---
title: "Stability of Change-of-Numéraire Reweighting: An Exact Wasserstein Boundary"
authors:
  - name: "Shaosai Huang"
    affiliation: "Kspectra Research Inc., Toronto, Canada"
date: "2026-08"
status: "Working paper"
url: https://kspectra.ai/papers/numeraire-reweighting-wasserstein-boundary/
doi: 10.2139/ssrn.7383120
ssrn: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7383120
---

# Stability of Change-of-Numéraire Reweighting: An Exact Wasserstein Boundary

Shaosai Huang — Working paper, version of August 2026.

Links: [Web page](https://kspectra.ai/papers/numeraire-reweighting-wasserstein-boundary/) · [SSRN](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7383120)

> Converted by Kspectra Research from the LaTeX of the posted version. Section, theorem, equation and reference numbers match the PDF. Formulas are LaTeX; the paper's own macros are defined below.

## How to cite

```bibtex
@misc{huang2026stability,
  author = {Huang, Shaosai},
  title  = {{Stability of Change-of-Numéraire Reweighting: An Exact Wasserstein Boundary}},
  year   = {2026},
  month  = aug,
  note   = {Working paper, version of August 2026},
  doi    = {10.2139/ssrn.7383120},
  url    = {https://kspectra.ai/papers/numeraire-reweighting-wasserstein-boundary/}
}
```

## Macros

The formulas use these definitions from the paper's preamble:

```latex
\newcommand{\R}{\mathbb R}
\newcommand{\E}{\mathbb E}
\newcommand{\one}{\mathbf 1}
\newcommand{\dd}{\,\mathrm d}
\newcommand{\cP}{\mathcal P}
\newcommand{\cK}{\mathcal K}
\newcommand{\Ws}{W_s}
\newcommand{\Hplus}{[0,\infty)\times\R}
\newcommand{\Hopen}{(0,\infty)\times\R}
```

## Abstract

Reweighting a probability law by a positive numéraire and pushing forward the payoff-to-numéraire ratio yields the change-of-numéraire reweighting: the swap-rate law under the annuity measure, the numéraire-inversion involution of martingale optimal transport, and the population form of self-normalized importance sampling. We characterize exactly when it is Wasserstein-stable. Along weakly convergent inputs with uniformly integrable numéraires and a strictly positive limiting numéraire mean, convergence of the reweighted laws is equivalent to a uniform integrability condition, computed under the inputs, on one explicit family in the payoff and numéraire. At first order the numéraire cancels, the sole obstruction being payoff mass where the numéraire vanishes; absent such mass, prices and first moments pass to the limit under a uniformly integrable payoff alone, with no assumption on the reciprocal numéraire. Two-atom examples with bounded inputs attain the threshold exactly. Applications characterize a standing moment assumption in the transport literature and delimit which annuity-measure statistics survive.

**Keywords.** Change of numéraire; annuity measure; Wasserstein distance; uniform integrability; weighted marginal; size-biasing; martingale optimal transport; self-normalized importance sampling; swaption.

**MSC 2020.** Primary 60B10; secondary 28A33, 49Q22, 91G20, 91G30.

**JEL classification.** G13; G12; C02; C65.

## 1 Introduction

Let $Q$ be a probability law carrying a pair $(A,B)$ with $A>0$ and $\E_Q[A]\in(0,\infty)$. The *change-of-numéraire reweighting* of $Q$ is the probability measure

$$
\begin{equation} \Gamma(Q) :=Z_\#\bigl(D\dd Q\bigr), \qquad D:=\frac{A}{\E_Q[A]}, \qquad Z:=\frac BA . \label{eq:nr-gamma}\tag{1} \end{equation}
$$

This object is ubiquitous. In interest-rate markets, $A$ an annuity and $B$ the difference of swap legs make $\Gamma(Q)$ the law of the swap rate under the annuity measure, the measure in which swaptions are quoted (Neuberger, 1990; Geman et al., 1995; Jamshidian, 1997). In martingale optimal transport, the case $B\equiv1$, $A=X$ is the numéraire-inversion involution $S(\mu)=\mathrm{law}\bigl(1/X\ \text{under}\ (X/\E[X])\dd\mu\bigr)$ of Campi et al. (2017), recently extended to weak transport and continuous time by Beiglböck et al. (2026) and, with a general reweighting function, by Backhoff-Veraguas et al. (2025); see also Huesmann and Stebegg (2018), where the weighted inversion is singled out among all monotonicity-preserving transformations of martingale transport. In statistics, (1) with $B\equiv A\cdot Z$ is the population form of a self-normalized importance-sampling target, and for $B=A^2$ it is size-biasing (Arratia et al., 2019).

### 1.1 Main results

We are not aware of a base-law characterization of Wasserstein $\Ws$-continuity for normalized $A$-reweighting followed by the quotient map $B/A$ that also allows the weak limit to charge $\{A=0\}$. This note gives such a characterization: an exact two-way boundary carried by a single scalar family. Under weak convergence of the input laws with uniformly integrable numéraires and a strictly positive limiting numéraire mean (Lemma 3.1), $\Ws$-convergence of the weighted marginals is *equivalent* to vanishing, uniformly in the sequence index, of the base-law ratio tails of

$$
\begin{equation} |B|^s A^{1-s} \label{eq:nr-boundary-family}\tag{2} \end{equation}
$$

on the event $\{|B_n|>RA_n\}$ (Theorem 3.3). When the limit does not charge $\{A=0\}$, this is equivalent to ordinary uniform integrability of the family (2) (Corollary 3.4). At $s=1$ that family collapses to $|B_n|$: uniform integrability of the payoff leg then gives $W_1$-convergence with *no moment condition on the inverse numéraire* $1/A_n$. More generally, the only residual $s=1$ boundary obstruction is nonzero payoff mass at $A=0$. This is an exact cancellation, not an estimate: at the level of call prices and first moments, the numéraire divides out identically.

The boundary is attained. For each prescribed $s>1$, a two-atom example with all input variables in $[0,1]^2$ — so every nonnegative-order moment of the primitive coordinates is uniformly bounded — has weighted marginals converging in $W_\sigma$ exactly for $1\leq\sigma<s$, failing at $s$, and having divergent $\sigma$-moments for $\sigma>s$ (Example 4.1). The mechanism is a vanishing-numéraire state: a scenario of probability $\varepsilon$ in which $A$ collapses while $B$ stays of order one carries the entire $s$-th weighted moment. Nothing escapes to infinity on the input side; the escape is created by the ratio-and-reweight map itself. Two further examples delimit the hypotheses: if the limit law charges the vanishing-numéraire boundary $\{A=0\}$ with nonzero payoff mass, the $s=1$ cancellation fails even for bounded inputs (Example 4.3); and without uniform integrability of the numéraires, even weak stability of the weighted marginals fails (Example 4.4).

The normalized reweighting itself is classical; the additional issue here is its interaction with the quotient. Before the quotient pushforward, $Q\mapsto A\,Q/\E_Q[A]$ is a size-biased law and admits a one-point Palm representation. Kallenberg’s continuity theorem gives a three-way relation for the retained random measure, its mean mass, and its normalized mass-biased law; the forward implication uses the uniform integrability supplied by weak convergence together with convergence of the mean masses (Kallenberg, 1974, Section 10, Theorem 10.5); see also Kallenberg (2017, Chapter 6). This is not a three-way equivalence for an arbitrary marked law of $(A,B)$: the one-point random-measure encoding identifies all states with $A=0$ and therefore cannot recover the mark $B$ there.

In sequential Monte Carlo the normalized-reweighting operator is the Boltzmann–Gibbs or selection map, whose stability is classical in total variation and in weighted total-variation norms (Del Moral, 2004, 2013; Del Moral et al., 2023) and for which genuine Wasserstein estimates exist (Whiteley, 2021; Borghi, 2026). Three features separate those selection/filter results from what is proved here. They are one-directional sufficient bounds rather than characterizations, and the bounds that give a Lipschitz constant for the reweighting itself obtain it from two-sided control of the weight — in the total-variation estimates the mean of the weight sits in a denominator, so those bounds degenerate exactly where Corollary 3.4 operates. Results reaching beyond bounded weights answer a different question: contraction in time for a fixed model, rather than continuity along a varying sequence of models. They reweight and retain the underlying variable; they do not push forward by the ratio $B/A$. Consequently they do not identify the extra boundary term created when the quotient pushforward meets $B$-mass on the zero-weight fiber $\{A=0\}$. What follows is accordingly an equivalence rather than a bound, for the ratio marginal, allowing unbounded weights and a limit that may charge the zero-weight fiber; Section 1.3 places it against each literature in turn.

Two consequences are immediate. In the change-of-numéraire literature, Campi et al. (2017) and Beiglböck et al. (2026) prove algebraic and structural theorems about $S$ and *assume* what they need about moments of the transformed marginal: the standing finite-second-moment hypothesis on $S(\nu)$ in Beiglböck et al. (2026) is precisely the $s=2$, $B\equiv1$ instance of (2), and Backhoff-Veraguas et al. (2025) impose sufficient inverse-moment conditions ($\mu\in\cP_{-1,1}$). Corollary 6.1 characterizes exactly when such hypotheses survive a limit. On the sampling side, Coste and Goldman (2026) prove sharp rates for the self-normalized importance-sampling *empirical* measure in $W_p$ under bounded densities; the present note is the population-level companion of that question, in the unbounded-weight regime where (2) binds.

### 1.2 Why the boundary matters in the motivating application

For annuity-weighted swap-rate marginals the dichotomy is not a technicality; it separates two computations practitioners treat very differently. Physically settled swaption prices and every $W_1$-continuous functional of the annuity-measure marginal sit at $s=1$, where the theorem says: for weakly convergent primitive laws with uniformly integrable annuities and legs and a boundary-free limit, prices move continuously with no separate inverse-annuity moment hypothesis. Related finite-model calculations, such as the discounting-switch comparison of Piterbarg (2020), express prices under a common annuity measure through an annuity-ratio density and approximate its conditional projection onto the swap rate, in the tradition of the annuity-mapping framework of Andersen and Piterbarg (2010, Chapter 16). Our result addresses a different, asymptotic question: convergence of the primitive laws implies convergence of physically settled prices under the stated uniform-integrability hypotheses. It neither prices a finite discounting switch nor bounds a ratio-approximation error.

The raw quadratic statistic $\int z^2\dd\Gamma=\E_Q[B^2/A]/\E_Q[A]$, which enters second-order CMS-type approximations (Hagan, 2003), sits at $s=2$. In its stylized annuity interpretation, Example 4.1 exhibits bounded annuities and legs for which the entire normalized call-price curve converges while this statistic stays wrong by a fixed amount. This is the same higher-moment integrability obstruction highlighted, in a different setting, by the moment-explosion analysis of Andersen and Piterbarg (2007); it also connects with the moment-extraction discussion of Trolle and Schwartz (2014, fn. 16). Cash-settled swaptions require a different interpretation. Provided $G(S)>0$ and $0<\E[G(S)]<\infty$, their payoff $G(S)(S-K)^+$ remains algebraically within the perspective structure after setting $A=G(S)$ and $B=S G(S)$, so the $s=1$ cancellation applies to the normalized $G$-weighted law. But $G(S)$ is not a traded numéraire, so this weighting is not supplied as a market pricing measure by the change-of-numéraire theorem. In the reverse no-arbitrage problem of recovering a forward density from quoted cash-settled prices, inverse-annuity integral conditions appear in Mercurio (2008).

### 1.3 Mathematical decomposition and related work

The map $Q\mapsto\Gamma(Q)$ first biases the input law by $A/\E_Q[A]$ and then pushes the retained pair $(A,B)$ forward by the quotient $B/A$. For nonnegative Borel $f$,

$$
\int f\,\dd\Gamma(Q) =\frac{\E_Q[A f(B/A)]}{\E_Q[A]}.
$$

Taking $f(z)=(z-K)^+$ or $f(z)=|z|^s$ gives the annuity-measure pricing identity or $\int|z|^s\dd\Gamma=\E_Q[|B|^sA^{1-s}]/\E_Q[A]$. The call identity follows from change of numéraire (Neuberger, 1990; Geman et al., 1995; Jamshidian, 1997; Andersen and Piterbarg, 2010; Hagan, 2003); analytically, the formula combines a perspective integrand with the defining identity of a weighted law (Combettes, 2018; Arratia et al., 2019). The continuity question concerns their composition, because $B/A$ can become large along states where the reweighting factor $A$ is small while $B$ does not vanish.

Under (S1)–(S2), Lemma 3.1 first gives $\Gamma_n\Rightarrow\Gamma$, even when the limiting base law charges $\{A=0\}$; the reweighting discards that fiber. Pulling Villani’s weak-plus-moment characterization of Wasserstein convergence through the moment identity then gives, for every $s\geq1$,

$$
\begin{aligned} &\Gamma\in\cP_s(\R),\quad\Gamma_n\in\cP_s(\R)\ \text{for every }n,\quad\Ws(\Gamma_n,\Gamma)\to0 \\
&\hspace{4em}\Longleftrightarrow\quad\lim_{R\to\infty}\sup_n \E\!\left[ |B_n|^sA_n^{1-s}\one_{\{|B_n|>RA_n\}} \right]=0. \end{aligned}
$$

The weak-convergence step is related to the change-of-law argument in Le Cam’s third lemma (van der Vaart, 1998, Theorem 6.6); the moment step uses Villani (2009, Theorem 6.9), with related formulations in Krätschmer et al. (2014) and Feinberg et al. (2020). If $Q(\{a=0\})=0$, the ratio-tail condition is equivalent to ordinary uniform integrability of $\{|B_n|^sA_n^{1-s}\}$. At $s=1$, uniform integrability of $\{|B_n|\}$ leaves exactly the boundary residue:

$$
W_1(\Gamma_n,\Gamma)\to0 \quad\Longleftrightarrow\quad\int_{\{a=0\}}|b|\,\dd Q=0.
$$

Thus mass at $(0,0)$ is harmless, whereas nonzero payoff mass on the zero-numéraire fiber remains in primitive first moments but is absent from the weighted limit.

Palm and size-bias results describe the first operation rather than the quotient composition. In the one-point specialization, Kallenberg relates a retained random measure, its mean mass, and its normalized mass-biased law (Kallenberg, 1974, Section 10, Theorem 10.5); see also Kallenberg (2017, Chapter 6). Zero-mass states are permitted and discarded, but the encoding identifies all $B$-marks on $\{A=0\}$ and does not apply $B/A$. For a family of nonnegative variables whose means are bounded away from zero, Arratia et al. (2019, Theorem 8.1) equate uniform integrability with tightness of the size-biased family, through the identity $\E[X;X>L]=\E[X]\,\mathbb P(X^*>L)$ that the proofs below also use; Theorem 3.3 is that criterion one level up. Gnedin (1998) likewise allow mass loss for size-biased permutations, representing it by an added zero component. These are weak or tightness statements for retained objects; the result here is a $\Ws$ criterion after the quotient pushforward.

Other neighboring theories contain parts of the same algebra. Latz’s condition (A5) places a prior-integrable envelope on $\|\theta\|^pA(y\mid\theta)$, which becomes $|B|^sA^{1-s}$ for $B=\theta A$ and $s=p$ (Latz, 2020, Theorem 19). That theory assumes a positive likelihood; its survey discusses noiseless degeneracies, including a subcase with vanishing normalization, whereas the limiting normalization here remains positive (Latz, 2023, §7.2). Related posterior results give sufficient stability bounds rather than this boundary equivalence (Sprungk, 2020; Cvetković and Lie, 2025; Dolera and Mainini, 2023). When $A$ and $B$ are densities, the mixed moment is a power integral adjacent to contiguity and Hellinger-process theory, although Hellinger integrals proper have orders in $(0,1)$ rather than $s\geq1$ (Jacod and Shiryaev, 2003, Chapter IV). Martingale-transport stability instead perturbs unweighted marginals of a coupling problem (Backhoff-Veraguas and Pammer, 2022; Wiesel, 2023; Beiglböck et al., 2023; Jourdain and Pammer, 2024); the instability examples of Brückerhoff and Juillet (2022) concern dimension rather than reweighting.

The boundary residue also has a perspective-functional interpretation. The closed extension of $a\phi(b/a)$, with $\phi(z)=|z|^s$, places a recession term on $a=0$ (Spector, 2011; Liero et al., 2018; Savaré and Sodini, 2024), whereas the limiting quotient law $\Gamma$ assigns that fiber zero weight. At $s=1$, the discrepancy between the perspective numerator and the weighted-law numerator is exactly $\int|b|\dd Q-\E_Q[A]\int|z|\dd\Gamma=\int_{\{a=0\}}|b|\dd Q$. Generalized Young-measure theory gives a related local two-sided criterion through vanishing concentration, with spatial tightness needed for a global conclusion (Alibert and Bouchitté, 1997, Theorem 2.9). For $s>1$, extended-valued lower-semicontinuity results accommodate the superlinear perspective, but do not yield a two-sided Wasserstein criterion for the normalized quotient law. The weak-plus-homogeneous-moment template appears in balanced transport (Ambrosio et al., 2008, Proposition 7.1.5) and in unbalanced transport (Savaré and Sodini, 2024, Theorem A.5). Here it yields the exact base-law ratio-tail equivalence, its ordinary-uniform-integrability form off the boundary, and the first-order boundary residue. The examples attain the exponent and separate boundary failure from failure of numéraire uniform integrability; Theorem 5.1 packages the criterion into joint continuity over compact model and parameter classes.

### 1.4 Structure

Section 2 fixes the setting and records the cancellation identities. Section 3 proves the weak stability lemma, the two-way boundary theorem, and its sharp ordinary-uniform-integrability form. Section 4 presents the three examples. Section 5 gives the parameterized joint-continuity theorem. Section 6 works out the two applications. An AI-use disclosure ends the paper.

## 2 The reweighting map and the cancellation identities

Everything below rests on a single identity: weighting by $A$ and dividing by $A$ cancel, so each moment of $\Gamma$ is a base-law expectation of an explicit integrand. We record it in the generality the stability theory will need, and fix the notation used throughout.

Fix a probability law $Q$ on a measurable space carrying measurable $A>0$ and $B\in\R$ with $A^0:=\E_Q[A]\in(0,\infty)$, and define $D$, $Z$, $\Gamma=\Gamma(Q)$ by (1). Since $\E_Q[D]=1$ and $D>0$, $\Gamma$ is a probability measure on $\R$.

**Proposition 2.1 (Perspective cancellation).**  The following hold as identities in $[0,\infty]$. For every $K\in\R$,

$$
\begin{equation} \int(z-K)^+\,\Gamma(\dd z) =\frac{\E_Q\bigl[(B-KA)^+\bigr]}{A^0}, \qquad\int|z|\,\Gamma(\dd z) =\frac{\E_Q|B|}{A^0}, \label{eq:nr-call-identity}\tag{3} \end{equation}
$$

and for every $s>1$,

$$
\begin{equation} \int|z|^s\,\Gamma(\dd z) =\frac{\E_Q\bigl[|B|^s A^{1-s}\bigr]}{A^0}. \label{eq:nr-moment-identity}\tag{4} \end{equation}
$$

More generally, for every measurable $f\geq0$,

$$
\begin{equation} \int f\,\dd\Gamma=\frac{\E_Q[A\,f(B/A)]}{A^0}. \label{eq:nr-general-identity}\tag{5} \end{equation}
$$

In particular calls and the first moment are finite whenever $\E_Q|B|<\infty$, with no condition on $1/A$; the $s$-moment is finite exactly when $\E_Q[|B|^sA^{1-s}]<\infty$.

*Proof.* Equation (5) is the definition of $\Gamma$. Positive homogeneity gives $A\,(B/A-K)^+=(B-KA)^+$ and $A\,|B/A|^s=|B|^sA^{1-s}$, proving (3) and (4).∎

**Remark 2.2 (Lineage).**  In the annuity instance, (3) is the classical swaption pricing identity (Neuberger, 1990; Geman et al., 1995; Jamshidian, 1997). Abstractly, the map $(A,B)\mapsto\E_Q[A\,f(B/A)]$ is a perspective-function integral (Combettes, 2018), and (5) is the definitional identity of a weighted law (Arratia et al., 2019). We claim no novelty for Proposition 2.1; the note pivots on reading it as a statement about which base-law functional controls which $\Gamma$-moment: at $s=1$ the numéraire cancels exactly, and for $s>1$ the exact cost is the single scalar family (2).

## 3 The two-way stability theorem

Proposition 2.1 converts each $\Gamma$-moment into a base-law expectation. This section turns that conversion into a stability theory: we fix a mode of convergence for the input laws, show that the weighted marginals then converge weakly at no further cost, and identify which additional condition upgrades weak convergence to $\Ws$. The identification is an equivalence, so it also says what cannot be weakened.

### 3.1 Standing setting

For $n\in\mathbb N$, let $Q_n$ be the law of a pair $(A_n,B_n)$ with values in $\Hopen$. Let $Q$ be a probability law on $\Hplus$, write $(A,B)$ for its coordinate map, and assume throughout this section:

- (S1) $Q_n\to Q$ weakly as laws on the closed half-plane $\Hplus$ (the limit is allowed to charge the boundary $\{0\}\times\R$ unless stated otherwise);
- (S2) the numéraires $\{A_n\}_{n<\infty}$ are uniformly integrable, and $m:=\int a\,\dd Q>0$.

Under (S1)–(S2), $\E[A_n]\to m$ (uniform integrability upgrades weak convergence of the nonnegative marginals to convergence of means). Define $\Gamma_n:=\Gamma(Q_n)$ by (1) for $n<\infty$, and define the limit object by zero-weighting the boundary:

$$
\begin{equation} \int f\,\dd\Gamma:=\frac1m\int_{\{a>0\}} a\,f(b/a)\,Q(\dd a,\dd b), \qquad f\in C_b(\R). \label{eq:nr-gamma-limit}\tag{6} \end{equation}
$$

Since $\int_{\{a>0\}}a\,\dd Q=\int a\,\dd Q=m$, $\Gamma$ is a probability measure; when $Q(\{a=0\})=0$ it is $\Gamma(Q)$. Let $\mu(\dd a,\dd b):=(a/m)\one_{\{a>0\}}Q(\dd a,\dd b)$ and define $T(a,b):=b/a$ for $a>0$ and $T(0,b):=0$. Then $T$ is Borel, $\mu(\{a=0\})=0$, and equality in (6) for $f\in C_b(\R)$ identifies $\Gamma=T_\#\mu$. Consequently, (6) holds for every nonnegative Borel $f$: first for bounded Borel $f$ by equality of the measures, and then in general by monotone convergence.

### 3.2 Weak stability

Weak convergence of the weighted marginals costs nothing beyond (S1)–(S2). The argument is a truncation, carried out in the boundary-permitting form (6): mass that the limit places on $\{a=0\}$ simply receives no weight. That convention is not a technical convenience, and Example 4.3 shows what it costs when the payoff does not vanish there.

**Lemma 3.1 (Weak stability under numéraire uniform integrability).**  Under (S1)–(S2), $\Gamma_n\to\Gamma$ weakly.

*Proof.* For $f\in C_b(\R)$ define $g_f:\Hplus\to\R$ by $g_f(a,b):=a\,f(b/a)$ for $a>0$ and $g_f(0,b):=0$. Then $g_f$ is continuous: at interior points this is clear, and if $(a_k,b_k)\to(0,b)$ then $|g_f(a_k,b_k)|\leq a_k\|f\|_\infty\to0$. Moreover $|g_f|\leq\|f\|_\infty\,a$. Fix $\varepsilon>0$ and choose $M$ with $\sup_n\E[A_n\one\{A_n>M\}]\leq\varepsilon$, possible by (S2). Let $\chi_M:[0,\infty)\to[0,1]$ be continuous with $\chi_M=1$ on $[0,M]$ and $\chi_M=0$ on $[2M,\infty)$. Then $g_f\cdot\chi_M(a)\in C_b(\Hplus)$, so $\E[g_f\chi_M(A_n,B_n)]\to\int g_f\chi_M\,\dd Q$ by (S1), while $|\E[g_f(A_n,B_n)]-\E[g_f\chi_M(A_n,B_n)]| \leq\|f\|_\infty\,\E[A_n\one\{A_n>M\}]\leq\|f\|_\infty\varepsilon$ uniformly in $n$. The same tail bound holds for the limit law by the lower-semicontinuous Portmanteau theorem applied to $a\one\{a>M\}$. Hence $\E[g_f(A_n,B_n)]\to\int g_f\,\dd Q$. Taking $f\equiv1$ gives $\E[A_n]\to m$, and therefore $\int f\,\dd\Gamma_n=\E[g_f(A_n,B_n)]/\E[A_n]\to\bigl(\int g_f\,\dd Q\bigr)/m=\int f\,\dd\Gamma$.∎

**Remark 3.2.**  Lemma 3.1 is a change-of-numéraire dress of Le Cam’s third lemma (van der Vaart, 1998, Theorem 6.6): uniform integrability of the normalized weights $D_n$ is the classical contiguity condition, and Example 4.4 below is the standard failure when it is dropped. We include the proof because the boundary-permitting formulation (6) — the limit may sit mass on $\{a=0\}$, which then receives zero weight — is used essentially in Example 4.3.

### 3.3 The boundary

Weak convergence being free, the entire question is the $s$-th moment, and Proposition 2.1 moves that question to the base laws, where it concerns the single family (2). The following theorem records the answer as an equivalence: the displayed ratio tails are not one sufficient condition among several, but exactly the price of $\Ws$-convergence.

**Theorem 3.3 (Exact two-way \(\Ws \) boundary).**  Assume (S1)–(S2) and fix $s\geq1$. Put

$$
W_n:=|B_n|^sA_n^{1-s}, \qquad I_s:=\int_{\{a>0\}}|b|^sa^{1-s}\,Q(\dd a,\dd b).
$$

The following are equivalent:

- (i) $\Gamma\in\cP_s(\R)$, $\Gamma_n\in\cP_s(\R)$ for every $n$, and $\Ws(\Gamma_n,\Gamma)\to0$;
- (ii) $W_n$ is integrable for every $n$, $I_s<\infty$, and $\E[W_n]\to I_s$;
- (iii) the base-side ratio tails vanish uniformly: $$
\begin{equation} \lim_{R\to\infty}\ \sup_{n\in\mathbb N} \E\bigl[W_n\one\{|B_n|>RA_n\}\bigr]=0. \label{eq:nr-condition-iii}\tag{7} \end{equation}
$$

*Proof.* Write $m_n:=\E[A_n]$. By (S1)–(S2), $m_n\to m\in(0,\infty)$; hence $0<\inf_n m_n\leq\sup_n m_n<\infty$. By Lemma 3.1, $\Gamma_n\to\Gamma$ weakly, while Proposition 2.1 and equation (6) give, in $[0,\infty]$, ∎

$$
\int|z|^s\,\dd\Gamma_n =\frac{\E[W_n]}{m_n}, \qquad\int|z|^s\,\dd\Gamma=\frac{I_s}{m},
$$

and

$$
\int_{\{|z|>R\}}|z|^s\,\dd\Gamma_n =\frac{\E[W_n\one\{|B_n|>RA_n\}]}{m_n}.
$$

The standard moment characterization of Wasserstein convergence gives (i)$\Leftrightarrow$(ii), and its uniform-tail form gives (i)$\Leftrightarrow$(iii) (Villani, 2009, Theorem 6.9). For completeness, (iii) also supplies the domains in (i): for some $R_0$ its supremum is finite, and $W_n\one\{|B_n|\leq R_0A_n\}\leq R_0^sA_n$, so every $W_n$ is integrable. The corresponding uniform $s$-moment tails of $\Gamma_n$, together with weak convergence and lower-semicontinuous portmanteau, imply $\Gamma\in\cP_s(\R)$.

The point of Theorem 3.3 is that the exact stability condition is a statement about the base laws: the single family (2), truncated on the ratio event $\{|B_n|>RA_n\}$, decides $\Ws$-convergence. The following gives both its ordinary-uniform-integrability form away from the boundary and the sharp residual obstruction at $s=1$.

**Corollary 3.4 (Sharp cancellation and ordinary uniform integrability).**  Assume (S1)–(S2).

- (a) Suppose $\{|B_n|\}$ is uniformly integrable. Then $\Gamma\in\cP_1(\R)$ and $\Gamma_n\in\cP_1(\R)$ for every $n$, and $$
\begin{equation} W_1(\Gamma_n,\Gamma)\to0 \quad\Longleftrightarrow\quad\int_{\{a=0\}}|b|\,Q(\dd a,\dd b)=0. \label{eq:nr-s1-boundary}\tag{8} \end{equation}
$$ Thus boundary mass at $(0,0)$ is harmless; in particular the conclusion holds when $Q(\{a=0\})=0$, with no hypothesis on $\{1/A_n\}$.
- (b) Suppose $Q(\{a=0\})=0$ and fix $s\geq1$. With $W_n:=|B_n|^sA_n^{1-s}$, the following are equivalent: - (i) $\Gamma\in\cP_s(\R)$, $\Gamma_n\in\cP_s(\R)$ for every $n$, and $\Ws(\Gamma_n,\Gamma)\to0$;
- (ii) $\{W_n:n\in\mathbb N\}$ is uniformly integrable.

*Proof.* For (a), uniform integrability and weak convergence give

$$
\E|B_n|\longrightarrow\int_{\Hplus}|b|\,\dd Q.
$$

Since $\E[A_n]\to m$, Proposition 2.1 yields

$$
\int|z|\,\dd\Gamma_n\longrightarrow\frac1m\int_{\Hplus}|b|\,\dd Q, \qquad\int|z|\,\dd\Gamma= \frac1m\int_{\{a>0\}}|b|\,\dd Q.
$$

Together with $\Gamma_n\to\Gamma$ weakly, the first-moment characterization of $W_1$ proves (8).

For (b), put $Z_n:=B_n/A_n$. Boundary-freeness and weak convergence imply ∎

$$
\begin{equation} \lim_{\delta\downarrow0}\sup_n Q_n(A_n\leq\delta)=0. \label{eq:nr-uniform-away-zero}\tag{9} \end{equation}
$$

Indeed, portmanteau controls all sufficiently large $n$, and each of the finitely many remaining laws assigns no mass to $\{A_n=0\}$. If $\{W_n\}$ is uniformly integrable, then for $M,R>0$,

$$
\E[W_n\one\{|Z_n|>R\}] \leq\E[W_n\one\{W_n>M\}] +M Q_n(A_n\leq MR^{-s}).
$$

First choose $M$ and then $R$, using (9), to obtain (7); apply Theorem 3.3. Conversely, if $\Ws(\Gamma_n,\Gamma)\to0$, then Theorem 3.3 gives (7), and

$$
\E[W_n\one\{W_n>M\}] \leq\E[W_n\one\{|Z_n|>R\}] +R^s\E[A_n\one\{A_n>M/R^s\}].
$$

Choose $R$ by (7) and then $M$ by (S2). Hence $\{W_n\}$ is uniformly integrable.

**Remark 3.5 (Ratio tails versus ordinary uniform integrability).**  Under (S2), the exact ratio-tail condition (7) always implies ordinary uniform integrability of $W_n=|B_n|^sA_n^{1-s}$, by the second inequality in the proof above. The converse holds when the limiting law does not charge $\{a=0\}$, by Corollary 3.4(b), but can fail at the boundary: Example 4.3 has $W_n=|B_n|\leq1$ while (7) fails. At $s=1$, Corollary 3.4(a) identifies the residual obstruction exactly as $\int_{\{a=0\}}|b|\,\dd Q$.

## 4 Sharpness

Neither the hypotheses of Theorem 3.3 and Corollary 3.4 nor the exponent in (2) is decorative. Three two-atom sequences, all computed in closed form, settle this. The first attains the threshold exactly at any prescribed $s>1$ while confining every primitive coordinate to $[0,1]$; the second shows that the $s=1$ cancellation is destroyed by payoff mass on the vanishing-numéraire boundary, and only by that; the third shows that numéraire uniform integrability cannot be dropped even for weak convergence.

**Example 4.1 (The boundary is attained: bounded inputs, exact threshold).**  Fix $s>1$. Let $\varepsilon_n\downarrow0$, $a_n:=\varepsilon_n^{1/(s-1)}$, and let $Q_n$ put mass $1-\varepsilon_n$ at $(A,B)=(1,0)$ and mass $\varepsilon_n$ at $(A,B)=(a_n,1)$. All inputs take values in $[0,1]^2$, so every nonnegative-order moment of the primitive coordinates is uniformly bounded and (S1)–(S2) hold with limit $Q=\delta_{(1,0)}$ and $Q(\{a=0\})=0$; also $\E[A_n]=1-\varepsilon_n+\varepsilon_na_n\to1$. The weighted marginal $\Gamma_n$ has an atom at $0$ of mass $(1-\varepsilon_n)/\E[A_n]$ and an atom at $1/a_n$ of mass $\varepsilon_na_n/\E[A_n]\to0$; hence $\Gamma_n\to\delta_0=\Gamma$ weakly. For every $\sigma\geq1$,

$$
\int|z|^\sigma\,\dd\Gamma_n =\frac{\varepsilon_n\,a_n^{1-\sigma}}{\E[A_n]} =\frac{\varepsilon_n^{(s-\sigma)/(s-1)}}{\E[A_n]} \longrightarrow\begin{cases} 0, & 1\leq\sigma<s, \\
[0.2em] 1, & \sigma=s, \\
[0.2em] +\infty, & \sigma>s. \end{cases}
$$

Thus $\Gamma_n\to\delta_0$ in $W_\sigma$ for every $\sigma<s$ (in particular the $s=1$ cancellation operates: the first moment is $\varepsilon_n/\E[A_n]\to0$, matching Proposition 2.1 exactly), while the $s$-th moment converges to $1\neq0$ and $\Ws$-convergence fails. Consistently, the family $\{|B_n|^\sigma A_n^{1-\sigma}\}$ is uniformly integrable for every $\sigma<s$: on the rare atom its tail expectation is at most $\varepsilon_n a_n^{1-\sigma} =\varepsilon_n^{(s-\sigma)/(s-1)}\to0$, and the finitely many remaining indices are harmless. It first fails to be uniformly integrable at $\sigma=s$, where the single rare atom carries $\E[W_n^{(s)}]=1$; here $W_n^{(\sigma)}:=|B_n|^\sigma A_n^{1-\sigma}$.

**Remark 4.2 (Reading of Example 4.1 ).**  Viewed on the $\Gamma$-side this is the classical escape-to-infinity moment example; the content is on the $Q$-side, where nothing escapes: both coordinates live in $[0,1]$, all nonnegative-order moments of the primitive coordinates are bounded, and the joint laws converge. The escape is manufactured entirely by the ratio-and-reweight map in a vanishing-numéraire state of probability $\varepsilon_n$ in which the payoff leg stays of order one. In a stylized annuity interpretation, this is a rare scenario in which the annuity collapses while the leg value does not. The entire normalized call-price curve converges (an $s=1$ object, protected by the cancellation). With $s=2$ and $a_n=\varepsilon_n$, however, the raw quadratic statistic $\int z^2\dd\Gamma_n$ stays asymptotically one unit away from its limiting value. This statistic enters second-order CMS-type approximations (Hagan, 2003); any such approximation with nonzero exposure to it can therefore fail to converge. A particular CMS correction must still be assessed by its payoff growth. Bounded primitive inputs do not imply stable weighted moments: normalized call-price curves can converge while a quadratic statistic stays wrong by a fixed amount.

**Example 4.3 (Nonzero payoff mass on the vanishing-numéraire boundary defeats \(W_1\) -stability).**  Fix $\varepsilon\in(0,1)$, let $\alpha_n\downarrow0$, and let $Q_n$ put mass $1-\varepsilon$ at $(1,0)$ and mass $\varepsilon$ at $(\alpha_n,1)$. Then (S1)–(S2) hold with limit $Q=(1-\varepsilon)\delta_{(1,0)}+\varepsilon\delta_{(0,1)}$, which charges $\{a=0\}$; $m=1-\varepsilon>0$; $\{|B_n|\}$ is bounded by $1$, hence uniformly integrable. Lemma 3.1 applies and indeed $\Gamma_n\to\Gamma=\delta_0$ weakly (the escaping atom at $1/\alpha_n$ has $\Gamma_n$-mass $\varepsilon\alpha_n/\E[A_n]\to0$). But

$$
\int|z|\,\dd\Gamma_n =\frac{\varepsilon}{\E[A_n]} \longrightarrow\frac{\varepsilon}{1-\varepsilon} \neq0=\int|z|\,\dd\Gamma,
$$

so $W_1$-convergence fails although the payoff legs are bounded by one. Consistently, (7) fails: $\E[|B_n|\one\{|B_n|>RA_n\}]=\varepsilon$ once $R\alpha_n<1$. The boundary-payoff condition (8) is therefore sharp: the cancellation protects against vanishing-numéraire states of vanishing probability, and boundary mass at $(0,0)$ is harmless, but nonzero payoff mass on $\{a=0\}$ destroys $W_1$-convergence.

**Example 4.4 (Numéraire uniform integrability is needed for weak stability).**  Let $Q_n$ put mass $1-\varepsilon_n$ at $(1,0)$ and mass $\varepsilon_n$ at $(1/\varepsilon_n,\,1/\varepsilon_n)$ with $\varepsilon_n\downarrow0$. Then $Q_n\to\delta_{(1,0)}$ weakly, but $\{A_n\}$ is not uniformly integrable ($\E[A_n]=1-\varepsilon_n+1\to2\neq1$). The weighted marginals are

$$
\Gamma_n =\frac{1-\varepsilon_n}{\E[A_n]}\,\delta_0 +\frac{1}{\E[A_n]}\,\delta_1 \longrightarrow\tfrac12\delta_0+\tfrac12\delta_1 \neq\delta_0=\Gamma:
$$

even weak stability fails. This is the classical failure of contiguity when the likelihood ratios are not uniformly integrable (van der Vaart, 1998, Chapter 6), in numéraire dress: half of the limiting weighted mass sits on an event the limiting base law does not see.

## 5 Joint continuity for parameterized contracts

In applications the pair $(A,B)$ carries a contract parameter — an expiry, a tenor, a settlement convention — and the model law and the parameter vary together. The following packages Theorem 3.3 and Corollary 3.4 in that setting.

**Theorem 5.1 (Joint continuity, compact image, affine slices).**  Let $\Omega$ be a Polish space, $\cK$ a compact set of Borel probability laws on $\Omega$ (weak topology), and $\mathsf A$ a compact metric space of contract parameters. Let

$$
(a,\omega)\longmapsto\bigl(A_a(\omega),B_a(\omega)\bigr) \in\Hopen
$$

be jointly continuous, and fix $s\geq1$. Assume the uniform weighted-tail condition

$$
\begin{equation} \lim_{R\to\infty}\ \sup_{(Q,a)\in\cK\times\mathsf A} \E_Q\bigl[\,|B_a|^sA_a^{1-s}\, \one\{|B_a|>R\,A_a\}\bigr]=0, \label{eq:nr-joint-tail}\tag{10} \end{equation}
$$

together with uniform integrability of $\{A_a:\ (Q,a)\in\cK\times\mathsf A\}$ and $\inf_{(Q,a)}\E_Q[A_a]>0$. Then:

- (i) the map $(Q,a)\mapsto\Gamma_a(Q):=\Gamma$ of (1) applied to the law of $(A_a,B_a)$ under $Q$ is continuous from $\cK\times\mathsf A$ into $(\cP_s(\R),\Ws)$;
- (ii) its image is $\Ws$-compact;
- (iii) on every calibration slice $\cK_c:=\{Q\in\cK:\E_Q[A_a]=c\}$ with $c>0$ and $a$ fixed, the map $Q\mapsto\Gamma_a(Q)$ is affine whenever the relevant mixture remains in $\cK_c$; on all of $\cK$ it is projective (a ratio of two affine maps). For $Q_1,Q_2$ and their mixture in $\cK$, mixtures reweight by relative numéraire value: with $w_1:=\lambda\,\E_{Q_1}[A_a]$ and $w_2:=(1-\lambda)\,\E_{Q_2}[A_a]$, $$
\Gamma_a\bigl(\lambda Q_1+(1-\lambda)Q_2\bigr) =\frac{w_1\,\Gamma_a(Q_1)+w_2\,\Gamma_a(Q_2)}{w_1+w_2}.
$$

*Proof.* (i) Let $(Q_n,a_n)\to(Q,a)$ in $\cK\times\mathsf A$. The product laws $\delta_{a_n}\otimes Q_n$ converge weakly to $\delta_a\otimes Q$ on $\mathsf A\times\Omega$, so by the continuous mapping theorem applied to the jointly continuous map $(a,\omega)\mapsto(A_a(\omega),B_a(\omega))$, the laws of $(A_{a_n},B_{a_n})$ under $Q_n$ converge weakly to the law of $(A_a,B_a)$ under $Q$. The latter charges $\Hopen$, so its $a$-marginal puts no mass at $0$. The assumed uniform integrability of the numéraires gives (S2), and (10) gives (7) along the sequence. Theorem 3.3 yields $\Ws(\Gamma_{a_n}(Q_n),\Gamma_a(Q))\to0$.

(ii) A continuous image of the compact set $\cK\times\mathsf A$ is compact.

(iii) Both $Q\mapsto\E_Q[A_a\,f(Z_a)]$ and $Q\mapsto\E_Q[A_a]$ are affine in $Q$; the displayed mixture formula follows by dividing, and on a slice of constant $\E_Q[A_a]$ the ratio itself is affine.∎

**Remark 5.2 (Why calibration slices are the natural domain).**  In the calibration instance the numéraire’s time-zero value is itself a quoted input: every candidate law prices the annuity identically, so the relevant $\cK$ is a slice $\cK_c$ and the map is genuinely affine there. Off the slice, affinity in $Q$ fails in general — the mixture formula reweights by relative numéraire value — which matters for barycentric constructions: averaging models does not average their annuity-measure marginals unless the annuity values agree.

## 6 Two applications

The two settings named in Section 1 are now instances. In the first, the boundary characterizes a moment hypothesis that the change-of-numéraire literature currently assumes; in the second, it separates the statistics of an annuity-measure marginal that survive a model limit from those that do not.

### 6.1 The change-of-numéraire involution of martingale optimal transport

For $\mu$ on $(0,\infty)$ with mean $m(\mu)\in(0,\infty)$, the change-of-numéraire transform

$$
S(\mu):=\text{law of }1/X\ \text{under}\ \tfrac{X}{m(\mu)}\dd\mu
$$

is the involution of Campi et al. (2017) (there under a unit-mean normalization), extended to weak martingale transport by Beiglböck et al. (2026) and, with general reweighting functions, by Backhoff-Veraguas et al. (2025). This is (1) with $(A,B)=(X,1)$, and Proposition 2.1 gives

$$
\begin{equation} \int y^s\,\dd S(\mu) =\frac{\E_\mu[X^{1-s}]}{m(\mu)}, \qquad s\geq1 . \label{eq:nr-cn-moment}\tag{11} \end{equation}
$$

**Corollary 6.1 (Exact \(\Ws \) -stability of the involution).**  Let $\mu_n\to\mu$ weakly as probability laws on $(0,\infty)$, assume $\{X_n\}$ is uniformly integrable and $m(\mu)>0$, and fix $s\geq1$. Then $S(\mu_n)\to S(\mu)$ weakly, and the following are equivalent:

- (i) $S(\mu)\in\cP_s(\R)$, $S(\mu_n)\in\cP_s(\R)$ for every $n$, and $\Ws(S(\mu_n),S(\mu))\to0$;
- (ii) every expectation below is finite and $\E_{\mu_n}[X_n^{1-s}]\to\E_\mu[X^{1-s}]$;
- (iii) $\{X_n^{1-s}:n\in\mathbb N\}$ is uniformly integrable;
- (iv) $$
\lim_{R\to\infty}\sup_n \E_{\mu_n}[X_n^{1-s}\one\{X_n<1/R\}]=0.
$$

At $s=1$ these conditions are automatic, so $W_1$-stability needs no assumption beyond the standing ones.

*Proof.* Apply Lemma 3.1, Theorem 3.3, and Corollary 3.4(b) with $(A_n,B_n)=(X_n,1)$; then $W_n=X_n^{1-s}$ and $\{|B_n|>RA_n\}=\{X_n<1/R\}$.∎

Two readings. First, the standing assumption “$S(\nu)$ has finite second moment” in Beiglböck et al. (2026) is, by (11), exactly $\E_\nu[X^{-1}]<\infty$; Corollary 6.1 upgrades this static observation to the exact stability statement: along weakly convergent marginals with converging means, the transformed marginals move continuously in $W_2$ precisely when the inverse moments $\E[X_n^{-1}]$ converge, equivalently when $\{X_n^{-1}\}$ is uniformly integrable. The characterization is attained already by the two-atom laws that put mass $1-\varepsilon_n$ at $1$ and mass $\varepsilon_n$ at $a_n=\varepsilon_n$; their transforms have the same exact $W_2$ threshold as Example 4.1. Second, the inverse-moment classes $\cP_{-1,1}$ imposed in Backhoff-Veraguas et al. (2025) are sufficient-condition territory for $s=2$; the corollary locates the exact boundary inside them.

### 6.2 Annuity-measure swap-rate marginals

Let $a$ range over a compact family of swap contracts, $A_a>0$ the annuity (a finite sum of accrual-weighted zero-coupon bonds — hence, for a contract class with uniformly bounded accrual sums, a common bond bound gives a common bound on the annuities and therefore their uniform integrability), and $B_a$ the difference of the legs. Then $\Gamma_a(Q)$ is the annuity-measure swap-rate law and (3) is the swaption quoting identity. Corollary 3.4 and Theorem 5.1 say:

- 1. *Prices and normalized call-price curves require no inverse-annuity moments.* Assume the compact model–contract setting and joint continuity of Theorem 5.1, together with uniform integrability, over the model–contract class, of the annuities and legs, positive normalization, and its boundary-free primitive range. These hypotheses supply the $s=1$ uniform ratio-tail condition in (10). Indeed, for $R\geq M/\delta$, $$
\E_Q\bigl[|B_a|\one\{|B_a|>RA_a\}\bigr] \leq2\,\E_Q\bigl[|B_a|\one\{|B_a|>M\}\bigr] +M\,Q(A_a\leq\delta),
$$ by splitting on $\{A_a>\delta\}$ and $\{A_a\leq\delta\}$; the first term is uniformly small in $M$ by the assumed uniform integrability of the legs, and the second is handled by $$
\sup_{(Q,a)\in\cK\times\mathsf A}Q(A_a\leq\delta) \xrightarrow[\ \delta\downarrow0\ ]{}0,
$$ which holds because $(Q,a)\mapsto Q(A_a\leq\delta)$ is upper semicontinuous (portmanteau on the closed set $\{A_a\leq\delta\}$, after the continuous-mapping step of Theorem 5.1) and decreases pointwise to $Q(A_a=0)=0$, so that for each $\eta>0$ the nested closed sets $\{(Q,a):Q(A_a\leq\delta)\geq\eta\}$ have empty intersection and one of them is therefore already empty by compactness of $\cK\times\mathsf A$. Theorem 5.1 then gives joint $W_1$-convergence of the annuity-measure marginals and uniform convergence of their normalized call-price curves. As the normalizations converge, the associated physically settled swaption prices converge at each fixed strike, uniformly over bounded strike sets. No additional moment assumption on $1/A_a$ is imposed. Related finite-model calculations, such as the discounting-switch analysis of Piterbarg (2020), use an annuity-ratio measure-change density and approximate its conditional projection onto the swap rate, in the tradition of Andersen and Piterbarg (2010, Chapter 16). The present result is an asymptotic stability statement, not a valuation formula for a finite switch or a bound on that approximation error.
- 2. *Quadratic CMS-type statistics are boundary-priced.* For the raw second moment, $$
\int z^2\dd\Gamma_a =\frac{\E_Q[B_a^2/A_a]}{\E_Q[A_a]};
$$ in the boundary-free regime its stability is equivalent to uniform integrability of $\{B_a^2/A_a\}$. At the level of the abstract primitive-law model, Example 4.1 shows convergence of the entire normalized call-price curve while this quadratic statistic remains asymptotically displaced by one. Thus a second-order CMS approximation depending nontrivially on that statistic need not converge (Hagan, 2003). A particular CMS correction must be assessed by its payoff growth; the theorem does not identify every correction with the second moment. This is the same higher-moment integrability obstruction highlighted, in a different setting, by Andersen and Piterbarg (2007), and it connects with the moment-extraction discussion of Trolle and Schwartz (2014, fn. 16).
- 3. *Algebraic cancellation and the settlement distinction.* For a cash-settled payoff $G(S)(S-K)^+$ with $G(S)>0$ and $0<\E[G(S)]<\infty$, setting $A=G(S)$ and $B=S G(S)$ gives the perspective identity $A(B/A-K)^+=G(S)(S-K)^+$. Thus Proposition 2.1 applies to the normalized $G$-weighted law; tradability is not one of its mathematical hypotheses. The distinction is instead that $G(S)$ is not a traded numéraire, so this normalized weighting is not automatically a market pricing measure. In the reverse problem of recovering a forward density from a quoted cash-settled price surface, inverse-annuity integral conditions appear in the no-arbitrage analysis of Mercurio (2008). Under its stated weak-convergence and integrability hypotheses, the present theorem gives forward stability of the specified weighted laws; it neither constructs a market pricing measure nor removes or characterizes those reverse conditions.

**AI-use disclosure.**

The author used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools. They assisted with exploratory discussion, testing and refinement of ideas, literature and source organization, code development and verification, mathematical error checking, and editorial revision.

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