=\Gamma^{2}$ (each square being at most $(\tilde{B}+\tilde{K})^{2}$ by the monotonicity of the square root), and the monotonicity of the square root gives $|g_\upsilon(x)|\le\Gamma$. The map $t\mapsto S_t$ has measurable components and values in $\Delta^l$, so by Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable the maps $t\mapsto\partial_\gamma\bar{\tilde{b}}^\upsilon(S_t)$ and $t\mapsto\tilde{b}^\upsilon(S_t)$ are measurable on $[0,T]$, the latter with values in $[\underline{b},\infty)$, so that $t\mapsto1/\tilde{b}^\upsilon(S_t)$ is measurable by the same lemma with $d=1$ (the reciprocal being sequentially continuous on the nonempty set $[\underline{b},\infty)$: if $b_n,b\in[\underline{b},\infty)$ with $|b_n-b|\to0$, the Euclidean distance on $\mathbb{R}^{1}$ being the absolute value of the difference, then $b_n\ne0$, $b\ne0$, and claim 4 of Arithmetic of Limits of Real Sequences applied to the constant sequence $1$ and to $(b_n)$ gives $1/b_n\to1/b$) with values in $(0,1/\underline{b}]$. As $\bar\psi$ has measurable components bounded by $\mathsf{M}$, the function $t\mapsto\bar\psi_t\cdot(\tilde{D}(S_t)\bar\psi_t)=\sum_\upsilon(g_\upsilon(S_t)\cdot\bar\psi_t)^{2}/\tilde{b}^\upsilon(S_t)$ is a finite sum of products of measurable functions, hence measurable (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); it is nonnegative as a sum of squares times positive factors; and it is bounded by $\tilde{l}\,\Gamma^{2}\mathsf{M}^{2}/\underline{b}$, since $|g_\upsilon(S_t)\cdot\bar\psi_t|\le\Gamma\mathsf{M}$ by Cauchy-Schwarz Inequality for the Euclidean Dot Product. Hence its integral over $[0,T]$ (of a bounded measurable function against the finite measure $\lambda_{[0,T]}$ of claim 1 of Restricted Lebesgue Measure and Integral Toolkit on a Compact Interval) is a well-defined nonnegative real number by monotonicity, and $\mathsf{Q}^{\mathrm{cl}}\ge0$: $\mathsf{e}_F\ge0$ and $\epsilon_\psi\ge0$ by their formulas in claims 2 and 3 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data, every ingredient being nonnegative and every denominator positive ($\Lambda,\mathsf{M},\varepsilon_S,\varepsilon_{\mathrm{ctl}},B,D,K,\tilde{K},\Lambda_1,\Lambda_2,\Lambda_3,\Lambda_{\mathcal{E}},\mu_{\max},\lVert w\rVert_1\ge0$; $A_0,\Gamma>0$ by (a); $\mu_{\min}>0$, the cell lengths being positive; $N\ge1$; $\underline{b}>0$; and $\exp>0$), hence $\kappa\ge0$ by its formula, and every remaining factor of $\mathsf{Q}^{\mathrm{cl}}$ is nonnegative, with $\zeta>0$. This proves the first sentence of claim 1.

Step 1 (claim 1, prior part). Fix $q\in\mathsf{L}$. By the definition of $\mathsf{j}_q$ in Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass, $\mathsf{j}_q=\sum_{i=1}^{\mathsf{m}}\binom{\mathsf{m}}{i}^{2}i!\,\mu_q^{-i}$, and by claim 3 of The Poisson Removal Ratio for Moves of Several Points: Move Score, Mean, Exact Second Moment, Move Information, and Pointwise Bounds, applied with the parameter $\mu_q>0$, the move size $\mathsf{m}$, the single move $\mathsf{m}$ and the single weight $1$ (unrelated to the injection weights $w$ of the present setting), this sum is the move information of that claim, whose upper bound (available since the single weight is nonzero) gives $\mathsf{j}_q\le\frac{\mathsf{m}^{2}}{\mu_q}+\frac{\mathsf{m}^{4}}{2\mu_q^{2}}\exp\Bigl(\frac{\mathsf{m}^{2}}{\mu_q}\Bigr)=\frac{\mathsf{m}^{2}}{\mu_q}\Bigl(1+\frac{\mathsf{m}^{2}}{2\mu_q}\exp\Bigl(\frac{\mathsf{m}^{2}}{\mu_q}\Bigr)\Bigr)\le\frac{\mathsf{m}^{2}}{\mu_q}\,\kappa_0 ,$ the last step because $\mu_q\ge\mu_{\min}$ and $\exp$ is increasing (claim 4 of Basic Properties of the Exponential Function). Multiplying by $\mu_q>0$ gives $\mu_q\mathsf{j}_q\le\kappa_0\mathsf{m}^{2}$, and dividing by $\mu_q\ge\mu_{\min}$ gives $\mathsf{j}_q\le\kappa_0\mathsf{m}^{2}/\mu_{\min}$ for every $q$, hence for the maximum $\bar{\mathsf{j}}$. Finally, the weights $w$ are the injection weights of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection formed from the mean-field label rates $\phi_c$ and the profile $\varpi$ (admissible by claim 1 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data), and $\mathcal{P}$ is the profile energy of that lemma (identified in claim 1 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data); claim 3 of Injection Weights from a Bounded Profile Along Mean-Field Clocks: Label-Rate Form of the Fluctuation Covariance, Weight Bounds, the Prior Quadratic Form, and the Step-Function Injection, applied with its constant taken to be $\kappa_0$ and with $\mathsf{h}_q=\mathsf{j}_q\ge0$, gives $\sum_qw_q^{2}\mathsf{j}_q\le\kappa_0N\mathcal{P}$.

Step 2 (claim 2). By claim 2 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound, each $\varrho_q$ is a random variable with $\mathbb{E}[\varrho_q]=1$ and $\mathbb{E}[\varrho_q^{2}]=1+\mathsf{j}_q<\infty$ (the identity by the definition of $\mathsf{j}_q$ in Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass); in particular $\varrho_q$ and $\varrho_q^{2}$ are integrable, hence so is $(\varrho_q-1)^{2}=\varrho_q^{2}-2\varrho_q+1$ by linearity, with $\mathbb{E}[(\varrho_q-1)^{2}]=(1+\mathsf{j}_q)-2+1=\mathsf{j}_q$.

A Cauchy-Schwarz inequality. Let $X,Y$ be random variables with $X^{2}$ and $Y^{2}$ integrable. We claim that $|XY|$ is integrable with $\mathbb{E}|XY|\le(\mathbb{E}[X^{2}])^{1/2}(\mathbb{E}[Y^{2}])^{1/2}$. Apply claim 1 of Weighted Cauchy-Schwarz Inequality on a Measure Space and the Symmetrised Score Functional: Bounds and Averaging on $(\Omega,\mathcal{F},P)$ with $\beta=X^{2}$ (measurable, nonnegative, finite integral) and $\alpha=|XY|$ (measurable, and $\alpha=0$ wherever $\beta=0$, i.e. wherever $X=0$). The function $q$ of that claim equals $\alpha^{2}/\beta=Y^{2}$ on $\{X\neq0\}$ and $0$ on $\{X=0\}$, so $0\le q\le Y^{2}$ pointwise and $\int q\,dP\le\mathbb{E}[Y^{2}]<\infty$ by monotonicity. Hence $\alpha$ is integrable and $(\mathbb{E}|XY|)^{2}\le\mathbb{E}[X^{2}]\int q\,dP\le\mathbb{E}[X^{2}]\,\mathbb{E}[Y^{2}]=\bigl((\mathbb{E}[X^{2}])^{1/2}(\mathbb{E}[Y^{2}])^{1/2}\bigr)^{2}$, and the claim follows from the monotonicity of the square root, $\mathbb{E}|XY|$ being nonnegative.

Fix $q,q'\in\mathsf{L}$. Pointwise, $\varrho_q\varrho_{q'}-1=\varrho_q(\varrho_{q'}-1)+(\varrho_q-1)$, so by the triangle inequality $|\varrho_q\varrho_{q'}-1|\le|\varrho_q(\varrho_{q'}-1)|+|\varrho_q-1|$. The Cauchy-Schwarz inequality with $X=\varrho_q$, $Y=\varrho_{q'}-1$ gives $\mathbb{E}|\varrho_q(\varrho_{q'}-1)|\le(1+\mathsf{j}_q)^{1/2}\mathsf{j}_{q'}^{1/2}$, and with $X=1$, $Y=\varrho_q-1$ (here $\mathbb{E}[X^{2}]=P(\Omega)=1$) it gives $\mathbb{E}|\varrho_q-1|\le\mathsf{j}_q^{1/2}$; both functions are integrable, so by linearity $|\varrho_q\varrho_{q'}-1|$ is integrable with $\mathbb{E}|\varrho_q\varrho_{q'}-1|\le(1+\mathsf{j}_q)^{1/2}\mathsf{j}_{q'}^{1/2}+\mathsf{j}_q^{1/2}$. Since $\mathsf{j}_q\le\bar{\mathsf{j}}$ and $\mathsf{j}_{q'}\le\bar{\mathsf{j}}$, the monotonicity of the square root gives $(1+\mathsf{j}_q)^{1/2}\mathsf{j}_{q'}^{1/2}+\mathsf{j}_q^{1/2}\le(1+\bar{\mathsf{j}})^{1/2}\bar{\mathsf{j}}^{1/2}+\bar{\mathsf{j}}^{1/2}=\mathsf{j}^{\star}$.

By claim 2 of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass, each $\omega\mapsto C^{\omega}_{qq'}$ is $\mathcal{F}$-measurable with finite values, so $\omega\mapsto\mathcal{C}^{\omega}$ is measurable as a finite linear combination; by claim 3 there, $|C^{\omega}_{qq'}|\le\mathsf{c}_N$ for $\omega\in G$, so $|\mathcal{C}^{\omega}|\le\sum_{q,q'}|w_q||w_{q'}|\mathsf{c}_N=\lVert w\rVert_1^{2}\mathsf{c}_N$ on $G$ by the triangle inequality. Claim 3 of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass also states that $\mathcal{C}=\sum_{q,q'}w_qw_{q'}\mathbb{E}[\mathbf{1}_G\varrho_q\varrho_{q'}C^{\omega}_{qq'}]$ is a well-defined real number, equal to that sum. For fixed $q,q'$ write, pointwise on $\Omega$, $\mathbf{1}_G\varrho_q\varrho_{q'}C^{\omega}_{qq'}=\mathbf{1}_GC^{\omega}_{qq'}+\mathbf{1}_G(\varrho_q\varrho_{q'}-1)C^{\omega}_{qq'} .$ The first function on the right is measurable and bounded by $\mathsf{c}_N$ in absolute value, hence integrable by domination ($P$ is finite, constants being integrable with $\mathbb{E}[c]=cP(\Omega)=c$ by Simple Function and Its Integral); the second is measurable with absolute value at most $\mathsf{c}_N|\varrho_q\varrho_{q'}-1|$, which is integrable, so by domination it is integrable with $|\mathbb{E}[\mathbf{1}_G(\varrho_q\varrho_{q'}-1)C^{\omega}_{qq'}]|\le\mathsf{c}_N\mathsf{j}^{\star}$. By linearity, $\mathbb{E}[\mathbf{1}_G\varrho_q\varrho_{q'}C^{\omega}_{qq'}]=\mathbb{E}[\mathbf{1}_GC^{\omega}_{qq'}]+\mathbb{E}[\mathbf{1}_G(\varrho_q\varrho_{q'}-1)C^{\omega}_{qq'}]$. Multiplying by $w_qw_{q'}$, summing over $q,q'$, and using linearity once more ($\mathbf{1}_G\mathcal{C}^{\omega}=\sum_{q,q'}w_qw_{q'}\mathbf{1}_GC^{\omega}_{qq'}$ is integrable with $\mathbb{E}[\mathbf{1}_G\mathcal{C}^{\omega}]=\sum_{q,q'}w_qw_{q'}\mathbb{E}[\mathbf{1}_GC^{\omega}_{qq'}]$) and the triangle inequality, $\mathcal{C}=\mathbb{E}[\mathbf{1}_G\mathcal{C}^{\omega}]+\sum_{q,q'}w_qw_{q'}\,\mathbb{E}[\mathbf{1}_G(\varrho_q\varrho_{q'}-1)C^{\omega}_{qq'}]\le\mathbb{E}[\mathbf{1}_G\mathcal{C}^{\omega}]+\sum_{q,q'}|w_q||w_{q'}|\,\mathsf{c}_N\mathsf{j}^{\star}=\mathbb{E}[\mathbf{1}_G\mathcal{C}^{\omega}]+\mathsf{c}_N\lVert w\rVert_1^{2}\mathsf{j}^{\star}.$

Step 3 (claim 3). Let $\omega\in G$; by (G) and (G$'$), $\omega\in G^{\mathsf{m}}\subseteq G_{L,D}$, $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$, and $\mathsf{K}_q(\omega)\ge\mathsf{m}$ for every $q$. We verify the hypotheses of Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum, claim 4, with the record space $(\mathbf{R},\mathcal{R},\rho)$, the base intensity of that lemma taken to be $\lambda^{\sharp,\omega}$ with bound $N\tilde{B}$ and lower bound $N\underline{b}$, the null set $\mathsf{N}=\mathbf{R}\setminus\mathsf{T}_\omega$, its perturbation parameter taken to be $\varepsilon_0$ (the letter $\eta$ of that lemma is not used here), $n=d$, and its perturbed intensities taken to be $\lambda^{-q,\omega}$ ($q\in\mathsf{L}$, identified with $\{1,\dots,d\}$ through the fixed bijection), each with bound $N\tilde{B}$ (the letters $\mu$, $\mu_q$ of that lemma denote intensities there and are not the cell lengths $\mu_q$ of the present setting). By claim 1 of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances, $\lambda^{\sharp,\omega}$ is a causal intensity with bound $N\tilde{B}$ and $\lambda^{\sharp,\omega,\upsilon}\ge N\underline{b}$ everywhere, where $N\underline{b}>0$ by (OC); each $\lambda^{-q,\omega}$ is a causal intensity with bound $N\tilde{B}$; and $\mathsf{T}_\omega\in\mathcal{R}$, so $\mathsf{N}\in\mathcal{R}$ with $\rho(\mathsf{N})=0$. By claim 2 of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances (applicable as $\omega\in G_{L,D}$ and $\mathsf{K}_q(\omega)\ge\mathsf{m}$), for every $q$, $r\in\mathsf{T}_\omega$, $\upsilon$ and $t$, $|\lambda^{-q,\omega,\upsilon}_t(r)-\lambda^{\sharp,\omega,\upsilon}_t(r)|\le\Gamma A_0=\varepsilon_0\,N\underline{b}\le\varepsilon_0\,\lambda^{\sharp,\omega,\upsilon}_t(r),$ so $\lambda^{-q,\omega}$ is a relative $\varepsilon_0$-perturbation of $\lambda^{\sharp,\omega}$ off $\mathsf{N}$ (the parameter $\varepsilon_0$ being positive by Step 0); and $\varepsilon_0\le\tfrac12\le1$ by (P). The pair exponents and pair covariances of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form for this base and this perturbed family are, by the definitions in Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances, exactly $E^{\omega}_{qq'}$ and $C^{\omega}_{qq'}$, the likelihood $\ell_{\lambda^{\sharp,\omega}}$ is $\ell^{\sharp,\omega}$ (claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record and claim 4 of The Record-Driven Causal Intensity of the Open-Loop Aggregate Solution: Joint Measurability of the Record-Frozen Control, the Regularised Recursion Path, Non-Anticipation, and Measurability of the Likelihood), and the quadratic form $\sum_{q,q'}w_qw_{q'}E^{\omega}_{qq'}$ is $\mathcal{Q}^{\omega}$ by its definition in claim 3 of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound. The two constants of Relative Perturbations of a Causal Intensity: Pair-Exponent Bound, the Pair Intensity as a Relative Perturbation, Replacement of the Pair Likelihood by the Base Likelihood, and the Weighted Pair-Covariance Sum (there written $\mathsf{E}_\eta$ and $\mathsf{e}_\eta$), formed with the bound $N\tilde{B}$ and the perturbation parameter $\varepsilon_0$, are then $\tilde{l}TN\tilde{B}\varepsilon_0^{2}=\mathsf{E}^{\star}_N$ and $\mathsf{e}^{\star}_N$. Claim 4 of that lemma therefore yields that $\mathcal{Q}^{\omega}$ is $\mathcal{R}$-measurable and bounded, that $\ell^{\sharp,\omega}\mathcal{Q}^{\omega}$ is integrable, and that $\mathcal{C}^{\omega}=\sum_{q,q'}w_qw_{q'}C^{\omega}_{qq'}\le\int_{\mathbf{R}}\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\,d\rho+\lVert w\rVert_1^{2}\mathsf{e}^{\star}_N$, the first inequality of claim 3.

For the second inequality, recall that $\mathcal{Q}^{\omega}(r)\ge0$ for every $r$ (claim 3 of Weighted Response and Pair-Exponent Quadratic Form on the Synthetic Copy: the Linearised Injection Equation, Comparison with the Profile Injection, and the Observation-Information Bound). Let $r\in\mathsf{R}^{\mathrm{cl}}_\omega$. Then $r\in\mathsf{T}_\omega$, $\omega\in G^{\mathsf{m}}$, and (CL) provides the closeness bounds $|\bar\Sigma^{\sharp,r}_t(\omega)-S_t|\le\varepsilon_S$ ($t\in[0,T]$) and $\mathsf{D}^{r}_{\mathrm{ctl}}\le\varepsilon_{\mathrm{ctl}}$, so claim 4 of Closeness to a Mean-Field Pair on the Synthetic Copy: Clock Discrepancy, Gronwall Comparison of the Weighted Response with the Profile Response, and the Observation-Information Bound with Mean-Field Data with the fixed $\zeta$ gives $\mathcal{Q}^{\omega}(r)\le\mathsf{Q}^{\mathrm{cl}}$. Let instead $r\in\mathsf{T}_\omega\setminus\mathsf{R}^{\mathrm{cl}}_\omega$. Then claim 3 of Uniform Pair-Exponent Bound on the Synthetic Copy: Removed-Clock Intensities, the Insertion Response on the Clock-Good Event, and Bounds on the Pair Covariances gives $|E^{\omega}_{qq'}(r)|\le\bar{E}_N$ for all $q,q'$, so by the triangle inequality $\mathcal{Q}^{\omega}(r)\le\sum_{q,q'}|w_q||w_{q'}|\bar{E}_N=\lVert w\rVert_1^{2}\bar{E}_N$. Consequently, pointwise on $\mathbf{R}$ (the three sets $\mathsf{R}^{\mathrm{cl}}_\omega$, $\mathsf{T}_\omega\setminus\mathsf{R}^{\mathrm{cl}}_\omega$ and $\mathsf{N}$ partition $\mathbf{R}$, and all terms are nonnegative), $\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\le\mathsf{Q}^{\mathrm{cl}}\,\ell^{\sharp,\omega}\mathbf{1}_{\mathsf{R}^{\mathrm{cl}}_\omega}+\lVert w\rVert_1^{2}\bar{E}_N\,\ell^{\sharp,\omega}\mathbf{1}_{\mathbf{R}\setminus\mathsf{R}^{\mathrm{cl}}_\omega}+\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\mathbf{1}_{\mathsf{N}} ,$ all functions being measurable and nonnegative ($\ell^{\sharp,\omega}$ is measurable with $\int_{\mathbf{R}}\ell^{\sharp,\omega}\,d\rho=1$ by claim 3 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record). The last integral vanishes: for every natural number $k$, the measurable function $\min(\ell^{\sharp,\omega}\mathcal{Q}^{\omega},k)\mathbf{1}_{\mathsf{N}}$ is at most $k\mathbf{1}_{\mathsf{N}}$, whose integral is $k\rho(\mathsf{N})+0\cdot\rho(\mathbf{R}\setminus\mathsf{N})=0$ (Simple Function and Its Integral, with its convention $0\cdot\infty=0$), so its integral is $0$ by monotonicity, and Monotone Convergence Theorem applied to the nondecreasing sequence $(\min(\ell^{\sharp,\omega}\mathcal{Q}^{\omega},k)\mathbf{1}_{\mathsf{N}})_k$, whose pointwise supremum is $\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\mathbf{1}_{\mathsf{N}}$, gives $\int_{\mathbf{R}}\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\mathbf{1}_{\mathsf{N}}\,d\rho=0$. Integrating the displayed inequality with monotonicity and the additivity of claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and using $\int\ell^{\sharp,\omega}\mathbf{1}_{\mathsf{R}^{\mathrm{cl}}_\omega}\,d\rho\le\int\ell^{\sharp,\omega}\,d\rho=1$ (monotonicity), $\mathsf{Q}^{\mathrm{cl}}\ge0$ and the definition of $\pi^{\mathrm{nc}}_\omega$, $\int_{\mathbf{R}}\ell^{\sharp,\omega}\mathcal{Q}^{\omega}\,d\rho\le\mathsf{Q}^{\mathrm{cl}}\cdot1+\lVert w\rVert_1^{2}\bar{E}_N\,\pi^{\mathrm{nc}}_\omega+0 .$

Step 4 (claim 4). By Step 3, for every $\omega\in G$, $\mathcal{C}^{\omega}\le\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\mathsf{e}^{\star}_N+\lVert w\rVert_1^{2}\bar{E}_N\pi^{\mathrm{nc}}_\omega$; hence pointwise on $\Omega$, $\mathbf{1}_G\mathcal{C}^{\omega}\le\bigl(\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\mathsf{e}^{\star}_N\bigr)\mathbf{1}_G+\lVert w\rVert_1^{2}\bar{E}_N\,\mathbf{1}_G\pi^{\mathrm{nc}}_\omega .$ Both sides are integrable: the left side by Step 2, and the right side because $\mathbf{1}_G$ is a bounded measurable function and $\mathbf{1}_G\pi^{\mathrm{nc}}$ is measurable by (CL) with values in $[0,1]$ ($P$ being finite, bounded measurable functions are integrable by domination). Taking expectations (linearity), with $\mathbb{E}[\mathbf{1}_G]=P(G)\le1$, $\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\mathsf{e}^{\star}_N\ge0$ and $\mathbb{E}[\mathbf{1}_G\pi^{\mathrm{nc}}]=\bar\pi^{\mathrm{nc}}$, $\mathbb{E}[\mathbf{1}_G\mathcal{C}^{\omega}]\le\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\mathsf{e}^{\star}_N+\lVert w\rVert_1^{2}\bar{E}_N\,\bar\pi^{\mathrm{nc}} .$ Combining with Step 2, $\mathcal{C}\le\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}(\mathsf{e}^{\star}_N+\bar{E}_N\bar\pi^{\mathrm{nc}}+\mathsf{c}_N\mathsf{j}^{\star})$, and with Step 1, $\sum_qw_q^{2}\mathsf{j}_q+\mathcal{C}\le\kappa_0N\mathcal{P}+\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}(\mathsf{e}^{\star}_N+\bar{E}_N\bar\pi^{\mathrm{nc}}+\mathsf{c}_N\mathsf{j}^{\star})$. Since $1+\delta>0$, inserting this into claim 3 of Assembly of the Symmetrised Move Information on the Synthetic Copy: Prior Part, Record Part, and the Bad Part with Exponentially Small Under-Likelihood Mass, $\mathsf{J}^{\mathrm{sym}}\le(1+\delta)\Bigl[\sum_qw_q^{2}\mathsf{j}_q+\mathcal{C}\Bigr]+2d\lVert w\rVert_1^{2}\mathsf{B}\le(1+\delta)\Bigl[\kappa_0N\mathcal{P}+\mathsf{Q}^{\mathrm{cl}}+\lVert w\rVert_1^{2}\bigl(\mathsf{e}^{\star}_N+\bar{E}_N\bar\pi^{\mathrm{nc}}+\mathsf{c}_N\mathsf{j}^{\star}\bigr)\Bigr]+2d\lVert w\rVert_1^{2}\mathsf{B},$ which is claim 4. $\blacksquare$