Proof 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

Throughout, fix $\omega\in\Omega$ and abbreviate $\mathsf{K}=\mathsf{K}(\omega)$; for $(c,j)\in\mathsf{L}$ with $\mathsf{K}_{c,j}\ge\mathsf{m}$ write $y^{(c,j)}=\mathsf{K}-\mathsf{m}e_{c,j}\in\mathbb{N}_0^{\mathsf{L}}$. Recall from 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 that $\lambda^{\sharp,\omega}$, $\lambda^{(y),\omega}$ and their likelihoods are the objects 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 for the clock families $\mathsf{P}^{\sharp}(\omega)$ and $\mathsf{P}^{(y)}(\omega)$ respectively, and that $\ell^{\sharp,\omega}=\ell_{\lambda^{\sharp,\omega}}$, $\ell^{(y),\omega}=\ell_{\lambda^{(y),\omega}}$.

Step 0 (measurability of $G_{L,D}$ and $\mathsf{T}_\omega$). Let $\mathsf{Q}=(\mathbb{Q}\cap[0,R])\cup\{R\}$, a countable set. We claim that a counting path $p$ satisfying $|p(u')-p(u)-(u'-u)|\le D$ for all $u\le u'$ in $\mathsf{Q}$ with $u'-u\le L$ satisfies the same bound for all real $0\le u\le u'\le R$ with $u'-u\le L$. Given such real $u\le u'$, choose for each natural number $n$ points $u_n,u'_n\in\mathsf{Q}$ with $u\le u_n<u+1/n$, $u'\le u'_n<u'+1/n$ and $u'_n-u_n\le u'-u$: if $u=u'$ take $u_n=u'_n$ (the bound is then trivial), so assume $u<u'$; if $u'<R$ take $u'_n$ rational in $[u',\min(u'+1/n,R))$ and then $u_n$ rational in $[u+(u'_n-u'),\min(u+1/n,u'_n))$, an interval which is nondegenerate for $n$ large because $u+(u'_n-u')<u'_n$ and $u'_n-u'<1/n$ (rationals being dense); if $u'=R$ take $u'_n=R$ and $u_n$ rational in $[u,\min(u+1/n,R))$. In all cases $u\le u_n\le u'_n$, $u_n<u+1/n$, $u'\le u'_n<u'+1/n$ and $u'_n-u_n\le u'-u$. By right-continuity and monotonicity of counting paths (clauses 2 and 3 of Counting Path and Its Jump Times), $p(u_n)\to p(u)$ and $p(u'_n)\to p(u')$: $p(u)\le p(u_n)\le p(s)$ for every $s>u$ once $u_n<s$, and $p(u)$ is the greatest lower bound of the $p(s)$, $s>u$, the values being integers. Since also $u'_n-u_n\to u'-u$, the bound passes to the limit by Order Properties of Limits of Real Sequences. Consequently $G_{L,D}=\Omega^{U}_0\cap\bigcap_{c\in\mathcal{L}}\ \bigcap_{u\le u'\in\mathsf{Q},\ u'-u\le L}\bigl\{\omega:\ |\mathsf{P}^{\sharp,c}_{u'}(\omega)-\mathsf{P}^{\sharp,c}_{u}(\omega)-(u'-u)|\le D\bigr\},$ a countable intersection of events ($\Omega^{U}_0\in\mathcal{U}\subseteq\mathcal{F}$ and each $\mathsf{P}^{\sharp,c}_u$ is $\mathcal{F}$-measurable by claims 1 and 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), so $G_{L,D}\in\mathcal{F}$. Next, $\mathsf{G}^{\sharp}\in\mathcal{R}\otimes\mathcal{F}$ and $\mathsf{G}^{(y)}\in\mathcal{R}\otimes\mathcal{U}\subseteq\mathcal{R}\otimes\mathcal{F}$ by claim 1 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 (applied as in 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 sections at $\omega$ of members of $\mathcal{R}\otimes\mathcal{F}$ lie in $\mathcal{R}$ (the sets whose section at $\omega$ lies in $\mathcal{R}$ form a $\sigma$-algebra containing the measurable rectangles, hence the product $\sigma$-algebra by Intersections of Sigma-Algebras and Minimality of the Generated Sigma-Algebra); $\mathsf{T}_\omega$ is a finite intersection of such sections, so $\mathsf{T}_\omega\in\mathcal{R}$.

Step 1 (claim 1). By claim 3 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, $\lambda^{\sharp,\omega}$ and every $\lambda^{(y),\omega}$ are causal intensities with bound $N\tilde{B}$, and by its last assertion together with (OC), $\lambda^{\sharp,\omega,\upsilon}\ge N\underline{b}$ everywhere. Hence each $\lambda^{-(c,j),\omega}$ is a causal intensity with bound $N\tilde{B}$. If $\mathsf{K}_{c,j}\ge\mathsf{m}$, then $\ell^{-(c,j),\omega}=\ell^{(y^{(c,j)}),\omega}=\ell_{\lambda^{-(c,j),\omega}}$ by the definition of the removed-clock likelihood in Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound; if $\mathsf{K}_{c,j}<\mathsf{m}$, then $\varrho_{c,j}(\mathsf{K}_{c,j})=0$ by The Poisson Removal Ratio for Moves of Several Points: Move Score, Mean, Exact Second Moment, Move Information, and Pointwise Bounds, so both sides of the asserted identity vanish. The hypotheses of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form hold with $\underline\mu=N\underline{b}>0$, $\bar\mu=\bar\mu_q=N\tilde{B}$ and $n=d$. For Good-Bad Splitting of the Integrated Symmetrised Score Functional: Chebyshev Bound for the Under-Likelihood Set, Transfer of Mass Between Densities, and the Split Bound: $\ell^{\sharp,\omega}>0$ by claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form, the likelihoods are nonnegative and $\mathcal{R}$-measurable with $\int\ell^{\sharp,\omega}\,d\rho=\int\ell_{\lambda^{-(c,j),\omega}}\,d\rho=1$ by 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 ratio factors $\varrho_{c,j}(\mathsf{K}_{c,j})$ are nonnegative reals. Its integrand $\Phi_w(\ell,(\mathsf{r}_q\ell_q)_q)$ then equals $\Phi_w(\ell^{\sharp,\omega}(r),(\varrho_{c,j}(\mathsf{K}_{c,j})\ell^{-(c,j),\omega}(r))_{(c,j)})=\Psi(r,\omega)$ by the identity just proved.

Step 2 (claim 2). Let $\omega\in G_{L,D}$, $(c,j)$ with $\mathsf{K}_{c,j}\ge\mathsf{m}$, $r\in\mathsf{T}_\omega$, and $y=y^{(c,j)}$. We apply Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect with the clock family $p=\mathsf{P}^{(y)}(\omega)$ (a family of counting paths by claim 2 of The Synthetic Copy: Independent Cell Structure, Deterministic-Count Clocks, the Copy Measure, and the Smoothed Joint Density of Parameter and Observation Record), the control path $a^r$ (a control path with values in $\mathcal{A}$ by claim 1 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 hypothesis on $h$), the point $x_0$, the label $c_0=c$, the move size $\mathsf{m}$, the extension $(U,W_\beta,\bar\beta)$, the real numbers $R$, $L$ and the tolerance $D+\mathsf{m}$, and the inserted points $u_1<\dots<u_{\mathsf{m}}$, the increasing arrangement of the points $U^{c,j}_i(\omega)$ with $\mathsf{K}_{c,j}-\mathsf{m}<i\le\mathsf{K}_{c,j}$. We check its hypotheses.

The inserted points. Since $\omega\in\Omega^{U}_0$, claim 1 of Pathwise Reduction of the Symmetrised Move Information on the Synthetic Copy: Removed-Clock Likelihoods, the Shifted Family, and the Averaging Bound shows that these $\mathsf{m}$ points are pairwise distinct, lie in $I_{c,j}\subseteq(0,R]$ (so $0<u_1<\dots<u_{\mathsf{m}}$), and differ from every point $U^{c,j'}_{i'}(\omega)$ with $1\le j'\le J_c$, $1\le i'\le y_{c,j'}$. The path $p^{c}=\mathsf{P}^{(y),c}(\omega)$ is $u\mapsto\sum_{j'}\sum_{i'\le y_{c,j'}}\mathbf{1}\{U^{c,j'}_{i'}(\omega)\le u\}$ (the factor $\mathbf{1}_{\Omega^{U}_0}$ in its definition being $1$ at $\omega$), a finite sum of indicators of distinct points; by clause 4 of Counting Path and Its Jump Times, $u>0$ is a jump time when $p^{c}(u)$ exceeds the least upper bound $p^{c}(u-)$ of the values on $[0,u)$, which happens exactly when $u$ is one of these points, so its jump times are exactly the points $U^{c,j'}_{i'}(\omega)$ ($i'\le y_{c,j'}$), and none of the $u_k$ is a jump time of $p^{c}$. By the same claim 1, $p^{+}=\mathsf{P}^{\sharp}(\omega)$: indeed $\mathsf{P}^{\sharp,c'}(\omega)=p^{c'}$ for $c'\neq c$ and $\mathsf{P}^{\sharp,c}_u(\omega)=p^{c}(u)+\#\{k:u_k\le u\}$ for all $u\ge0$.

The discrepancy hypothesis. Let $c'\in\mathcal{L}$ and $0\le u\le u'\le R$ with $u'-u\le L$. If $c'\neq c$, then $p^{c'}=\mathsf{P}^{\sharp,c'}(\omega)$ and the definition of $G_{L,D}$ gives $|p^{c'}(u')-p^{c'}(u)-(u'-u)|\le D\le D+\mathsf{m}$. If $c'=c$, then $p^{c}(u')-p^{c}(u)=\mathsf{P}^{\sharp,c}_{u'}(\omega)-\mathsf{P}^{\sharp,c}_{u}(\omega)-\#\{k:u<u_k\le u'\}$, and the last count lies in $\{0,\dots,\mathsf{m}\}$, so $|p^{c}(u')-p^{c}(u)-(u'-u)|\le D+\mathsf{m}$. Thus (D) holds with tolerance $D+\mathsf{m}$, and $A_0$ of the statement is the constant of the insertion lemma for this tolerance.

Conflict-freeness. Since $r\in\mathsf{T}_\omega$, $(r,\omega)\in\mathsf{G}^{(y)}$ and $(r,\omega)\in\mathsf{G}^{\sharp}$, i.e. the data $(p,a^r,x_0)$ and $(p^{+},a^r,x_0)$ are conflict-free; by claim 2(c) 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 the open-loop aggregate solutions for these data are $\Sigma=\bar\Sigma^{(y),r}(\omega)$ and $\Sigma^{+}=\bar\Sigma^{\sharp,r}(\omega)$.

Conclusion. By (W), $\Lambda_1TA_0<L$, so claim 2 of Shared-Clock Point Insertion into the Open-Loop Aggregate Solution: Exact Response Identity, Crude Bound, and Linearisation Defect gives $|N(\Sigma^{+}_t-\Sigma_t)|\le A_0$ for every $t\in[0,T]$, which is the first bound. For the left limits at $t\in(0,T]$: by claim 2(b) 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 there are $\delta,\delta'>0$ such that $\bar\Sigma^{\sharp,r}(\omega)$ is constant equal to $\bar\Sigma^{\sharp,r}_{t-}(\omega)$ on $[t-\delta,t)\cap[0,T]$ and $\bar\Sigma^{(y),r}(\omega)$ is constant equal to $\bar\Sigma^{(y),r}_{t-}(\omega)$ on $[t-\delta',t)\cap[0,T]$; evaluating the first bound at any $s$ in the nonempty intersection of these intervals gives the second bound; at $t=0$ both left limits equal $x_0$. Finally, both left limits lie in $\mathbb{G}_N\subseteq\Delta^l$, where $\tilde{b}^\upsilon$ agrees with $\bar{\tilde{b}}^\upsilon$ (claim (i) of Regularity and Derivative Bounds of the Extended Aggregate Observation Drift) and the latter satisfies $|\bar{\tilde{b}}^\upsilon(\Sigma)-\bar{\tilde{b}}^\upsilon(\Sigma')|\le\Gamma\,|\Sigma-\Sigma'|$ (claim (ii) there, the Euclidean distance of $\Sigma$ and $\Sigma'$ being $|\Sigma-\Sigma'|$); hence $|\lambda^{-(c,j),\omega,\upsilon}_t(r)-\lambda^{\sharp,\omega,\upsilon}_t(r)|=N\,\bigl|\tilde{b}^\upsilon(\bar\Sigma^{(y),r}_{t-}(\omega))-\tilde{b}^\upsilon(\bar\Sigma^{\sharp,r}_{t-}(\omega))\bigr|\le N\Gamma\frac{A_0}{N}=\Gamma A_0 .$ If $\mathsf{K}_{c,j}<\mathsf{m}$, then $\lambda^{-(c,j),\omega}=\lambda^{\sharp,\omega}$ by definition.

Step 3 (claim 3). Let $\omega\in G_{L,D}$, $q=(c,j)$, $q'=(c',j')$ and $r\in\mathsf{T}_\omega$. By claim 3 of Integer Power Products of Causal-Intensity Likelihoods: Pathwise Identity, Integral Bounds, and the Pair Identity (as recalled in Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form), $E^{\omega}_{qq'}(r)=\int_{[0,T]}\sum_{\upsilon\in V}\frac{\bigl(\lambda^{-q,\omega,\upsilon}_s(r)-\lambda^{\sharp,\omega,\upsilon}_s(r)\bigr)\bigl(\lambda^{-q',\omega,\upsilon}_s(r)-\lambda^{\sharp,\omega,\upsilon}_s(r)\bigr)}{\lambda^{\sharp,\omega,\upsilon}_s(r)}\,ds .$ By claim 2 (in both cases $\mathsf{K}_{c,j}\ge\mathsf{m}$ and $\mathsf{K}_{c,j}<\mathsf{m}$, the difference being $0$ in the latter) and Step 1, each summand has absolute value at most $(\Gamma A_0)^{2}/(N\underline{b})$ for every $s\in[0,T]$, so the integrand $f(s)$ satisfies $|f(s)|\le\tilde{l}\Gamma^{2}A_0^{2}/(N\underline{b})=:\bar{f}$ for all $s\in[0,T]$; it is measurable on $[0,T]$ because each intensity is $\mathcal{B}_{[0,T]}\otimes\mathcal{R}$-measurable (condition (i) of Causal Intensity on the Observation Record Space), so its section at $r$ is measurable in $s$ (the section argument of Step 0, with the roles of the two factors exchanged), and sums, products and quotients by the positive denominator are measurable (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions and Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable for the reciprocal); the integral defining $E^{\omega}_{qq'}(r)$ is well defined by claim 1 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form. Applying the monotonicity of claim 2 of Linearity and Monotonicity of the Lebesgue Integral to $f\le\bar f$ and to $-f\le\bar f$ gives $|E^{\omega}_{qq'}(r)|\le\int_{[0,T]}\bar f\,ds=T\bar f=\bar{E}_N$.

Step 4 (claim 4). Let $\omega\in G_{L,D}$ with $\rho(\mathbf{R}\setminus\mathsf{T}_\omega)=0$. By Step 0, $\mathsf{Z}=\mathbf{R}\setminus\mathsf{T}_\omega\in\mathcal{R}$, and $\rho(\mathsf{Z})=0$. By Step 3, $|E^{\omega}_{qq'}|\le\bar{E}_N$ off $\mathsf{Z}$, and claim 3 of Pair Expansion of the Weighted Likelihood-Ratio Square: Pair Covariances of Causal-Intensity Likelihoods, the Exact Expansion, and Its First-Order Form with $\bar{E}=\bar{E}_N$ gives the first and third bounds; the second follows from the first with $q'=q$ and from $\mathrm{Var}^{\omega}_{q}=C^{\omega}_{qq}\ge0$ (claim 1 of that lemma). $\blacksquare$