--- title: "Joint Calibration of Caplet and Swaption Smiles: A Finite-Panel Impossibility Theorem and Its Payment-Timing Completion" authors: - name: "Shaosai Huang" affiliation: "Kspectra Research Inc., Toronto, Canada" date: "2026-08" status: "Working paper" url: https://kspectra.ai/papers/caplet-swaption-finite-panel-impossibility/ doi: 10.2139/ssrn.7383160 ssrn: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7383160 --- # Joint Calibration of Caplet and Swaption Smiles: A Finite-Panel Impossibility Theorem and Its Payment-Timing Completion Shaosai Huang — Working paper, version of August 2026. Links: [Web page](https://kspectra.ai/papers/caplet-swaption-finite-panel-impossibility/) · [SSRN](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7383160) > 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{huang2026joint, author = {Huang, Shaosai}, title = {{Joint Calibration of Caplet and Swaption Smiles: A Finite-Panel Impossibility Theorem and Its Payment-Timing Completion}}, year = {2026}, month = aug, note = {Working paper, version of August 2026}, doi = {10.2139/ssrn.7383160}, url = {https://kspectra.ai/papers/caplet-swaption-finite-panel-impossibility/} } ``` ## 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{\cA}{\mathcal A} \newcommand{\cB}{\mathcal B} \newcommand{\cC}{\mathcal C} \newcommand{\cD}{\mathcal D} \newcommand{\cF}{\mathcal F} \newcommand{\cK}{\mathcal K} \newcommand{\cM}{\mathcal M} \newcommand{\cP}{\mathcal P} \newcommand{\cQ}{\mathcal Q} \newcommand{\cS}{\mathcal S} \newcommand{\cV}{\mathcal V} \newcommand{\conv}{\operatorname{conv}} \newcommand{\ri}{\operatorname{ri}} \newcommand{\dd}{\mathrm{d}} \newcommand{\Abar}{\overline A} \newcommand{\Fbar}{\overline F} \newcommand{\Zbar}{\overline Z} ``` ## Abstract On declared supports containing a one-parameter family of terminal-segment curve deformations, and under the printed integrability and quote-Slater hypotheses, no finite book of caplets, floorlets, physical or cash-settled swaptions, or standard-timing CMS products on any finite strike–tenor panel bounds any superlinear moment of the annuity pricing weight: exact joint caplet–swaption calibration fails an observable well-posedness criterion. One payment-timing quote per date repairs this. The in-advance claim’s price is affinely a second moment of the inverse terminal bond price, yielding a market-implied coercive gauge, a compact closed weighted-smile image, and an exact support-function dual; a single settlement family on the compounded tape carries every date’s budget, while plain compounded futures provably cannot. Deterministic fixtures certify each mechanism. **Keywords:** joint calibration; caplet and swaption smiles; CMS convexity; payment timing; support-function duality. **MSC (2020):** 91G20; 91G30; 60G44; 90C46. **JEL:** G12; G13; E43. ## 1 Introduction Caplets and swaptions are two central vanilla option markets of the interest-rate world, written on overlapping segments of one discount curve. Parametric models can and do fit selected caplet and swaption panels jointly, while the literature documents both the modeling difficulty and economically relevant cap–swaption relative-value deviations (Henry-Labordère, 2008; Hunt et al., 2000; Longstaff et al., 2001). This paper does not claim those restricted calibration tasks are unsolved. It addresses the sharper nonparametric question: on an untruncated state space, retaining the natural numeraire and annuity weights, does there exist an arbitrage-free joint law matching both quoted smile families, is the weighted image closed, and which market information makes that problem well posed? The equity analogue of this question—the joint S&P 500/VIX smile calibration puzzle—was solved by Guyon (2020, 2024): the set of martingale laws matching the SPX marginals, the VIX marginal, and the consistency constraint linking them is characterized by a duality, and, relative to a chosen reference law, a minimum-entropy member can be selected and computed by Sinkhorn iteration (Bourgey and Guyon, 2024). A structural feature supports that program: the prescribed coordinate marginals are ordinary, unit-weight marginals. They pin the corresponding laws and provide the tightness backbone; Guyon’s results then close the conditional constraints and prove the relevant duality under their printed moment assumptions. This is the familiar martingale-transport setting: existence under fixed marginals in convex order is classical (Strassen, 1965), and fixed marginals with suitable moments supply the main compactness input (Beiglböck et al., 2013). The rates problem transposes differently, and the difference is the subject of this paper. Written in units of a terminal (anchor-maturity) bond $P(\cdot,T_\star)$, the normalized price of a payer swaption at strike $K$ is $$ \E_Q\bigl[\Abar_e\,(S_e-K)^+\bigr], \qquad\Abar_e=\sum_j\delta_j\,\Zbar_e(q_j), $$ where $e$ is the exercise date, $S_e$ the swap rate, $q_j$ the fixed-leg payment dates with accrual factors $\delta_j>0$, and $\Zbar_t(u):=P_t(u)/P_t(T_\star)$ the price at $t$, in units of the anchor bond, of one currency unit paid at $u$; thus $\Abar_e$ is the annuity in anchor units. A caplet price is $\E_Q[\Zbar_\theta(p)(L-K)^+]$, carrying a bond weight at the fixing date $\theta$ for payment at $p$; a cash-settled swaption carries yet another curve-dependent weight. Differentiating twice in the strike à la Breeden and Litzenberger (1978), a quoted smile family determines not the law of the rate but the *weighted marginal* $(Y)_\#(W\,Q)$—the pushforward of the measure $W\,\dd Q$ under the observable $Y$, with a *state-dependent, unbounded* weight $W$. Weighted masses and weighted first moments are then pinned by martingality alone ($\E_Q[\Abar_e]$ is the initial annuity, for every candidate law), but an $L^1$ bound is not uniform integrability: nothing in the quoted data controls the joint tail of the weight and the state. The question this paper answers is whether a finite standard-timing rates book supplies that missing control, and which additional quote does so when it does not—equivalently, whether the weighted-smile image of the calibrated class can be made closed, so that feasibility has a duality characterization and calibration is well posed on the declared untruncated class. ### 1.1 Results Fix a finite tenor calendar $0=t_0<\dots0$ is the contractual accrual factor of $[b,T_\star]$ and $\pi_b$ the claim’s normalized ask; one quoted number is affinely one second moment, with no convexity adjustment or expansion. Theorem 4.3 converts a strip of such asks into a closed-form second-moment bound for every linearly dominated pricing weight (Cauchy–Schwarz plus conditional Jensen; no rate floor); Theorem 4.6 builds from the strip a continuous, inf-compact gauge whose budget is a market quote; and Theorem 4.7 proves that on the strip-calibrated class the weighted-smile image is compact, convex, and closed, with an exact support-function characterization of feasibility. The well-posedness input thus becomes a *price*, not a model parameter. Proposition 5.2 locates the boundary: the in-advance claim is statically equivalent to a natural-timing FRA plus an *infinite*-strike cap/floor strip, and every finite-strike truncation restores the negative theorem (Proposition 5.3); a finite-strike implementation therefore requires a wing extrapolation or another tail assumption. **Implementation side.** Section 6 removes the need for a strip at every date. The rolled overnight deposit on the published compounded fixing tape has a martingale anchor value $V^{(i)}$ with $V^{(i)}_{t_i}=\zeta_{t_i}$, so conditional Jensen chains every calendar date’s budget to a *single* settlement date (Theorem 6.2); the account-convexity claims $\mathrm{AC}(t_i)$, Borel functions of the published tape and one terminal bond fixing, carry the entire gauge budget in one settlement family (Corollary 6.5). A *designed* margined settlement index prices these claims exactly with no sign or correlation condition (Corollary 6.7), whereas *plain* compounded-rate futures quotes are provably uninformative for the upper budget: they are *exactly invariant* under a terminal escape surgery that blows the budget up, and—given quote-Slater room for the panel augmented by the futures quotes themselves—the surgery-plus-correction leaves any standard book and all futures quotes exact while the budget diverges (Proposition 6.8); an imposed positive-association condition bounds the required moment from the wrong side. A deterministic numerical certificate (Section 7) exhibits two exactly calibrated laws, indistinguishable on a six-instrument standard book to ten digits, whose second annuity moments differ by a factor $3.2\times10^4$; the strip separates them by the same order. **The finite-panel boundary.** The negative and positive sides meet at a sharp three-regime boundary, and the impossibility is directional, not a growth count. Finite panels fail at any strikes: the standard class includes payoffs of *arbitrary* growth in the swap-segment rates (class (S2) of Definition 3.4), and they fail identically, because every standard claim weights its convexity by a numeraire-*linear* factor (Proposition 3.5). The idealized *all-strike* far-segment smile does span the missing moment (Proposition 5.2), and every finite-strike truncation of it restores the failure (Proposition 5.3); one payment-timing quote per date, or one settlement family at a single date, replaces the continuum exactly (Lemma 4.2 and Theorem 6.2). The impossibility theorem therefore concerns practical finite quote panels—the data that exist—while the completion closes the full weighted-smile image; neither statement reduces to the other. ### 1.2 Related literature *Robust pricing and martingale transport.* In the model-independent literature, existence under fully prescribed *unit-weight* marginals goes back to Strassen (1965), and the compactness of the calibrated class is obtained from those marginals with suitable moments (Beiglböck et al., 2013); finitely many co-maturing quotes are treated by Davis and Hobson (2007). None of this applies verbatim to rates smiles, whose constraints carry unbounded state-dependent weights; the present paper can be read as the statement that, on the declared escape-rich supports and under the printed Slater hypotheses, the weighted-marginal analogue of that compactness theory *fails*, and cannot be repaired by a finite standard-timing quote book. Closest in mechanism is the observation of Acciaio et al. (2017), refined by Nutz et al. (2023), that semistatic outcome spaces can fail to be closed for *integrability* reasons; the annuity escape is an integrability failure of the same species, realized inside the term structure and shown here, under the printed support and quote-Slater hypotheses, to be immune to every finite standard-timing remedy. Campi et al. (2017) study how martingale transport interacts with a numeraire change in the symmetric two-marginal case; the rates problem is the asymmetric case in which each constraint family lives in its own numeraire and no change of measure removes the weights simultaneously. *The equity precedent.* The joint SPX/VIX literature (Guyon, 2020, 2024; Bourgey and Guyon, 2024) supplies existence, duality, reference-dependent entropy selection, and computation in its ordinary-marginal setting. Our negative theorem identifies the rates obstruction at the closedness step, and the positive theorems recover compactness, a closed weighted image, and the support-function feasibility dual once the missing tail budget is quoted. Entropy dual potentials and a non-atomic numerical scheme are not proved here. The instrument analogy is structural rather than literal: VIX derivatives reveal a conditional log-contract dispersion quantity, whereas the rates completion must control the numeraire/annuity tail directly. A variance-style contract written only on a rate can remain blind to the escape (Remark 3.12). *CMS convexity and the practitioner folklore.* The classical CMS literature prices convexity by replication against the swaption smile under an annuity-mapping (terminal swap rate) assumption (Hagan, 2003; Mercurio and Pallavicini, 2006; Andersen and Piterbarg, 2010); it is folklore that the mapping is a model input not identified by vanilla quotes. More specifically, Cedervall and Piterbarg (2012) model the joint covariance of the swap rate, its annuity, and the payment-delay bond, linking CMS and payment convexity to swaption volatilities across relevant tenors and CMS-spread prices while controlling potential arbitrage. Their construction supplies a dependence model. Building directly on that line, Bang and Daboussi (2022, Proposition 3.1 and Corollary 3.1.1), see also Bang and Daboussi (2023), introduce an annuity-due measure for a common-expiry family of RFR swap rates. In their single-curve setting, idealized full swaption smiles determine the individual rate marginals, while a specified dependence structure supplies a common joint law. Their Corollary 3.1.1 prices the receipt of a swap rate at reset against payment of the same rate at maturity: its present value is model-independent and equals the initial annuity times the annuity-measure second moment of that rate. Applied to the one-period swap spanning $[b,T_\star]$, that contract is the in-advance claim minus its natural-timing FRA, and their identity is then, after accrual and anchor normalization, exactly Lemma 4.2 below: our one-period identity is a specialization and re-expression of their Corollary 3.1.1 in anchor units, not a novelty claimed here. Those works supply dependence constructions and a contract-level identity; the result here instead concerns the full nonparametric class calibrated to a finite standard-timing panel and, under the printed escape-rich-support, common-integrability, and quote-Slater hypotheses, proves that the panel does not control the numeraire-weight tail. Theorem 3.6 and Corollary 3.11 turn that folklore into a theorem about the calibrated *class*: the non-identified direction carries, quantitatively, an unbounded superlinear annuity moment, and no finite family of CMS or cash-settled quotes (Mercurio, 2008) at any strikes can bound it. The hinge decomposition behind the in-advance replication is classical, while the identity used here is the pathwise anchor-unit form of the payment-timing precursor just described. Its economic motivation is the same marking-to-market convexity studied in the classical forward–futures literature (Cox et al., 1981); the contribution here is its exact placement as a quoted uniform-integrability budget, together with a recession-direction theorem showing why finite standard-timing books do not play that role. *Transport methods in rates.* Roome (2026) calibrates the joint law of two CMS rates to CMS and spread options by a constrained Schrödinger problem and computes no-arbitrage bounds for bivariate exotics. Posed under a single payment measure with unit weights, finitely many strikes, and a truncated compact state grid, it does not meet the weighted-tail obstruction studied here. The two meet at the bounds: for payoffs growing superlinearly in the annuity weight, the untruncated escape-rich class gives infinite bounds (Theorem 3.6), so the grid is the tail assumption, not merely a computational device. ### 1.3 Scope and organization *Scope.* The analysis is single-collateral on a fixed finite calendar; Bermudan exercise, collateral optionality, and moving calendars lie outside it. The negative results require supports rich enough to contain the escape family—a declared hard cap on the numeraire coordinate evades them by construction, at the price of becoming the tail assumption itself. No liquidity or execution claim is made for the completing instruments; whether sufficiently close structures are quoted and usable is empirical, so absent that binding the honest output of an exact joint calibration is relative to a declared envelope. Each boundary is stated precisely in Section 8. *Organization.* Section 2 sets the market, the calibrated class, and the two properness notions. Section 3 proves the negative results. Sections 4 to 6 develop the in-advance strip, its replication boundary, and the chaining to a single settlement family, with the lower-power and capped contract variants recorded in Section 4.4. Section 7 reports the numerical certificate, Section 8 collects boundaries and open problems, and Section A discusses adoption economics and contract design of the budget carrier; the three longest proofs are deferred to Section B. ## 2 The finite-calendar market and the properness target Every result in the paper plays out on a single fixed arena, which this section constructs. We declare the finite calendar and path space, express all prices in the anchor units $\Zbar_t(u)$ and $\zeta_t$ in which term-structure laws become martingale families, and define the calibrated class $\cM(\cB)$ of laws matching a finite quote book, together with the carry-closure convention without which conditional-mean arguments fail outright (Lemma 2.5). The section culminates in the target that both sides of the paper address: *observable properness* for linearly dominated pricing weights (Definition 2.8)—a uniform-integrability property that free first moments cannot supply (Remark 2.11), that Section 3 proves no standard-timing book supplies, and whose market-implied restoration is the business of Sections 4 and 6. The section closes by exhibiting a canonical host class—curves with a fixed log-forward-rate floor—realizing the support and one-date richness hypotheses outright, and the multi-date branch hypothesis in the localized form the surgery uses (Section 2.3). A separate finite-dimensional lemma explains the genericity of relative-interior targets; it does not assert that the observed target is interior. ### 2.1 Calendar, states, and anchor prices Fix a finite master calendar $$ 0=t_00$. All pricing is in anchor units. The anchor price at $t$ of one currency unit paid at $u\ge t$ is $$ \begin{equation} \Zbar_t(u)=\frac{P_t(u)}{P_t(T_\star)}, \qquad\zeta_t:=\Zbar_t(t)=\frac1{P_t(T_\star)}. \label{eq:pc1-anchor-price}\tag{1} \end{equation} $$ Only finitely many maturities enter the book, its carries, and the strip below. The master calendar is enlarged, if necessary, so that every retained maturity not later than $T$ is itself a calendar date. Let $U_t\subset[t,T_\star]$ be the finite set of live maturities at date $t$, with $t\in U_t$, and let the date-$t$ price state be the vector $$ x^t=\bigl(\Zbar_t(u)\bigr)_{u\in U_t}\in\R_{>0}^{k_t}. $$ Two standing conventions govern the maturity inventory. *Initial completeness and carry closure*: every maturity that appears in some $U_t$ already belongs to $U_0$, and, for every such $u$, one has $u\in U_s$ at each calendar date $s\le\min\{u,T\}$. Thus adding a maturity needed by a later book or strip only enlarges the finite state vector; it cannot create an unconstrained coordinate midway through its carry chain. *Stopped coordinates*: if $u0}^{k_t}$. The deterministic initial state is part of the support declaration itself: every $\omega\in\Omega$ has $x^{t_0}(\omega)=x^0$. Thus $\Omega$ lies in the corresponding closed initial-state cylinder; it is Polish, carries the raw filtration $(\cF_{t_i})_i$ generated by the price states, and encodes the declared contractual support identities of the market (for example, the tape–curve consistency identities of Section 6). Non-price production coordinates (fixing tape, settlement marks) are suppressed from the notation, and their status must be stated precisely, because the gauge of Section 4 is coercive in the price coordinates only. Three regimes are compatible with the arguments below: (i) compact-valued coordinates, such as marks confined to a declared bounded settlement domain, change nothing; (ii) coordinates that are continuous functions of the retained price coordinates through the support identities inherit pathwise control from the price state—the compounded fixing tape is of this type, each realized period rate being a ratio of price coordinates with positive denominator on $\Omega$ (Section 6); (iii) a genuinely unbounded coordinate not controlled by the price state would require its own coercive gauge term with its own quoted budget, and no claim of this paper covers it. **Assumption 2.1 (Declared support).** $\Omega$ is closed in the ambient Euclidean product $\prod_{i=0}^N\R^{k_{t_i}}$. Equivalently, every bounded subset of $\Omega$ is relatively compact in $\Omega$. Since $\Omega$ lies in the open positive orthants, ambient closedness is a genuine support declaration: it forbids admissible states from accumulating, at bounded price levels, at degenerate curves with a vanishing bond price. It plays, *pathwise*, the role that inverse-price terms in coercive gauges ( $\Psi=1+M^p+m^{-q}+\cdots$ ) play in expectation: control of the zero-price boundary. It does not bound $\zeta$ above, and the escape family of Section 3, whose coordinates diverge rather than degenerate, respects it. Closedness of $\Omega$ merely in the relative topology of the open orthant would *not* suffice for the gauge theorem below. **Assumption 2.2 (Currency-price bound).** There is $C_P<\infty$ with $P_t(u)\le C_P$ for all $t\le u$ on $\Omega$. Equivalently, $\Zbar_t(u)\le C_P\,\zeta_t$ for every $u\in U_t$. This is a mild deterministic floor on negative rates (for example $C_P=e^{|r_{\min}|T_\star}$ under a rate floor $r_{\min}$). No upper rate bound is imposed: $\zeta_t$ is unbounded on the supports of interest. **Assumption 2.3 (Observation regularity).** Each weight $W_a$ and observable $Y_a$ of the finite observation family $\{(W_a,Y_a)\}_{a\in\cA}$ used below is continuous on $\Omega$, and both $W_a$ and $W_a(1+|Y_a|)$ are linearly dominated in the sense of Definition 2.6 below. For the vanilla family (4) this holds automatically: the weights are positive combinations of coordinates, the observables are ratios of coordinate combinations with denominators positive on $\Omega$, the cash-settlement domain obeys the explicit multiplier bound stated below, and the dominations are exhibited after Definition 2.6. **Definition 2.4 (Candidate law class).** $\cM$ is the set of probability laws $Q$ on $\Omega$ (whose deterministic initial state is already fixed pathwise by the support declaration) under which, for every retained maturity $u$ and every adjacent pair $t_i0$ the accruals and $H^a_e$ the strike-independent cash multiplier on the declared settlement domain. For a cash observable $\widehat S_e$ we require, on that domain, $0\le H^a_e\le C_H$ and $H^a_e(1+|\widehat S_e|)\le C_{H,1}<\infty$; the latter is the precise condition controlling the cash-rate first-moment test. **Definition 2.6 (Linearly dominated weight).** A weight $W\ge0$, measurable at some date $e_W$, is linearly dominated if there are finitely many constants $c_j\ge0$ and maturities $d_j\ge e_W$ in $U_{e_W}$ (each therefore carry-closed to its own maturity) with $$ \begin{equation} W\le\sum_jc_j\,\Zbar_{e_W}(d_j) \qquad\text{pathwise on }\Omega. \label{eq:pc1-linear-domination}\tag{5} \end{equation} $$ All weights in (4) are linearly dominated: the annuity with $c_j=\delta_j$ and equality; the caplet weight trivially; the cash weight with the single term $C_H\Zbar_e(\sigma)$. The associated observables $Y_a$ satisfy $W_a|Y_a|\le(\text{another linear domination})$ for the vanilla book (for the annuity, $W|Y|=|\Fbar_e|$ is a signed combination of anchor bond prices, dominated by the sum of their absolute coefficients), so control of second moments of linearly dominated weights controls the full test family $W_a(1+|Y_a|)$. For the cash observation the second domination uses $H^a_e(1+|\widehat S_e|)\le C_{H,1}$ explicitly. **Remark 2.7 (The test family \(W_a(1+|Y_a|)\) sits at the cancellation boundary).** The pair controlled here is exactly the borderline one, and it is worth recording why no inverse-weight moment appears. Write $A:=W_a$ for the weight and $B:=W_aY_a$, so that the observable is the ratio $Y_a=B/A$ on $\{A>0\}$ and the weighted marginal $\Gamma_a(Q)=(Y_a)_\#(W_aQ)$ is the finite measure obtained by pushing the $A$-weighted measure forward through that ratio, with total mass $\E_Q[A]$. In these coordinates the two dominations required above are domination of $A$ and of $$ W_a\bigl(1+|Y_a|\bigr)=A+|B|, $$ that is, of the weight together with the first-order functional $|B|$, the case $s=1$ of $|B|^sA^{1-s}$. For the family (4) the dominating combinations are positive combinations of carried bond prices, whose expectations are common deterministic market inputs across the class, so the total masses $\E_Q[A]$ are uniformly controlled and the observation topology’s separate mass and first-absolute-moment coordinates are exactly the $s=1$ data. That the inverse weight is absent at first order is sharp rather than incidental. More precisely, let $(A_n,B_n)$ take values in $(0,\infty)\times\R$, suppose their joint laws converge weakly on $[0,\infty)\times\R$ to $Q$, assume $\{A_n\}$ is uniformly integrable, and set $m:=\int a\,Q(\dd a,\dd b)>0$. If $\{|B_n|\}$ is uniformly integrable, then the normalized $A_n$-weighted laws of $B_n/A_n$ converge in Wasserstein-$1$ to the normalized limit obtained from $Q$ by zero-weighting $\{a=0\}$ if and only if $$ \int_{\{a=0\}}|b|\,Q(\dd a,\dd b)=0; $$ thus, for a boundary-free limit, this convergence requires no hypothesis on $\{1/A_n\}$. For $s>1$, under the weak-convergence, numeraire-UI, and positive-mean hypotheses above, membership of the normalized marginals and their limit in $\cP_s(\R)$ together with Wasserstein-$s$ convergence is equivalent to $$ \lim_{R\to\infty}\sup_n \E\!\left[ |B_n|^sA_n^{1-s}\one\{|B_n|>RA_n\} \right]=0. $$ If $Q(\{a=0\})=0$, this ratio-tail condition is equivalent to ordinary uniform integrability of $\{|B_n|^sA_n^{1-s}\}$ (Huang, 2026, Theorem 3.3 and Corollary 3.4). It can fail even for bounded $(A_n,B_n)$ when a rare state keeps $B_n$ of order one while $A_n\downarrow0$. The second-moment bounds used below strengthen the paper’s separate weight-tail target in Definition 2.8; they are not a substitute for this $s>1$ ratio-tail condition. The first-moment test family above therefore does not silently assume inverse-annuity control. **Definition 2.8 (Weight-tail target).** The book $\cB$ meets the weight-tail target for the family $\{W_a\}$ if $$ \sup_{Q\in\cM(\cB)}\E_Q\bigl[\Phi(W_a)\bigr]<\infty\quad\text{for some superlinear increasing convex }\Phi\text{ and every }a. $$ By de la Vallée–Poussin this is exactly uniform integrability of $\{W_a\}$ over $\cM(\cB)$. **Definition 2.9 (Admissible gauge; observable properness).** An admissible gauge is a function $\chi:\Omega\to[1,\infty)$, continuous and inf-compact, such that every weighted observation test $W_a\varphi(Y_a)$ with $\varphi$ continuous of at most linear growth, and every martingale test of Definition 2.4, is $\chi$-sublinear: bounded in absolute value by $C_\varepsilon+\varepsilon\chi$ for every $\varepsilon>0$. An admissible gauge is observably proper on the class $\cM$ if every output-compact set of smile data pulls back to a bounded gauge budget: for every compact $K$ in the observation output space, $$ \sup\bigl\{\E_Q[\chi]:\ Q\in\cM,\ \E_Q\chi<\infty,\ \Gamma(Q)\in K\bigr\}<\infty. $$ Observable properness is the natural compactness hypothesis for feasibility theorems on the unbudgeted class $\cM$: gauge sublevels $\{\E_Q\chi\le M\}$ are weakly compact by inf-compactness, and properness transfers that compactness to the preimages of output-compact sets, closing the weighted image without a declared budget. **Remark 2.10 (Two properness notions).** The weight-tail target and observable properness must not be conflated. Because weighted tests are $\chi$-sublinear, observable properness implies the weight-tail target on output-compact sets, but the converse fails in general, and neither is a statement about the other’s gauge. The results below interact with the two notions in precisely delimited ways: negatively, Theorem 3.6 refutes the weight-tail target for standard books, and Corollary 3.9 separately refutes observable properness itself, for every admissible gauge, on escape-rich supports; positively, Theorem 4.6 does not verify observable properness but bypasses it, supplying an explicit fixed envelope with quoted budget under which the closed-image theorem (Theorem 4.7) needs no properness at all. **Remark 2.11 (Why first moments are free and useless).** Every anchor coordinate has its mean pinned by (3): $\E_Q[\Zbar_e(u)]=\Zbar_0(u)$, hence $\E_Q[\Abar_e]=\Abar_0$ for every $Q\in\cM$. Thus the entire initial curve worth of first-moment constraints comes for free and contributes nothing to Definition 2.8: an $L^1$ bound is not uniform integrability. The question is which quoted convexity upgrades it. ### 2.3 A canonical host class The support and richness hypotheses used below are structural declarations, and one may ask whether any recognizable nonparametric class satisfies them. The following *rate-floor host* realizes the support and one-date richness hypotheses outright, and the multi-date branch hypothesis in the localized, interior-margin form that the surgery actually uses. The quote-Slater condition remains a separate target hypothesis. Fix a floor $r_{\min}\le0$ and let $K^{\rm host}_t$ be the set of date-$t$ anchor states $(\Zbar_t(u))_{u\in U_t}$ with $\Zbar_t(T_\star)=1$ and $$ \begin{equation} \Zbar_t(u')\le e^{-r_{\min}(u'-u)}\,\Zbar_t(u), \qquad u\le u',\quad u,u'\in U_t\cup\{T_\star\}, \label{eq:pc1-host}\tag{6} \end{equation} $$ the set of curves whose average continuously compounded, or *log-forward*, rates $$ f^{\log}_t(u,u'):=-\frac{1}{u'-u} \log\frac{P_t(u')}{P_t(u)},\qquad u0$ be the smallest gap between consecutive points of the calendar-and-maturity grid. Then: - 1. $\Omega_{\rm host}$ satisfies Assumption 2.1 and Assumption 2.2 with $C_P=e^{|r_{\min}|T_\star}$; moreover every coordinate is bounded below, $\Zbar_t(u)\ge e^{r_{\min}(T_\star-u)}>0$, so the support lies in the open positive orthant while remaining ambient-closed. - 2. $\cM\neq\emptyset$ on $\Omega_{\rm host}$: the deterministic law frozen at the initial curve satisfies every martingale row pathwise. - 3. For every nondecreasing ramp $\kappa:[e,T_\star]\to[0,1]$ vanishing on $[e,T_m]$, every base path in $\Omega_{\rm host}$, and every $\Lambda\ge1$, replace the date-$e$ curve by $$ P^\Lambda(u):=P^0(u)\Lambda^{-\kappa(u)} $$ and freeze it thereafter. The resulting path again lies in $\Omega_{\rm host}$, and $\Lambda\mapsto\omega^\Lambda$ is continuous. In particular every retained log-forward rate is weakly raised and the support-richness hypothesis of Section 3 (Assumption 3.3) holds at every base path, for the escape ramp there, for every capped variant used in Theorem 5.5, and for the terminal ramp of the multi-date surgery. - 4. For $\gamma>0$ call a state $\gamma$-interior if $f^{\log}_s(u,u')\ge r_{\min}+\gamma$ for every retained pair $u0$, with the uniform slack $\varepsilon_0(B)=1-e^{-\gamma\delta_*}$ independent of the bounded set $B$. The localization is not removable: at a state with a floor-binding retained pair $u\kappa(u)$, the compensator branch exits the host for every $\Lambda>1$ and every $p>0$ with $p\Lambda<1$. Such states belong to the host, so no slack uniform over all bounded state sets exists on the host. *Proof.* (i) The constraints (6) are weak linear inequalities among nonnegative coordinates, so each $K^{\rm host}_t$, and then the product cylinder, is closed in the ambient Euclidean product. Taking $u'=T_\star$ in (6) gives $1\le e^{-r_{\min}(T_\star-u)}\,\Zbar_t(u)$, the positive lower bound; chaining (6) from the own-maturity coordinate gives $P_t(u)=\Zbar_t(u)/\zeta_t\le e^{-r_{\min}(u-t)}\le e^{|r_{\min}|T_\star}$. (ii) Freezing every anchor coordinate at its initial value makes each increment in (3) vanish pathwise, and the frozen states inherit (6) from the initial curve. (iii) In currency terms the deformation multiplies each ratio $P(u')/P(u)$ by $\Lambda^{\kappa(u)-\kappa(u')}\le1$, so every retained log-forward rate weakly increases and (6) is preserved; along the frozen continuation the later curves are the ratios $\bigl(P^0(u)/P^0(t)\bigr)\Lambda^{\kappa(t)-\kappa(u)}$, again with raised forwards. Positivity, the price bound, and continuity in $\Lambda$ are immediate. (iv) $D_\Lambda$ raises log-forward rates, by the computation in (iii) applied to the terminal ramp, and lowers prices, so it preserves $K^{\rm host}_s$ and the price bound outright. The compensator multiplies the date-$s$ currency curve by $u\mapsto(1-p\Lambda^{1-\kappa(u)})/(1-p\Lambda)$, a nondecreasing factor with values in $[1,(1-p)/(1-p\Lambda)]$, so each retained log-forward rate is lowered by at most $-\delta_*^{-1}\ln(1-p\Lambda)\le\gamma$ whenever $p\Lambda\le1-e^{-\gamma\delta_*}$; a $\gamma$-interior state therefore stays above the floor, and the price bound follows from the floor chain as in (i).∎ The following finite-dimensional fact shows that relative-interior targets are dense in any chosen attainable affine image. It neither asserts that the observed target is interior nor preserves separately fixed target coordinates unless it is applied to the corresponding convex slice. **Lemma 2.13 (Relative-interior targets are generic).** Let $\cC$ be an affine map from a nonempty convex set $\cD$ of laws to $\R^m$ and let $A=\cC(\cD)$. Then: - 1. there are $r\le m+1$ laws $Q_1,\ldots,Q_r\in\cD$ whose images form an affine basis of $\operatorname{aff}A$, and their uniform mixture $\bar Q:=r^{-1}\sum_jQ_j\in\cD$ has $\cC(\bar Q)\in\ri A$; - 2. for every $Q_0\in\cD$ and every $\delta\in(0,1]$, $\cC\bigl((1-\delta)Q_0+\delta\bar Q\bigr)\in\ri A$; - 3. consequently the targets in $\ri A$ form a convex dense subset of $A$ containing $(1-\delta)A+\delta\,\cC(\bar Q)$ for every $\delta\in(0,1]$: a relative-interior hypothesis on the target can fail only on the relative boundary of the attainable set, and any attainable target is moved into the relative interior by an arbitrarily small fixed mixture, at a target perturbation of order $\delta$. *Proof.* (i) $\operatorname{aff}A$ is the affine hull of the image points, so some $r\le m+1$ of them form an affine basis; convexity of $\cD$ and affinity of $\cC$ give $\cC(\bar Q)=r^{-1}\sum_j\cC(Q_j)$, the barycentre of that basis, which lies in the relative interior of the simplex $\Delta:=\conv\{\cC(Q_j)\}$. Since $\Delta\subseteq A$ and $\operatorname{aff}\Delta=\operatorname{aff}A$, a relative neighbourhood of the barycentre inside $\Delta$ is one inside $A$, so $\cC(\bar Q)\in\ri A$. (ii) $\cC((1-\delta)Q_0+\delta\bar Q) =(1-\delta)\cC(Q_0)+\delta\,\cC(\bar Q)$, and the segment from a point of $A$ to a point of $\ri A$ lies in $\ri A$ except possibly at the $A$-endpoint (Rockafellar, 1970, Section 6). (iii) Immediate from (ii), with density as $\delta\downarrow0$.∎ ## 3 Failure of standard-timing books Remark 2.11 closed the market section with a precise question: which quoted convexity upgrades the free first-moment pins to a genuine tail control? The negative side of the paper, occupying this section, answers: none within the standard-timing universe. The escape family of Definition 3.1 supplies the mechanism, Proposition 3.5 isolates the structural reason—every standard claim weights its rate convexity by a numeraire-*linear* factor—and Theorem 3.6 converts the two into unbounded superlinear annuity moments over the exactly calibrated class, with observable properness itself failing in its wake (Corollary 3.9) and CMS and cash-settled panels offering no way out (Corollary 3.11). Throughout this section $e=t_1$ is the first calendar date, the initial state $x^0$ is deterministic, and every book claim is $\cF_e$-measurable. This covers the configuration of the joint problem itself—a caplet–swaption panel referencing one common fixing/exercise date after a deterministic initial segment—and is the scope of the base theorem below. The general case—books fixing and paying across the whole calendar—is proved in Section 3.4 by a different device: a conditional-mean- preserving surgery exploiting the diagonal linearity of the escape deformation. We write $p^0$ for a base currency curve at $e$ and construct from it a one-parameter family of admissible deformations. ### 3.1 The numeraire-segment escape family The family is a single ramp in maturity: it steepens the curve only beyond the last annuity payment, and leaves everything the swap market quotes untouched. **Definition 3.1 (Escape deformation).** For $\Lambda\ge1$ define $$ \begin{equation} \kappa(u)=\frac{(u-T_m)^+}{T_\star-T_m}\in[0,1], \qquad P^\Lambda(u)=P^0(u)\,\Lambda^{-\kappa(u)}, \qquad u\in[e,T_\star]. \label{eq:pc1-deformation}\tag{7} \end{equation} $$ Thus $P^\Lambda=P^0$ on $[e,T_m]$, $P^\Lambda(T_\star)=P^0(T_\star)/\Lambda$, and $P^\Lambda$ is positive, equals one at $e$, and is nonincreasing whenever $P^0$ is. The deformation raises forward rates only inside the numeraire segment $(T_m,T_\star)$ and leaves the swap-segment currency curve unchanged. **Lemma 3.2 (Scaling of the state).** Under (7) the anchor coordinates transform as $$ \begin{equation} \Zbar^\Lambda_e(u)=\Lambda^{1-\kappa(u)}\,\Zbar^0_e(u), \label{eq:pc1-coordinate-scaling}\tag{8} \end{equation} $$ so every swap-segment coordinate ($u\le T_m$), including $\zeta_e$ and each $\Zbar_e(q_j)$, is multiplied exactly by $\Lambda$, and every coordinate is multiplied by at most $\Lambda$. Consequently $\Abar^\Lambda_e=\Lambda\,\Abar^0_e$ and $\Fbar^\Lambda_e=\Lambda\,\Fbar^0_e$, while every currency-curve functional of the restriction to $[e,T_m]$—in particular the swap rate $S_e$, every swap-segment simple rate, the currency annuity $A^{ccy}_e$, and any cash-settlement rate and multiplier—is unchanged. Along the frozen continuation of Assumption 3.3, the later currency prices are $P_t(u)=\bigl(P^0(u)/P^0(t)\bigr)\Lambda^{\kappa(t)-\kappa(u)}$, so swap-segment currency prices observed at any later date ($t,u\le T_m$, where $\kappa=0$) are unchanged as well, and every forward rate for an accrual $[a,b]$ observed at any date up to $a$ coincides with its date-$e$ value. *Proof.* Direct substitution in (1): $\Zbar^\Lambda(u) =P^0(u)\Lambda^{-\kappa(u)}\big/\bigl(P^0(T_\star)\Lambda^{-1}\bigr)$. For $u\le T_m$, $\kappa(u)=0$. The annuity and floating leg are positive, respectively signed, combinations of swap-segment anchor coordinates, and every listed rate functional is a ratio of currency prices on $[e,T_m]$, which (7) does not touch.∎ **Assumption 3.3 (Support richness).** There is a base path $\omega^0\in\Omega$ whose price state at $e$ is $p^0$, and for every $\Lambda\ge1$ the path $\omega^\Lambda$—identical before $e$, carrying the curve $P^\Lambda$ at $e$, and continued deterministically after $e$ with all anchor coordinates frozen at their date-$e$ values—belongs to $\Omega$. The map $\Lambda\mapsto\omega^\Lambda$ is continuous in the declared finite-dimensional path topology. The frozen continuation is admissible for Assumption 2.2: the implied later currency prices are $P_t(u)=\Zbar_e(u)/\Zbar_e(t) =\bigl(P^0(u)/P^0(t)\bigr)\Lambda^{\kappa(t)-\kappa(u)} \le P^0(u)/P^0(t)$ since $\kappa$ is nondecreasing. Along $\omega^\Lambda$ every post-$e$ price increment vanishes, so $\delta_{\omega^\Lambda}$ satisfies every conditional martingale row after $e$ pathwise. A declared hard support that caps $\zeta_e$—a finite tree over bounded states, say—violates Assumption 3.3 by construction; such truncations are the standing alternative to everything below, and then the cap itself is the tail assumption (Section 8). Escape-richness is an explicit structural hypothesis: it asks that the declared support be closed under the particular terminal-segment steepenings above. An unbounded-rate class need not have that closure property, and unboundedness alone does not imply the hypothesis. It is satisfiable in a canonical class: by Proposition 2.12 (iii), the rate-floor host contains the escape family launched from *every* base path, and indeed every capped-ramp and terminal-ramp variant used later. ### 3.2 Standard-timing claims are numeraire-linear With the escape admissible, what matters is how quoted payoffs behave along it. One growth computation settles that, once the traded universe is delimited. **Definition 3.4 (Standard-timing claim).** A claim $(C,b)$ with payment date $b\in U_e\cup\{T_\star\}$ is standard-timing if one of the following holds on $\Omega$: - (S1) $|C|\le c_C<\infty$ (bounded currency payoff); - (S2) $C$ is a Borel function, of arbitrary growth, of the currency curves observed at dates up to $b$ and restricted to maturities in the swap segment $[e,T_m]$; - (S3) $C=h\bigl(L(a,b)\bigr)$ with $h$ of at most linear growth, where $L(a,b)=\bigl(P(a)/P(b)-1\bigr)/\delta_{ab}$ is the simple rate for an arbitrary accrual $[a,b]\subseteq[e,T_\star]$, observed at any date up to $a$ and paid at its accrual end $b$; along the frozen continuation this rate coincides with its date-$e$ forward value (Lemma 3.2). The standard class $\cS$ consists of finite portfolios of such claims. Class (S1) contains every physical swaption: indeed $(F^{ccy}_e-KA^{ccy}_e)^+\le|F^{ccy}_e|+|K|A^{ccy}_e$, and both legs are finite linear combinations of currency bond prices bounded by Assumption 2.2. It also contains every cash-settled swaption ($\mathfrak a(\widehat S)(\widehat S-K)^+ \le\sum_j\delta_j\widehat S/(1+\delta_j\widehat S)^{\,j}\le m$ on $\widehat S\ge0$; more generally use the declared multiplier bound following (4)), and every bond option. Class (S2) contains every CMS caplet, floorlet, or linear CMS coupon of any tenor contained in the swap segment and any payment lag, and every swap-segment caplet. Class (S3) contains caplets and floorlets on arbitrary accruals, including accruals inside the numeraire segment, at their natural payment dates. **Proposition 3.5 (Uniform numeraire-linearity).** For every standard-timing claim $(C,b)$ there is a constant $K_{C}<\infty$, depending only on the claim and the base state, such that its anchor payoff along the escape family satisfies $$ \begin{equation} \bigl|\Zbar^\Lambda_e(b)\,C(\omega^\Lambda)\bigr| \le K_{C}\,\Lambda, \qquad\Lambda\ge1. \label{eq:pc1-linear-bound}\tag{9} \end{equation} $$ *Proof.* By (8), $\Zbar^\Lambda_e(b)=\Lambda^{1-\kappa(b)}\Zbar^0_e(b)\le\Lambda\,\Zbar^0_e(b)$. For (S1), $|C|\le c_C$ gives (9) with $K_C=c_C\Zbar^0_e(b)$. For (S2), Lemma 3.2 gives $C(\omega^\Lambda)=C(\omega^0)$, a constant. For (S3), the deformed rate is ∎ $$ L^\Lambda_e(a,b) =\frac1{\delta_{ab}} \Bigl(\frac{P^0(a)}{P^0(b)}\,\Lambda^{\kappa(b)-\kappa(a)}-1\Bigr), $$ so with $|h(x)|\le\alpha+\beta|x|$, $$ \bigl|\Zbar^\Lambda_e(b)h(L^\Lambda)\bigr| \le\Zbar^0_e(b)\Bigl[ \bigl(\alpha+\tfrac{\beta}{\delta_{ab}}\bigr)\Lambda^{1-\kappa(b)} +\tfrac{\beta}{\delta_{ab}}\frac{P^0(a)}{P^0(b)}\, \Lambda^{1-\kappa(a)}\Bigr] \le K_C\,\Lambda, $$ because $\kappa\ge0$. The exponent bookkeeping in the last display is the telescoping: the rate’s gain of $\Lambda^{\kappa(b)-\kappa(a)}$ is exactly cancelled by the discount’s loss of $\Lambda^{-\kappa(b)}$, leaving at most one full power of $\Lambda$, attained only when the accrual starts in the swap segment. Finite portfolios inherit the bound by the triangle inequality. ### 3.3 The negative theorem Let $\cB$ be a finite standard book: constraints $\ell_i\le\E_Q[\zeta_{b_i}C_i]\le u_i$, $i\le n$, with every $(C_i,b_i)\in\cS$, together with the martingale rows. Because $x^0$ is deterministic and $e=t_1$, the rows over $[0,e]$ reduce to the finitely many equalities $\E_Q[\Zbar_e(u)]=\Zbar_0(u)$, $u\in U_e$. It is convenient to treat these initial-curve rows as quote components. Let $$ \cM':=\Bigl\{Q\text{ on }\Omega:\ \text{initial state }x^0,\ \text{all conditional martingale rows at dates}\ge e\Bigr\} \supseteq\cM, $$ i.e. the class obtained from Definition 2.4 by releasing only the finitely many unconditional rows over $[0,e]$. Write $$ \cM'_{\rm f} :=\bigl\{Q\in\cM':\ \E_Q\bigl|\zeta_{b_i}C_i\bigr|<\infty,\ i\le n,\ \E_Q[\Zbar_e(u)]<\infty,\ u\in U_e\bigr\}, $$ a convex subset of $\cM'$, and collect the book expectations *and* those released rows into the finite *affine* map $\cC:\cM'_{\rm f}\to\R^{n+k_e}$. No topology on laws is used in this section: the argument below is finite-dimensional convex geometry, and $\cC$ is affine on the convex domain $\cM'_{\rm f}$; weak continuity of $\cC$ neither holds for arbitrary Borel standard-timing payoffs nor is needed. A law $Q\in\cM'_{\rm f}$ with $\cC(Q)=c^*$, where the initial-curve components of $c^*$ equal $(\Zbar_0(u))_{u\in U_e}$, automatically lies in $\cM$. Fix a calibrated $Q^*\in\cM(\cB)$—which lies in $\cM'_{\rm f}$ by the book integrability in the definition of $\cM(\cB)$ and the initial-row integrability of $Q^*\in\cM$—and set $c^*=\cC(Q^*)$. By Assumption 3.3 and the frozen continuation, which satisfies every conditional row pathwise, and because a Dirac mass has finite book values, $\delta_{\omega^\Lambda}\in\cM'_{\rm f}$ for every $\Lambda\ge1$. **Theorem 3.6 (No finite standard book bounds a superlinear weight moment).** Assume Assumptions 2.2 and 3.3 and the quote-Slater condition $c^*\in\ri\cC(\cM'_{\rm f})$, with the relative interior attained by laws of finite gauge budget whenever a gauge is under discussion. Then for every superlinear increasing convex $\Phi\ge0$, $$ \sup\bigl\{\E_Q[\Phi(\Abar_e)]: Q\in\cM,\ \cC(Q)=c^*\bigr\}=\infty. $$ In particular the calibrated class $\cM(\cB)$, which contains $\{Q\in\cM:\cC(Q)=c^*\}$, contains laws with arbitrarily large superlinear annuity moments. Since the weight-tail target of Definition 2.8 requires a uniform bound for every member of the weight family (4), divergence of the annuity member alone shows that $\cB$ fails the target. Divergence for an individual non-annuity weight is not claimed in general: it transfers, by the same proof with the probability schedule retuned to that weight, exactly when the weight scales proportionally to $\Lambda$ with positive base value along the escape—so for every weight loading positively on a swap-segment maturity—whereas a weight concentrated on numeraire-segment maturities scales like $\Lambda^{1-\kappa}$, sublinearly, and is not covered. *Proof.* Write $v_\Lambda=\cC(\delta_{\omega^\Lambda})\in\R^{n+k_e}$ for the functional vector of the escape atom; its book components are bounded by $\max_iK_{C_i}\Lambda$ by Proposition 3.5, and its initial-curve components equal $\Zbar^\Lambda_e(u)\le\Lambda\Zbar^0_e(u)$ by Lemma 3.2. Hence $|v_\Lambda|\le K\Lambda$ for a constant $K$. We first fix, once and for all, a finite family of correcting laws. Let $V=\operatorname{dir}\operatorname{aff}\cC(\cM'_{\rm f})$ be the direction space and $r=\dim V$. Since $c^*\in\ri\cC(\cM'_{\rm f})$, there is $\rho>0$ with $c^*+d\in\cC(\cM'_{\rm f})$ for every $d\in V$, $|d|\le\rho$ (Rockafellar, 1970, Section 6). Choose an orthonormal basis $e_1,\ldots,e_r$ of $V$ and laws $Q_i^{\pm}\in\cM'_{\rm f}$ with $\cC(Q_i^{\pm})=c^*\pm\rho e_i$. Let $g_\Lambda:=\Phi(\Lambda\Abar^0_e)/(\Lambda\Abar^0_e) \to\infty$, as superlinearity requires, and put ∎ $$ p_\Lambda:=\frac1{\Lambda\sqrt{g_\Lambda}}, \qquad\text{so}\qquad p_\Lambda\Lambda=\frac1{\sqrt{g_\Lambda}}\to0, \qquad p_\Lambda\,\Phi(\Lambda\Abar^0_e) =\Abar^0_e\sqrt{g_\Lambda}\to\infty. $$ The correction the atom requires is $$ d_\Lambda:=\frac{c^*-p_\Lambda v_\Lambda}{1-p_\Lambda}-c^* =\frac{p_\Lambda}{1-p_\Lambda}\,(c^*-v_\Lambda)\in V, \qquad|d_\Lambda|=O(p_\Lambda\Lambda)\to0, $$ where $d_\Lambda\in V$ because $c^*$ and $v_\Lambda$ both lie in $\cC(\cM'_{\rm f})$ ($Q^*\in\cM'_{\rm f}$ and $\delta_{\omega^\Lambda}\in\cM'_{\rm f}$). Expand $d_\Lambda=\sum_i\alpha_i(\Lambda)e_i$ and set $\mu_i^{\pm}(\Lambda):=\alpha_i^{\pm}(\Lambda)/\rho\ge0$, so that $d_\Lambda=\sum_i\mu_i^+\,(\rho e_i)+\mu_i^-\,(-\rho e_i)$ and $s_\Lambda:=\sum_i(\mu_i^++\mu_i^-)=O(p_\Lambda\Lambda)\to0$. Define the *fixed-mixture correction* $$ R_\Lambda:=(1-s_\Lambda)\,Q^* +\sum_{i=1}^r\bigl(\mu_i^+Q_i^{+}+\mu_i^-Q_i^{-}\bigr)\in\cM'_{\rm f}, \qquad\cC(R_\Lambda)=c^*+d_\Lambda, $$ which is a legitimate convex mixture for all large $\Lambda$, and put $$ Q_\Lambda:=(1-p_\Lambda)R_\Lambda+p_\Lambda\, \delta_{\omega^\Lambda}. $$ Then $Q_\Lambda$ is a probability law on $\Omega$ with the correct initial state; the conditional martingale rows at dates $\ge e$ are affine in the law and hold for every mixture component, and finite convex mixtures of finite-book-value laws have finite book values, so $Q_\Lambda\in\cM'_{\rm f}$; and $\cC(Q_\Lambda)=(1-p_\Lambda)(c^*+d_\Lambda)+p_\Lambda v_\Lambda=c^*$ exactly. The initial-curve components of $c^*$ are the released rows, so $Q_\Lambda\in\cM$ and $Q_\Lambda$ matches every book band. Finally, by Lemma 3.2 and $\Phi\ge0$, $$ \E_{Q_\Lambda}\bigl[\Phi(\Abar_e)\bigr] \ge p_\Lambda\,\Phi\bigl(\Lambda\Abar^0_e\bigr) \longrightarrow\infty. \qedhere $$ **Remark 3.7 (The Slater condition).** The hypothesis $c^*\in\ri\cC(\cM'_{\rm f})$ asks for relative-interior room in the joint attainable set of book values and initial-curve rows over the prediction class—the standard quote-Slater pattern of exact finite-panel calibration, here applied with the curve rows counted among the quotes. It excludes targets on the relative boundary of the attainable convex set, which need be neither closed nor polyhedral. No conclusion is made here for such targets; a separate face analysis would be required and may reveal a boundary constraint that blocks the escape. By Lemma 2.13, applied with $\cD=\cM'_{\rm f}$ and the joint book-and-released-row map $\cC$, relative-interior targets form a convex dense subset of that full attainable image. The small mixture in the lemma generally perturbs both the book values and the released initial-curve row targets. It therefore neither proves that the observed $c^*$ is interior nor preserves the fixed initial curve; the quote-Slater condition above remains an explicit hypothesis. If one instead fixes the released rows first, the lemma may be applied to that convex slice and its book-only image, but that is a different relative-interior statement from the one used in Theorem 3.6. **Lemma 3.8 (Output-space compactness criterion).** Equip the finite nonnegative Borel measures on $\R$ with the metric $$ d(\nu,\mu)=d_{\rm BL}(\nu,\mu)+\bigl|\nu(\R)-\mu(\R)\bigr| +\Bigl|\int|y|\,\dd\nu-\int|y|\,\dd\mu\Bigr|, $$ $d_{\rm BL}$ the bounded-Lipschitz distance. A family $\cV$ of such measures is relatively compact if - 1. masses are uniformly bounded, $\sup_{\nu\in\cV}\nu(\R)<\infty$; and - 2. first moments are uniformly integrable, $\sup_{\nu\in\cV}\int_{\{|y|>R\}}|y|\,\dd\nu\to0$ as $R\to\infty$. *Proof.* Conditions (i)–(ii) give uniform tightness by Markov, $\nu(|y|>R)\le R^{-1}\int_{\{|y|>R\}}|y|\,\dd\nu$, and uniformly bounded first moments, $\int|y|\,\dd\nu\le R_0\,\nu(\R)+\varepsilon_0$. Let $(\nu_n)\subseteq\cV$. If $\liminf_n\nu_n(\R)=0$, pass to a subsequence with $\nu_{n_k}(\R)\to0$; then $d_{\rm BL}(\nu_{n_k},0)\le\nu_{n_k}(\R)\to0$ and $\int|y|\,\dd\nu_{n_k}\le R\,\nu_{n_k}(\R) +\sup_{\cV}\int_{\{|y|>R\}}|y|\,\dd\nu$, which vanishes on letting $k\to\infty$ then $R\to\infty$; so $\nu_{n_k}\to0$ in $d$. Otherwise pass to a subsequence with $\nu_n(\R)\to m>0$ and normalize: the probability measures $\hat\nu_n=\nu_n/\nu_n(\R)$ are uniformly tight, so Prokhorov’s theorem (Kallenberg, 2021) yields $\hat\nu_{n_k}\Rightarrow\hat\mu$ weakly, hence $\nu_{n_k}\Rightarrow\mu:=m\hat\mu$, giving $d_{\rm BL}(\nu_{n_k},\mu)\to0$ and mass convergence. For the moment coordinate put $\psi_R(y)=|y|\wedge R$: $\int\psi_R\,\dd\nu_{n_k}\to\int\psi_R\,\dd\mu$ for each $R$ by weak convergence, while $0\le\int(|y|-\psi_R)\,\dd\nu_{n_k} \le\sup_{\cV}\int_{\{|y|>R\}}|y|\,\dd\nu\to0$ uniformly in $k$; a three-epsilon argument with monotone convergence in $R$ gives $\int|y|\,\dd\mu<\infty$ and $\int|y|\,\dd\nu_{n_k}\to\int|y|\,\dd\mu$. Hence $\nu_{n_k}\to\mu$ in $d$.∎ **Corollary 3.9 (Failure of observable properness).** Let $\chi$ be any admissible gauge (Definition 2.9)—in particular, taking the annuity test with $\varphi\equiv1$, $\Abar_e\le C_\varepsilon+\varepsilon\chi$ for every $\varepsilon>0$. Under Assumption 2.3 and the hypotheses of Theorem 3.6, with the correcting laws $Q^*,Q_i^{\pm}$ of its proof chosen with finite $\chi$-budget, there is a compact subset $K$ of the observation output space such that $$ \sup\bigl\{\E_Q[\chi]: Q\in\textstyle\bigcup_{M<\infty}\{\,Q\in\cM:\E_Q\chi\le M\,\},\ \Gamma(Q)\in K\bigr\}=\infty. $$ Thus observable properness fails for every admissible gauge on an escape-rich generic support: not only does the standard book fail to verify it, no admissible gauge satisfies it. *Proof.* Take $\Phi(x)=x^2$ in Theorem 3.6 and retain its family $Q_\Lambda=(1-p_\Lambda)R_\Lambda+p_\Lambda\delta_{\omega^\Lambda}$, except that $p_\Lambda$ is now chosen from the gauge: put $h(\Lambda):=\chi(\omega^\Lambda)$. Sublinearity with $\varepsilon=1$ gives $h(\Lambda)\ge\Abar_e(\omega^\Lambda)-C_1 =\Lambda\Abar^0_e-C_1\to\infty$, and for every $\varepsilon$, $\Lambda\Abar^0_e\le C_\varepsilon+\varepsilon h(\Lambda)$, so $\Lambda/h(\Lambda)\to0$. Choose $p_\Lambda:=(\Lambda h(\Lambda))^{-1/2}$; then $p_\Lambda\Lambda=(\Lambda/h)^{1/2}\to0$, so every step of the theorem’s proof is unchanged, while $$ \E_{Q_\Lambda}[\chi] \ge p_\Lambda\,h(\Lambda) =\bigl(h(\Lambda)/\Lambda\bigr)^{1/2}\longrightarrow\infty. $$ Each $Q_\Lambda$ has finite $\chi$-budget: the atom contributes $p_\Lambda h(\Lambda)<\infty$ and the mixture components were chosen with finite budget. It remains to place $\{\Gamma(Q_\Lambda)\}$ in one compact output set for the observation topology of Section 2; by Lemma 3.8 it suffices to check, coordinate by coordinate, uniformly bounded masses and uniformly integrable first moments for $\{\Gamma_a(Q_\Lambda):\Lambda\ge\Lambda_0\}$. Masses are uniformly bounded by pinned coordinate means over the whole class: $\Gamma_a(Q)(\R)=\E_Q[W_a]\le\sum_jc_j\Zbar_0(d_j)$ for every $Q\in\cM$, by linear domination and the martingale rows. For the first-moment tails $\int_{\{|y|>R\}}|y|\,\dd\Gamma_a(Q_\Lambda)$, split the mixture. The correction part lies in the convex hull of the *fixed* finite set $\{\Gamma_a(Q^*),\Gamma_a(Q_i^{\pm})\}$, each member of finite first moment since $\E[W_a|Y_a|]$ is bounded by pinned linear envelopes; their tails vanish as $R\to\infty$ by dominated convergence, and a tail of a convex combination is the convex combination of the tails. The atom part has *total* first moment $p_\Lambda W_a|Y_a|(\omega^\Lambda)\le p_\Lambda\Lambda\,K_a'\to0$, by Lemma 3.2 and linear domination of $W_a(1+|Y_a|)$ together with the frozen swap-segment observables; so given $\varepsilon>0$ there is $\Lambda_1$ beyond which the atom’s whole moment is below $\varepsilon$, while for the compact range $\Lambda\in[\Lambda_0,\Lambda_1]$ the atom is supported at the single point $Y_a(\omega^\Lambda)$, bounded because $\Lambda\mapsto\omega^\Lambda$ is continuous into $\Omega$ and $Y_a$ is continuous, so enlarging $R$ empties those tails entirely. The tails therefore vanish uniformly, $K:=\overline{\{\Gamma(Q_\Lambda):\Lambda\ge\Lambda_0\}}$ is compact by Lemma 3.8 applied in each of the finitely many coordinates, and $\Gamma(Q_\Lambda)\in K$ by construction.∎ **Remark 3.10 (What the corollary does and does not say).** The corollary upgrades a single escape example to a statement about the properness hypothesis itself: on any support rich enough to contain the deformation family, and for any book with quote-Slater room, the hypothesis is not merely unverified but false, for every admissible gauge. It does not touch supports that exclude the family—hard caps on $\zeta$ remain the coherent alternative—and it is proved here only in the $e=t_1$, $\cF_e$-book scope stated above. The phrase “for every admissible gauge” includes the load-bearing requirement that the fixed Slater correcting laws can be chosen with finite budget for that gauge. **Corollary 3.11 (CMS and cash-settled quotes cannot close the image).** CMS caplets, floorlets, and coupons of any strike, tenor, and payment lag within the swap segment, and cash-settled swaptions of any strike, are standard-timing claims. Add any finite family of them to a finite standard book and suppose the augmented finite-integrability domain and quote map still satisfy the quote-Slater hypothesis of Theorem 3.6. Then that theorem remains in force: their rare-escape-mixture contribution is $O(p_\Lambda\Lambda)=o(1)$ before the exact finite correction, while every superlinear annuity moment diverges. *Proof.* Classes (S2) and (S1) of Definition 3.4, as itemized there; then Theorem 3.6.∎ **Remark 3.12 (Diagnosis: convexity in the wrong variable).** Along the escape the swap rate, every swap-segment rate, and the cash-settlement functional are frozen; only the numeraire ratio $\zeta$ moves. CMS and cash-settled convexity is convexity in the rate and is therefore blind to the move, while the claims’ anchor weights are exactly linear in $\zeta$. Properness for the annuity weight is a tail condition on the numeraire direction. Under the quote-Slater mixing hypothesis, and excluding a boundary or support constraint that removes the escape, a nonnegative quoted instrument can block the escape through a finite ask only if its anchor payoff grows superlinearly in $\Lambda$ along Definition 3.1; by Proposition 3.5 this requires either payment before the accrual end of a rate it references, or explicit superlinear dependence on far-segment inverse bond prices. This is the exact content of the next section. Collapsing the swap-segment data to a pinned pair $(B\equiv1,\ A)$ yields an abstract two-coordinate version of the escape; the family of Definition 3.1 realizes it on an admissible curve with positive forwards and identifies which market segment carries it. ### 3.4 The multi-date extension The preceding theorem treats books of $\cF_e$-measurable claims at the first calendar date. A general book fixes and pays across the whole calendar, and the natural pasting attempt fails at once: an escape atom inserted at an interior date violates the *conditional* martingale rows at every earlier date along its history—a continuum of constraints, unreachable by any finite-dimensional relative-interior correction. The extension rests on one structural fact: the escape deformation is *diagonal-linear* on the live coordinates, $D_\Lambda x=(\Lambda^{1-\kappa(u)}x_u)_u$. It can therefore be paired with the compensating diagonal map $$ \begin{equation} S_{\Lambda,p}:=\frac{I-p\,D_\Lambda}{1-p}, \qquad(1-p)\,S_{\Lambda,p}+p\,D_\Lambda=I \quad\text{\emph{as an identity of maps}}, \label{eq:pc1-compensator}\tag{10} \end{equation} $$ well defined with positive entries whenever $p\Lambda<1$, acting on the live coordinates of the date-$s$ state and copying stopped coordinates. Replacing each date-$s$ state $x$ by the two-point branch $\{D_\Lambda x$ (probability $p$), $S_{\Lambda,p}x$ (probability $1-p)\}$, each followed by the frozen continuation $F_s$, preserves the conditional mean of every live coordinate at *every* history simultaneously and exactly. The continuum of conditional rows is disposed of pointwise; only the finitely many book values move, and those the Slater correction fixes. The fixture `PC1-MD-2P `(`pc1_multidate_regression.py`) verifies the mechanism numerically; see Section 7. Fix a calendar date $s=t_j$ with $t_1\le s\le T_m$. A coupon date $q_j\ge s$ is *live* at $s$ with $\kappa(q_j)=0$; a coupon date $q_j0, $$ so that the surgery below scales $\Abar_s^{\rm live}$ by exactly $\Lambda$ and *copies* $\Abar_s^{\rm hist}\ge0$; for $s\le q_1$ the historical part vanishes and $\Abar_s=\Abar_s^{\rm live}$. By carry closure and Lemma 2.5, $\E_{Q}[\Abar^{\rm live}_s]=\Abar_0^{\rm live}(s)$ for every $Q\in\cM$. **Assumption 3.13 (Multi-date support richness, localized).** There is a Borel set $S_s\subseteq K_s$ of date-$s$ states—the surgery region, declared with the support—such that for every bounded $B\subseteq S_s$ there is $\varepsilon_0(B)>0$ with the following property: for every $\omega\in\Omega$ with $x^s(\omega)\in B$, every $\Lambda\ge1$, and every $p$ with $p\Lambda\le\varepsilon_0(B)$, the two modified paths—agreeing with $\omega$ before $s$, carrying $D_\Lambda(x^s(\omega))$ respectively $S_{\Lambda,p}(x^s(\omega))$ on the live coordinates at $s$ with stopped coordinates copied, and frozen thereafter—belong to $\Omega$ and satisfy Assumption 2.2. The resulting two branch maps are Borel. The deformation lowers far-segment currency prices and fixes the rest; the compensator perturbs currency prices by a factor in $[1,1+2p\Lambda]$, so the price bound needs the vanishing slack the assumption grants. The localization to a declared surgery region with region-dependent slack is deliberate, and is what the proof actually uses: its truncation step works on one bounded subset of $S_s$ at a time. It is also what a support carrying a hard constraint boundary can honestly provide: at a state of the rate-floor host of Proposition 2.12 with a floor-binding retained pair on which the terminal ramp is strict, the compensator branch exits the support for every $\Lambda>1$ and every $p>0$ with $p\Lambda<1$. Thus a globally uniform slack over all bounded state sets is unavailable, while every interior-margin region works with a slack depending only on the margin (Proposition 2.12 (iv)). **Assumption 3.14 (Frozen surgery envelopes).** For every book claim paying at a date $b\ge s$ there is a nonnegative Borel envelope $\varphi_i(h,x)$ such that its absolute anchor payoff on the continuation frozen from the date-$s$ prefix $(h,x)$ is at most $\varphi_i(h,x)$, $\E_{Q^*}[\varphi_i(h,x^s)]<\infty$, and, throughout the richness window, $$ \begin{equation} \varphi_i(h,S_{\Lambda,p}x)\le\varphi_i(h,x), \qquad\varphi_i(h,D_\Lambda x)\le\Lambda\varphi_i(h,x). \label{eq:pc1-frozen-envelope}\tag{11} \end{equation} $$ The original anchor payoffs are required to be absolutely integrable only on the finite-integrability domain used below; no weak continuity is asserted for this Borel book. **Lemma 3.15 (Sufficient envelope cases).** Every positive affine envelope $$ \varphi(h,x)=a(h)+\sum_u c_u(h)x_u, $$ with nonnegative $\cF_{s^-}$-measurable coefficients and finite $Q^*$-expectation, satisfies (11). Consequently Assumption 3.14 holds for every bounded-currency claim (S1) ($\varphi=c_C\,x_b$, constant coefficient); every natural-timing rate claim (S3) with $|h(L)|\le\alpha+\beta|L|$ ($\varphi=(\alpha+\beta/\delta_{ab})\,x_b+(\beta/\delta_{ab})\,x_a$, constant coefficients); and every claim whose payoff is fixed strictly before $s$ ($\varphi=|C(h)|\,x_b$ with $\cF_{s^-}$-measurable coefficient, integrable since $\E_{Q^*}[|C|\,x_b]=\E_{Q^*}[|C|\,\zeta_b]<\infty$). A swap-segment claim (S2) fixing exactly at $s$ is also covered, without a payoff cap: take $\varphi(h,x)=x_b|C(h,x)|$. Both $D_\Lambda$ and $S_{\Lambda,p}$ multiply all live swap-segment coordinates by a common factor, so the referenced currency curves and $C$ are unchanged, while $x_b$ scales by at most $\Lambda$, respectively at most one. Its envelope is integrable because, conditionally on $\cF_s$, carry closure for the retained payment maturity gives $x_b=\E_{Q^*}[\zeta_b\mid\cF_s]$, and the book anchor payoff is absolutely integrable. A (S2) claim fixing strictly after $s$ is covered under the same formula whenever its frozen anchor payoff is $Q^*$-integrable; a declared bound on its currency payoff is a simple sufficient, but not necessary, condition. *Proof.* Each coordinate multiplier of $S_{\Lambda,p}$ lies in $(0,1]$ and each coordinate multiplier of $D_\Lambda$ lies in $[1,\Lambda]$, so the first assertion is immediate. For (S3), on frozen paths and with the frozen rate equal to the date-$s$ forward, ∎ $$ \zeta_b\,\bigl|h(L(a,b))\bigr| \le\alpha x_b+\frac{\beta}{\delta_{ab}}\,|x_a-x_b| \le\Bigl(\alpha+\frac{\beta}{\delta_{ab}}\Bigr)x_b +\frac{\beta}{\delta_{ab}}\,x_a . $$ **Theorem 3.16 (Multi-date negative theorem).** Let $\cB$ be a finite standard book, with claims fixing and paying at arbitrary calendar dates, satisfying Assumption 3.14, and let $X_i:=\zeta_{b_i}C_i$ denote its anchor payoffs. Let $c=(c_1,\ldots,c_n)$ be the finite affine quote map $c_i(Q):=\E_Q[X_i]$ on the convex domain $\cM_{\rm f}:=\{Q\in\cM:\E_Q|X_i|<\infty,\ i\le n\}$ of laws with absolutely finite book values. Assume Assumptions 2.1, 2.2 and 3.13, let $Q^*\in\cM(\cB)\subseteq\cM_{\rm f}$ with $c^*=c(Q^*)$ and $Q^*(x^s\in S_s)=1$—the base law loads the surgery region fully—and assume the quote-Slater condition $c^*\in\ri c(\cM_{\rm f})$. Then for every superlinear increasing convex $\Phi\ge0$, $$ \sup\bigl\{\E_Q[\Phi(\Abar_s)]:Q\in\cM,\ c(Q)=c^*\bigr\}=\infty. $$ The proof is deferred to Section B. **Corollary 3.17 (Multi-date failure of observable properness).** Let $\chi$ be an admissible gauge (Definition 2.9), with $Q^*$ and the correcting laws of finite $\chi$-budget. Under Assumption 2.3 and the hypotheses of Theorem 3.16 there is a compact output set $K$ with $$ \sup\bigl\{\E_Q[\chi]:\E_Q\chi<\infty,\ Q\in\cM,\ \Gamma(Q)\in K\bigr\}=\infty. $$ *Proof.* Use the trigger set $C$, raw surgery law $\widetilde Q$, fixed correcting laws $Q_i^{\pm}$, correction weights $\mu_i$, and exact laws $Q_\Lambda$ constructed in the proof of Theorem 3.16 in Section B. The live annuity is $\chi$-sublinear. Indeed, carry closure writes it as a finite linear combination of initial constants and martingale increments with the bounded test $\phi\equiv1$, and every such test is $\chi$-sublinear by Definition 2.9. Consequently, for every $\varepsilon>0$ there is $C_\varepsilon<\infty$ such that $\Abar_s^{\rm live}\le C_\varepsilon+\varepsilon\chi$ pathwise. Define the actual deformed-branch budget ∎ $$ H_\Lambda:=\E_{Q^*}\bigl[\one_C \chi(F_s(D_\Lambda x^s))\bigr]. $$ It is finite: for fixed $\Lambda$, the corresponding branch paths form a bounded subset of $\Omega$ (the prefix lies in $C$ and the post-$s$ coordinates are frozen transformed coordinates). Its closure is compact by Assumption 2.1, so the continuous $\chi$ is bounded there. Since the live annuity is multiplied by $\Lambda$ on that branch, $$ H_\Lambda\ge\frac{\Lambda\E_{Q^*}[\Abar_s^{\rm live}\one_C] -C_\varepsilon Q^*(C)}{\varepsilon} \ge\frac{\Lambda\Abar_0^{\rm live}(s)/2-C_\varepsilon}{\varepsilon}. $$ It follows that $g_\Lambda:=H_\Lambda/\Lambda\to\infty$: for every fixed $\varepsilon$, its limit inferior is at least $\Abar_0^{\rm live}(s)/(2\varepsilon)$, and then $\varepsilon\downarrow0$. For all sufficiently large $\Lambda$ take $$ p=(\Lambda g_\Lambda^{1/3})^{-1},\qquad q=g_\Lambda^{-1/3}. $$ Then $p\Lambda\to0$, $q\to0$, and the deformed branch contributes $qpH_\Lambda=g_\Lambda^{1/3}\to\infty$. The correction weight is $O(q)$, so eventually $1-\sum_i\mu_i\ge1/2$ and $\E_{Q_\Lambda}\chi\ge\tfrac12g_\Lambda^{1/3}\to\infty$. Each $Q_\Lambda$ has finite budget: both modified branch sets are bounded subsets of $\Omega$ with compact closure by Assumption 2.1, while the other mixture components have finite budget by hypothesis. For output compactness we verify the hypotheses of Lemma 3.8 coordinatewise, as in Corollary 3.9. Masses are uniformly bounded over all of $\cM$ by linear domination and pinned coordinate means. For the first-moment tails, decompose $\Gamma_a(Q_\Lambda)=(1-\sum_i\mu_i)\,\Gamma_a(\widetilde Q) +\sum_i\mu_i\Gamma_a(Q_i^{\sigma_i})$. The correction part lies in the convex hull of finitely many *fixed* laws of finite $\E[W_a|Y_a|]$, whose tails vanish as $R\to\infty$. The $\widetilde Q$ part splits into the unmodified paths, whose tail is at most the fixed tail of $Q^*$, and the surgery branches, whose *total* contribution to $\E[W_a|Y_a|]$ is at most $q(1+p\Lambda)\times$(pinned linear-domination envelopes), vanishing along the schedule; and for any residual compact parameter range $[\Lambda_0,\Lambda_1]$ the branch paths lie in a compact subset of $\Omega$ after taking their closure: the prefixes lie in $C$, the parameters are bounded, and Assumption 2.1 makes every bounded subset of $\Omega$ relatively compact. On that compact closure the continuous $W_a|Y_a|$ is bounded, so enlarging $R$ empties those tails. Hence the tails vanish uniformly, $K:=\overline{\{\Gamma(Q_\Lambda):\Lambda\ge\Lambda_0\}}$ is compact, and it contains every $\Gamma(Q_\Lambda)$. **Remark 3.18 (Consistency and three structural observations).** (i) The convexity quotes of Sections 4 and 6 block the surgery, as they must: the deformed branch moves an in-advance or account-convexity quote by order $qp\Lambda^2\to\infty$ along the second-moment schedule, so augmented books leave no calibrated escape. (ii) The one-date Theorem 3.6 is the degenerate case $s=t_1$ with deterministic history; the present proof is independent of the released-rows device used there. (iii) If every book maturity lies in the swap segment and every currency payoff depends only on currency curve ratios (hence is invariant under common rescaling), then $D_\Lambda$ and $S_{\Lambda,p}$ are scalar on the relevant live state, each anchor payoff is positively homogeneous of degree one, and the surgery preserves the book exactly. This covers the usual rate-based (S1)–(S3) claims, but not an arbitrary bounded (S1) payoff that explicitly reads the anchor scale. The Slater correction is therefore needed for claims mixing swap- and numeraire-segment maturities, and more generally for any claim lacking this scale invariance; the regression fixture exhibits the former case to machine precision. ## 4 The in-advance convexity strip The diagnosis of Remark 3.12 is constructive in disguise: of the two ways a quoted instrument can grow superlinearly along the escape, the simplest is payment timing. The positive side of the paper begins here, with the instrument that spends that observation. Paying the segment rate *at its fixing date* rather than at the natural payment date weights the payoff by one further factor of $\zeta_b$, and Lemma 4.2 turns the resulting quote into the second moment $\E_Q[\zeta_b^2]$ by an exact identity. From this single input the section derives a uniform second-moment certificate for every linearly dominated weight (Theorem 4.3), a market-implied inf-compact gauge under which the weighted-smile image is closed with an exact support-function dual (Theorems 4.6 and 4.7), and, in Section 4.4, lower-power and capped contract variants showing how much design freedom the certificate tolerates. ### 4.1 The instrument and the exact price identity The instrument is elementary to state, and its price identity is the whole of the input the positive side needs. **Definition 4.1 (In-advance claim).** For a calendar date $b$ and contractual accrual factor $\tau_b>0$ for the segment $[b,T_\star]$, the claim $\mathrm{IA}(b)$ pays at $b$ the currency amount $$ \begin{equation} X_b=\frac1{\tau_b}\Bigl(\frac1{P_b(T_\star)}-1\Bigr) =\frac{\zeta_b-1}{\tau_b}, \label{eq:pc1-inadvance-payoff}\tag{12} \end{equation} $$ the simple rate fixed at $b$ for the full segment $[b,T_\star]$, paid at the start of its own accrual. Under Assumption 2.2, $X_b\ge(1/C_P-1)/\tau_b$ is bounded below, while its anchor payoff $\zeta_bX_b=(\zeta_b^2-\zeta_b)/\tau_b$ has negative part bounded by $1/(4\tau_b)$. Its expectation is therefore well defined with value in $[-1/(4\tau_b),\infty]$; imposing a finite ask makes that expectation finite. No nonnegativity of the claim is assumed. The payment date precedes the accrual end of the rate the claim references; by Definition 3.4 it is not standard-timing, and it is exactly the payment-timing mismatch of the classical Libor-in-arrears and futures-margining convexity complex. **Lemma 4.2 (Exact second-moment identity).** One currency unit at $b$ is worth $\zeta_b$ anchor units, so the anchor payoff of $\mathrm{IA}(b)$ is $\zeta_bX_b=(\zeta_b^2-\zeta_b)/\tau_b$. Consequently, for every $Q\in\cM$, the following equality holds as an extended-real identity in $[-1/(4\tau_b),\infty]$: $$ \begin{equation} \E_Q\bigl[\zeta_bX_b\bigr] =\frac{\E_Q[\zeta_b^2]-\Zbar_0(b)}{\tau_b}. \label{eq:pc1-identity}\tag{13} \end{equation} $$ In particular, if the book contains the ask constraint $\E_Q[\zeta_bX_b]\le\pi_b$ with $\pi_b<\infty$, then every calibrated law satisfies $$ \begin{equation} \E_Q\bigl[\zeta_b^2\bigr]\le\tau_b\pi_b+\Zbar_0(b). \label{eq:pc1-second-moment-budget}\tag{14} \end{equation} $$ *Proof.* At $b$, one currency unit buys $1/P_b(T_\star)=\zeta_b$ anchor bonds; this is the definition (1) of the anchor price of currency. Multiplying (12) by $\zeta_b$ gives the anchor payoff. The function $\zeta_b^2-\zeta_b\ge-\tfrac14$ is bounded below, so its extended expectation belongs to $[-1/(4\tau_b),\infty]$. The right-hand side is also well defined because $\E_Q[\zeta_b]=\Zbar_0(b)<\infty$: the $b$-maturity bond’s anchor price is a true $Q$-martingale with terminal value $\Zbar_b(b)=\zeta_b$ (Definition 2.4). Rearranging proves (13), and the ask constraint gives (14).∎ The identity is exact and model-free within the class: no convexity adjustment, expansion, or lognormal proxy enters. One quoted number is affinely one second moment. Lemma 4.2 is not new: it is a specialization and re-expression, in anchor units, of the timing-spread identity of Bang and Daboussi (2022, Corollary 3.1.1), which prices the receipt of a swap rate at reset against payment of the same rate at maturity as the initial annuity times its annuity-measure second moment; taking their swap to be the one-period swap spanning $[b,T_\star]$ and normalizing by the anchor bond gives (13), the contract-level form being Proposition 5.2 below. What is claimed here is not the identity but its use: the quoted ask becomes a uniform-integrability budget for the calibrated class (Theorems 4.3, 4.6 and 4.7). ### 4.2 The certificate for linearly dominated weights That use begins here. A quoted second moment at the right maturities transfers, by domination, to every weight the quoted smiles carry. **Theorem 4.3 (Second-moment certificate).** Let $W$ have a linear domination at maturities $e_W\le d_j\le T$: $$ 0\le W\le\sum_{j\le r}c_j\Zbar_{e_W}(d_j), \qquad c_j\ge0. $$ Let the book contain the finite asks $\E_Q[\zeta_{d_j}X_{d_j}]\le\pi_j$ for $\mathrm{IA}(d_j)$, whose accrual factors are denoted $\tau_j$. Thus, by Lemma 4.2, the bounds (14) hold at all the $d_j$. Then every calibrated $Q$ satisfies $$ \begin{equation} \E_Q\bigl[W^2\bigr] \le\Bigl(\sum_{j}c_j\Bigr) \sum_{j}c_j\bigl(\tau_j\pi_j+\Zbar_0(d_j)\bigr). \label{eq:pc1-certificate}\tag{15} \end{equation} $$ In particular, for the physical annuity, $$ \begin{equation} \E_Q\bigl[\Abar_e^{\,2}\bigr] \le\Bigl(\sum_j\delta_j\Bigr) \sum_j\delta_j\bigl(\tau_j\pi_j+\Zbar_0(q_j)\bigr), \label{eq:pc1-annuity-certificate}\tag{16} \end{equation} $$ and the analogous single-term bounds hold for the caplet weight $\Zbar_\theta(p)$ and the cash weight $\Zbar_e(\sigma)H^a_e\le C_H\Zbar_e(\sigma)$. *Proof.* By the Cauchy–Schwarz inequality for the finite positive weights $c_j$, ∎ $$ W^2\le\Bigl(\sum_jc_j\Zbar_{e_W}(d_j)\Bigr)^{\!2} \le\Bigl(\sum_jc_j\Bigr)\sum_jc_j\,\Zbar_{e_W}(d_j)^2 . $$ By carry closure ($d_j\in U_{e_W}$) and Lemma 2.5, $\Zbar_{e_W}(d_j)=\E_Q[\zeta_{d_j}\mid\cF_{e_W}]$ almost surely, and conditional Jensen gives $\Zbar_{e_W}(d_j)^2\le\E_Q[\zeta_{d_j}^2\mid\cF_{e_W}]$. Taking expectations and inserting (14) termwise proves (15). No lower rate bound and no bound on the currency annuity is used. **Corollary 4.4 (Uniform integrability of the weight family).** Let $\cB^+$ be any book containing the strip ask at every domination maturity used for $W_a$ and $W_a(1+|Y_a|)$, for each $a\in\cA$. Then $$ \sup_{Q\in\cM(\cB^+)}\E_Q\bigl[W_a^2\bigr]<\infty $$ for every weight of (4), with the explicit constants of Theorem 4.3; hence $\{W_a\}$ and, for the vanilla book, $\{W_a(1+|Y_a|)\}$ are uniformly integrable over $\cM(\cB^+)$, the latter because $W_a(1+|Y_a|)$ is itself linearly dominated (Assumption 2.3). The weight-tail target of Definition 2.8 holds with $\Phi(x)=x^2$. ### 4.3 A market-implied gauge and the closed image Uniform integrability of the weight family is what the fixed-envelope theorems consume. Quoted at every date, the strip upgrades it to a coercive gauge whose budget is itself a price. **Assumption 4.5 (Quoted strip).** The book $\cB^+$ contains, for every post-initial calendar date $t_i$, $i=1,\ldots,N$, the ask constraint of $\mathrm{IA}(t_i)$ with finite normalized ask $\pi_i$. **Theorem 4.6 (Market-implied gauge).** Under Assumptions 2.1 to 2.3 and 4.5, define on $\Omega$ $$ \chi:=1+\sum_{i=1}^N\zeta_{t_i}^2, \qquad M_{\rm quoted}:=1+\sum_{i=1}^N \bigl(\tau_i\pi_i+\Zbar_0(t_i)\bigr). $$ Then $\chi$ is continuous and inf-compact on $\Omega$; every weighted test $W_a\varphi_a(Y_a)$ with $\varphi_a$ of linear growth and every martingale test of Definition 2.4 is $\chi$-sublinear, i.e. bounded by $C_\varepsilon+\varepsilon\chi$ for every $\varepsilon>0$; and every law calibrated to the strip satisfies $\E_Q\chi\le M_{\rm quoted}$. Thus $\chi$ is an admissible gauge in the sense of Definition 2.9, and the pair $(\chi,M_{\rm quoted})$ is a coercive envelope whose budget is a market quote. *Proof.* $\chi$ is a finite sum of squared coordinate functions, hence continuous. On $\{\chi\le c\}$ each $\zeta_{t_i}\le\sqrt c$ for $i\ge1$, and by Assumption 2.2 every noninitial live price coordinate obeys $\Zbar_{t_i}(u)\le C_P\zeta_{t_i}\le C_P\sqrt c$. The complete initial vector is the fixed $x^0$ on $\Omega$ by the support declaration of Section 2, so the level set is bounded in the ambient Euclidean product. It is also closed there: $\chi$ is continuous and $\Omega$ is ambient-closed by Assumption 2.1. A bounded, ambient-closed set in a finite product of Euclidean spaces is compact, and its points lie in $\Omega$, hence in the open positive orthants; without Assumption 2.1 the level set could accumulate at zero-coordinate boundary points and inf-compactness would fail, since $\chi$ controls the coordinates only from above. For sublinearity, any test in question is bounded in absolute value by $a+b\sum_i\zeta_{t_i}\le a+b'\sqrt\chi$ for constants depending on the test (weighted tests via Assumption 2.3 and Assumption 2.2; martingale tests via $|\Delta\Zbar(u)|\le C_P(\zeta_{t_i}+\zeta_{t_{i+1}})$), and $\sqrt\chi\le\varepsilon\chi+1/(4\varepsilon)$. The budget is Lemma 4.2 summed over the strip.∎ **Theorem 4.7 (Closed image and exact support dual on the calibrated class).** Let $\cM_{\rm str} :=\{Q\in\cM:\E_Q[\zeta_{t_i}^2]\le\tau_i\pi_i+\Zbar_0(t_i),\ i\le N\}$ be the class cut by the strip asks. Assume $\cM_{\rm str}\neq\emptyset$: feasibility of the augmented book is a hypothesis of this theorem, not a conclusion—an empty $\cM_{\rm str}$ is a conflict of the augmented quote panel and lies outside this theorem’s scope; on an infinite state space its detection is a separate question, not a finite-dimensional linear program. Let $\Gamma(Q)=\bigl((Y_a)_\#(W_aQ)\bigr)_{a\in\cA}$ be the finite observation family of Section 2, with linearly dominated weights, valued in the product of finite positive measures carrying the observation topology (weak convergence plus first absolute moments). Then: - 1. $\cM_{\rm str}$ is convex and weakly compact; - 2. $\Gamma$ is affine and continuous on $\cM_{\rm str}$, and $\Gamma(\cM_{\rm str})$ is compact, convex, and closed; - 3. a target $\gamma$ in the ambient product of finite positive measures with finite first absolute moments satisfies $\gamma\in\Gamma(\cM_{\rm str})$ if and only if $$ \sum_{a}\int\varphi_a\,\dd\gamma_a \le\sup_{Q\in\cM_{\rm str}} \E_Q\Bigl[\sum_aW_a\varphi_a(Y_a)\Bigr] $$ for every tuple of continuous $\varphi_a$ of at most linear growth. Consequently every conclusion available on a fixed coercive envelope holds on the strip-calibrated class with the envelope budget quoted rather than declared. Every particular infeasible target admits a certificate consisting of one finite tuple of continuous linear-growth tests. This is an existential separation statement, not an a priori finite-dimensional linear program, and the separating tuple need not be a finitely supported traded portfolio. Adjoining further quote constraints to $\cM_{\rm str}$ preserves compactness whenever their payoffs are continuous and $\chi$-sublinear—the truncation argument of the proof applies verbatim—but arbitrary Borel constraints, or unbounded constraints without this $\chi$-sublinear control, are not covered by this theorem. This bypasses observable properness on the unbudgeted class $\cM$ rather than verifying it (Remark 2.10); by Corollary 3.9, on escape-rich supports that hypothesis could not have been verified by any admissible gauge. The proof is deferred to Section B. **Remark 4.8 (What is quoted, what is declared).** In a declared-envelope implementation the gauge pair $(\chi,M)$ is a modeling input; Theorems 4.6 and 4.7 replace it by strip asks and the initial curve, and nothing else in the fixed-envelope chain changes. If the strip is quoted only on a subset of dates, the theorems hold with the calendar restricted accordingly; if it is not quoted at all, the envelope reverts to a declared field, and should be reported as such. Only ask sides are used: bids may be imposed for calibration but play no role in the tail budget, and no continuity of $Q\mapsto\E_Q[\zeta^2]$ is ever needed, which is why no fourth-moment instrument is required. ### 4.4 Contract-design variants: lower powers and caps The preceding remark separated what is quoted from what is declared. A second freedom concerns the shape of the quoted contract itself. The certificate and closure machinery is tolerant of contract engineering: neither the exponent $2$ nor the uncapped payoff is essential. The two variants below license, respectively, thin-tailed and bounded-payoff contract designs. **Proposition 4.9 ( \(p\) -power strips suffice).** Fix $p\in(1,2]$. Let $\mathrm{IA}_p(b)$ pay at $b$ the currency amount $X_{b,p}:=(\zeta_b^{\,p-1}-1)/\tau_b$, so that its anchor payoff is $(\zeta_b^{\,p}-\zeta_b)/\tau_b$, with negative part bounded by $1/(4\tau_b)$. Then: - 1. for every $Q\in\cM$, as an extended-real identity in $[-1/(4\tau_b),\infty]$, $\E_Q[\zeta_bX_{b,p}] =(\E_Q[\zeta_b^{\,p}]-\Zbar_0(b))/\tau_b$; a finite ask $\pi_{b,p}$ gives $\E_Q[\zeta_b^{\,p}]\le\tau_b\pi_{b,p}+\Zbar_0(b)$; - 2. if $0\le W\le\sum_jc_j\Zbar_{e_W}(d_j)$ with $c_j\ge0$ and retained maturities $e_W\le d_j\le T$ carrying finite asks $\pi_{j,p}$, then $$ \E_Q\bigl[W^p\bigr] \le\Bigl(\sum_jc_j \bigl(\tau_j\pi_{j,p}+\Zbar_0(d_j)\bigr)^{1/p}\Bigr)^{\!p}, $$ so the weight-tail target of Definition 2.8 holds with $\Phi(x)=x^p$; - 3. with $p$-power asks at every post-initial calendar date, $\chi_p:=1+\sum_i\zeta_{t_i}^{\,p}$ is an admissible gauge with quoted budget $M_{{\rm quoted},p}=1+\sum_i(\tau_i\pi_{i,p}+\Zbar_0(t_i))$, and Theorems 4.6 and 4.7 hold verbatim with $(\chi_p,M_{{\rm quoted},p})$ in place of $(\chi,M_{\rm quoted})$. *Proof.* (i) On $(0,1]$ and for $p\le2$ one has $z^p\ge z^2$, so $z-z^p\le z-z^2\le\tfrac14$; for $z\ge1$, $z^p\ge z$. Hence $\zeta^p-\zeta\ge-\tfrac14$ and both sides are well defined; the mean $\E_Q[\zeta_b]=\Zbar_0(b)$ is pinned by Lemma 2.5, and rearranging gives the identity. (ii) Minkowski’s inequality in $L^p(Q)$ gives $\|W\|_p\le\sum_jc_j\|\Zbar_{e_W}(d_j)\|_p$. By carry closure and Lemma 2.5, $\Zbar_{e_W}(d_j)=\E_Q[\zeta_{d_j}\mid\cF_{e_W}]$, and conditional Jensen for the convex $|\cdot|^p$ gives $\|\Zbar_{e_W}(d_j)\|_p^p\le\E_Q[\zeta_{d_j}^{\,p}] \le\tau_j\pi_{j,p}+\Zbar_0(d_j)$ by (i). (iii) The gauge proof of Theorem 4.6 transfers with three substitutions: on $\{\chi_p\le c\}$ every noninitial $\zeta_{t_i}\le c^{1/p}$ and every live coordinate is at most $C_Pc^{1/p}$, the initial vector being fixed, so level sets are bounded and, by ambient closedness, compact; sublinearity of a linear-growth test follows from $x\le\varepsilon x^{p}+C_{\varepsilon,p}$ for $p>1$; and the budget is (i) summed over the strip. The closed-image proof of Theorem 4.7 uses only inf-compactness of the gauge, $\chi_p$-sublinearity of the tests, and monotone truncation of the continuous nonnegative $\zeta_{t_i}^{\,p}$; all three persist.∎ **Proposition 4.10 (Capped strip and governed residual).** Fix a cap $K>1$ and let $\mathrm{IA}^K(b)$ pay at $b$ the currency amount $X_b^K:=\bigl((\zeta_b\wedge K)^2/\zeta_b-1\bigr)/\tau_b$. Then: - 1. $X_b^K$ is a bounded currency payoff, of class (S1): $-1/\tau_b< X_b^K\le(K-1)/\tau_b$. In particular, by Theorem 3.6, capped claims alone cannot restore the weight-tail target on escape-rich supports; - 2. the capped identity is exact: for every $Q\in\cM$, $\E_Q[\zeta_bX_b^K] =(\E_Q[(\zeta_b\wedge K)^2]-\Zbar_0(b))/\tau_b$, and a finite ask $\pi^K_b$ gives $\E_Q[(\zeta_b\wedge K)^2]\le\tau_b\pi^K_b+\Zbar_0(b)$; - 3. pathwise $\zeta_b^2=(\zeta_b\wedge K)^2+(\zeta_b^2-K^2)^+$, so for any declared tail field $\Theta_b(K)\ge\sup_{Q}\E_Q[(\zeta_b^2-K^2)^+]$ over the class in use, $$ \E_Q\bigl[\zeta_b^2\bigr] \le\tau_b\pi^K_b+\Zbar_0(b)+\Theta_b(K), $$ and Theorems 4.6 and 4.7 hold with the quoted-plus-declared budget $M=1+\sum_i(\tau_i\pi^K_i+\Zbar_0(t_i)+\Theta_i(K))$; - 4. the residual has the layer-cake representation $\E_Q[(\zeta_b^2-K^2)^+]=2\int_K^\infty\lambda\, Q(\zeta_b>\lambda)\,\dd\lambda$, valued in $[0,\infty]$ and nonincreasing in $K$; whenever it is finite at some $K_0$—equivalently $\E_Q[\zeta_b^2]<\infty$, as on any class carrying a finite declared field $\Theta_b(K_0)$—it is finite for all $K\ge K_0$ with right derivative $-2K\,Q(\zeta_b>K)$: on such classes the single governance parameter $K$ carries a computable sensitivity. *Proof.* (i) For $\zeta_b\le K$, $X^K_b=(\zeta_b-1)/\tau_b\in(-1/\tau_b,(K-1)/\tau_b]$; for $\zeta_b>K$, $X^K_b=(K^2/\zeta_b-1)/\tau_b\in(-1/\tau_b,(K-1)/\tau_b)$. (ii) Multiply by $\zeta_b$: $\zeta_bX^K_b=((\zeta_b\wedge K)^2-\zeta_b)/\tau_b$, and $|\zeta_bX^K_b|\le(K^2+\zeta_b)/\tau_b\in L^1(Q)$ because $(\zeta_b\wedge K)^2\le K^2$ and the mean $\E_Q[\zeta_b]=\Zbar_0(b)$ is pinned by Lemma 2.5; both sides are finite, and taking means and rearranging gives the identity. (iii) The pathwise square decomposition is immediate from considering $\zeta_b\le K$ and $\zeta_b>K$; the displayed bound follows from (ii), and the envelope theorems require of the budget only finiteness of $\E_Q[\zeta_{t_i}^2]$ per date, which quoted-plus-declared supplies. (iv) $(\zeta^2-K^2)^+=\int_{K^2}^{\infty}\one_{\{\zeta^2>u\}}\dd u =2\int_K^\infty\lambda\one_{\{\zeta>\lambda\}}\dd\lambda$ and Tonelli give the representation in $[0,\infty]$ and monotonicity; under finiteness at $K_0$ the tail integral is finite for $K\ge K_0$, and right-continuity of $\lambda\mapsto Q(\zeta_b>\lambda)$ gives the right derivative.∎ ## 5 Replication boundary, futures, and the anchor lever Does the in-advance quote carry information beyond the finite panel the market actually observes, or is it a repackaging of that panel? The question is one of information content relative to the observed quotes, not of historical priority: the underlying payment-timing identity is not new here, its precursor being recorded in Section 1.2 and beside Lemma 4.2. This section answers: both, and the tension is precisely locatable. Proposition 5.2 writes $\mathrm{IA}(b)$ as a natural-timing FRA plus an *infinite*-strike cap/floor strip on its segment, so the required convexity is present in an idealized all-strike smile; Proposition 5.3 shows that every finite-strike truncation restores the negative theorem, so no traded finite panel substitutes for the quote. The boundary between the paper’s negative and positive sides thus runs along the wing (Remark 5.4). The section then sets up the margining conventions under which Section 6 makes its contract-level statements, and closes with the one free coordinate not yet examined: the anchor $T_\star$ itself. Theorem 5.5 shows that no anchor placement strictly beyond the last annuity payment removes the obstruction and that the printed strip budgets every such placement at once, while Remark 5.6 reads the residual placement question as model risk rather than normalization. ### 5.1 The in-advance claim is an infinite-strike wing The wing enters through one elementary identity for the square. **Lemma 5.1 (Quadratic hinge decomposition).** For every $x\in\R$, $$ \begin{equation} x^2=2\int_0^\infty\bigl[(x-K)^++(-x-K)^+\bigr]\dd K . \label{eq:pc1-quadratic-hinge}\tag{17} \end{equation} $$ *Proof.* For $x\ge0$ the first integrand contributes $\int_0^x(x-K)\dd K=x^2/2$ and the second vanishes; for $x<0$ the roles are exchanged.∎ **Proposition 5.2 (Static equivalence with a natural-timing strip).** Let $L_b:=(\zeta_b-1)/\tau_b$ be the simple rate for $[b,T_\star]$ fixed at $b$. Then, pathwise, $$ \begin{equation} \zeta_bX_b =\frac{\zeta_b^2-\zeta_b}{\tau_b} =L_b+\tau_bL_b^2 , \label{eq:pc1-lia-identity}\tag{18} \end{equation} $$ so the anchor payoff of $\mathrm{IA}(b)$ coincides, as a random variable, with the anchor payoff of the portfolio paying at $T_\star$ the currency amount $L_b+\tau_bL_b^2$; recall that a currency payment at $T_\star$ has anchor weight $\zeta_{T_\star}=1$. By Lemma 5.1, that portfolio is the natural-timing FRA on $[b,T_\star]$ plus $2\tau_b$ times the equal-weight all-strike strip of caplets and floorlets on the same segment, paid at $T_\star$ under the terminal-settlement convention of Section 2. Prices below are normalized per unit rate payoff: for $K\ge0$ set $\pi^{\rm cap}_b(K):=\E_Q[(L_b-K)^+]$ and $\pi^{\rm floor}_b(-K):=\E_Q[(-K-L_b)^+]$; the usual deterministic notional and accrual factors can be restored on both sides. Under every $Q\in\cM$, and for equality quotes, the prices obey $$ \begin{equation} \pi_b =\pi^{\rm FRA}_b +2\tau_b\int_0^\infty\bigl(\pi^{\rm cap}_b(K)+\pi^{\rm floor}_b(-K)\bigr)\dd K . \label{eq:pc1-synthetic-price}\tag{19} \end{equation} $$ With bid–ask quotes, the right-hand side evaluated at ask prices is a synthetic ask $\pi^{\rm syn}_b$, and every calibrated law satisfies $\E_Q[\zeta_b^2]\le\tau_b\pi^{\rm syn}_b+\Zbar_0(b)$, provided the ask surface is integrable in $K$. *Proof.* With $\zeta_b=1+\tau_bL_b$, $(\zeta_b^2-\zeta_b)/\tau_b=\zeta_b(\zeta_b-1)/\tau_b =(1+\tau_bL_b)L_b$, which is (18). Both claims’ prices are $\E_Q$ of the same random variable, by Lemma 4.2 for the left side and by $\zeta_{T_\star}=1$ for the right; thus $\pi_b=\E_Q[L_b]+\tau_b\E_Q[L_b^2]$. Insert (17) at $x=L_b$ and Tonelli to obtain $\E_Q[L_b^2]=2\int_0^\infty(\E_Q[(L_b-K)^+]+\E_Q[(-L_b-K)^+])\dd K$, which is (19) under equality quotes. Under ask constraints, each hinge expectation is bounded by its ask, and monotone integration gives the synthetic-ask inequality; combining with Lemma 4.2 gives the second-moment budget. The factor bookkeeping is fixed by $\pi_b=(\E_Q[\zeta_b^2]-\Zbar_0(b))/\tau_b$: no additional $\tau_b$ multiplies the left-hand side of (19).∎ This is an elementary pathwise payment-timing identity in anchor units; the hinge decomposition is the classical payoff-spanning mechanism of Breeden and Litzenberger (1978) and Carr and Madan (2001). It reconciles the two sides of this paper: the properness information *is* present in an idealized all-strike far-segment smile, and the in-advance quote compresses the entire wing integral into one quoted number. **Proposition 5.3 (Every finite-strike truncation restores the failure).** Fix $K_{\max}<\infty$ and replace the strip in Proposition 5.2 by its truncation to strikes $K\le K_{\max}$, quoted either at finitely many cap strikes $K\in[0,K_{\max}]$ together with floor strikes $-K$, or as the single combined aggregated payoff $$ 2\int_0^{K_{\max}} \bigl[(L_b-K)^++(-L_b-K)^+\bigr]\dd K . $$ This equals $2K_{\max}|L_b|-K_{\max}^2$ when $|L_b|\ge K_{\max}$ and is therefore one claim of at-most-linear tail growth in class (S3). In either reading the truncated strip is a finite standard book. If $b=e=t_1$ and all claims are $\cF_e$-measurable, Theorem 3.6 applies directly; for arbitrary fixing dates or a finite panel across the calendar, the same conclusion follows from Theorem 3.16 under its support-richness, surgery-envelope, finite-integrability, and quote-Slater hypotheses. A continuum of separate strike equalities is not a finite book and is not covered. For the explicit base-case wing asymptotic, take $b=e$ (hence $b\le T_m$ and $\kappa(b)=0$). Along the escape family the truncated cap-side payoff then obeys $$ 2\int_0^{K_{\max}}(L^\Lambda_b-K)^+\dd K =(L^\Lambda_b)^2-\bigl((L^\Lambda_b-K_{\max})^+\bigr)^2 , $$ its price perturbation is $O(p_\Lambda\Lambda)\to0$, and the discarded wing term carries the whole divergence $p_\Lambda\bigl((L^\Lambda_b-K_{\max})^+\bigr)^2\asymp p_\Lambda\Lambda^2\to\infty$. *Proof.* The displayed cap-side integral identity is elementary for $L\ge0$; the corresponding floor-side identity follows by replacing $L$ by $-L$. Each finitely quoted hinge, and the combined aggregated payoff as a function of $L_b$, is a natural-timing claim of at-most-linear growth on $[b,T_\star]$ paid at $T_\star$, hence belongs to (S3). Therefore Proposition 3.5 bounds its escape payoff by $K_C\Lambda$, and the choice of $p_\Lambda$ in Theorem 3.6 sends the price perturbation to zero. Under the displayed $b=e$ condition, $L^\Lambda_b\asymp\Lambda\zeta^0_b/\tau_b$, so the wing term is of order $\Lambda^2$ with probability $p_\Lambda$, and $p_\Lambda\Lambda^2\to\infty$ for the second-moment choice $\Phi(x)=x^2$.∎ **Remark 5.4 (Where the tail envelope really lives).** In any finite-strike implementation, a wing extrapolation or another tail assumption is required. Within the declared escape-rich far-segment setting, Propositions 5.2 and 5.3 show that this choice is the tail envelope of the calibration problem. Quoting the in-advance claim, or its margined relative, would replace that modeling input by a price. **The boundary in bond-price coordinates.** A put on the terminal bond, paying $(K-P_b(T_\star))^+$ at $b$, is a currency payoff bounded by $K$, hence class (S1); a call is bounded through the declared price bound (Definition 3.4 and Assumption 2.2). In anchor units the mechanism is visible: the put pays $\zeta_b(K-1/\zeta_b)^+=(K\zeta_b-1)^+$ and the call $(1-K\zeta_b)^+\le1$, so every finite panel of terminal-bond options has anchor payoff of at most linear growth along Definition 3.1 and inherits the negative theorem. The all-strike limit reproduces the completion in bond-price coordinates: for a top strike $\bar K>0$, $$ \begin{equation} \int_0^{\bar K}(K\zeta_b-1)^+\,\frac{2\,\dd K}{K^{3}} =\bigl((\zeta_b-1/\bar K)^+\bigr)^{2} \;\nearrow\;\zeta_b^{2} \qquad(\bar K\uparrow\infty), \label{eq:pc1-putstrip}\tag{20} \end{equation} $$ the second-moment claim of Lemma 4.2 in bond-price coordinates. The two truncations behave asymmetrically, and the asymmetry locates the wing. Truncating the *top* strike is harmless: $0\le\zeta_b^2-((\zeta_b-1/\bar K)^+)^2\le2\zeta_b/\bar K$, a defect whose expectation is bounded by $2\Zbar_0(b)/\bar K$ under every $Q\in\cM$ by the pinned mean. Truncating the *bottom* strike is fatal: on $\{\zeta_b>1/K_{\min}\}$ the strip restricted to $K\ge K_{\min}$ is affine in $\zeta_b$, so any strike set bounded away from zero—in particular every finite panel—is tail-linear and restores the negative result. The smile wing of Propositions 5.2 and 5.3 reappears here as the deep out-of-the-money bond puts, with strikes accumulating at zero. Fixture R12 of the regression suite realizes the identity, both truncation behaviours, and the growth dichotomy numerically. ### 5.2 Futures margining and the practical strip Under the idealized discrete-resettlement and rolled-margin convention of Lemma 6.6, the futures price has the rolled-account expectation derived there; the classical forward–futures literature provides the economic motivation (Cox et al., 1981). Section 6 then makes the paper’s precise contract-level statements in both directions: a *designed* settlement index on the compounded tape prices the required second moments exactly, with no sign condition (Corollary 6.7), whereas *plain* compounded-rate futures quotes are exactly invariant under a terminal escape surgery, so no upper budget follows from them, alone or—given quote-Slater room for the panel augmented by the futures quotes themselves—jointly with any finite standard book (Proposition 6.8); an imposed positive-association condition bounds the needed moment from below—the wrong side for a certificate—whereas a negative-association *class* hypothesis would reverse it. Neither sign follows merely because the variables share the factor $\zeta_T$. ### 5.3 Anchor robustness One may ask whether the obstruction is an artefact of the anchor placement: the escape deforms a segment whose location the choice of $T_\star$ fixes. Two statements must be distinguished. Within the fixed-$T_\star$ formulation, the obstruction persists after normalizing the annuity by every retained maturity strictly beyond the last annuity payment. If that maturity also lies beyond the full calendar, the same construction transfers by change of numeraire and gives a literal alternative-anchor result. The quoted strip budgets all such normalizations at once under the hypotheses printed below. The boundary normalization $u^*=T_m$, which empties the invisible segment, is *not* covered and is the separate regime of Remark 5.6, where the residual escape must deform inside quoted accrual ranges. For a normalization maturity $u^*\in U_e\cup\{T_\star\}$ with $T_mT$ and every book payment occurs by $T$ (so the special $T_\star$-settlement convention is not used), define for each such $Q$ the $u^*$-forward law on $\cF_T$ by $$ \frac{dQ^{u^*}}{dQ}\bigg|_{\cF_T} :=\frac{\Zbar_T(u^*)}{\Zbar_0(u^*)}. $$ Then, for every retained $d$, the relative bond prices $P_t(d)/P_t(u^*)$ are martingales under $Q^{u^*}$ through $t\le\min\{d,T\}$; the corresponding $u^*$-denominated book and released-row targets equal $\cC(Q)/\Zbar_0(u^*)$, and $$ \sup\bigl\{\E_{Q^{u^*}}[\Phi(\Abar^{u^*}_e)]: Q\in\cM,\ \cC(Q)=c^*\bigr\}=\infty. $$ Hence every genuine whole-calendar anchor $u^*>T$ in this range inherits the obstruction. - 2. (Universality.) On any support with log-forward floor $r_{\min}$ in the sense of (6)—in particular the rate-floor host—fix an observation date $e_W$ and a normalization maturity $u^*\in U_{e_W}\cup\{T_\star\}$ with $e_W\le u^*$. Let $d_j\in U_{e_W}$ satisfy $e_W\le d_j\le T$, let the book contain a finite $\mathrm{IA}(d_j)$ ask at every $d_j$, and suppose $$ 0\le W\le\sum_jc_j\,\Zbar^{u^*}_{e_W}(d_j),\qquad c_j\ge0,\qquad\Zbar^{u^*}_t(d):=\frac{P_t(d)}{P_t(u^*)}. $$ Then $\Zbar_t(u^*)\ge e^{r_{\min}(T_\star-u^*)}$ pathwise, so $W$ is $T_\star$-anchored linearly dominated with constants $e^{|r_{\min}|(T_\star-u^*)}c_j$, and those quoted asks bound $\E_Q[W^2]$ through Theorem 4.3 with those constants. Under a nonnegative log-forward floor the constants are unchanged: one quoted strip at the longest anchor budgets every such normalization’s weights, and no per-normalization quote is required. *Proof.* (i) The capped ramp is nondecreasing, has range $[0,1]$, and vanishes on $[e,T_m]$, and the proof of Theorem 3.6 uses no other property of the ramp. In detail: the computation of Lemma 3.2 gives $\Zbar^\Lambda_e(u)=\Lambda^{1-\kappa_{u^*}(u)}\Zbar^0_e(u)$, so swap-segment coordinates scale by $\Lambda$ exactly, every coordinate by at most $\Lambda$, every currency-curve functional of the swap segment is unchanged, and the frozen continuation respects the price bound because the ramp is nondecreasing. The three estimates of Proposition 3.5 use only the frozen swap segment, the bound $\Zbar^\Lambda_e(b)\le\Lambda\Zbar^0_e(b)$, and ramp values in $[0,1]$, so $|\cC(\delta_{\omega^\Lambda})|\le K\Lambda$ as before. The target scales exactly, $\Abar^{u^*}(\omega^\Lambda)=\Lambda\,\Abar^{u^*}(\omega^0)$, since each $\Zbar_e(q_j)$ carries ramp value $0$ while $\Zbar_e(u^*)$ carries ramp value $1$. With these three inputs, the relative-interior correction, the probability schedule $p_\Lambda=1/(\Lambda\sqrt{g_\Lambda})$ built from $g_\Lambda=\Phi(\Lambda\Abar^{u^*}(\omega^0)) /(\Lambda\Abar^{u^*}(\omega^0))$, and the fixed-mixture construction in the proof of Theorem 3.6 apply verbatim, giving $\E_{Q_\Lambda}[\Phi(\Abar^{u^*}_e)] \ge p_\Lambda\,\Phi\bigl(\Lambda\Abar^{u^*}(\omega^0)\bigr) \to\infty$. If $u^*>T$, carry closure makes $\Zbar_t(u^*)/\Zbar_0(u^*)$ a strictly positive density martingale through the full calendar. Bayes’ formula therefore makes every $P_t(d)/P_t(u^*)$ a $Q^{u^*}$-martingale and, for a claim $C$ paid at $b\le T$, ∎ $$ \E_{Q^{u^*}}\!\left[\frac{C}{P_b(u^*)}\right] =\frac{1}{\Zbar_0(u^*)}\E_Q[\zeta_bC]. $$ Thus the transformed book and released rows are calibrated exactly. On the escape atom the frozen continuation and $\kappa_{u^*}(u^*)=1$ give $\Zbar_T^\Lambda(u^*)=\Zbar_e^0(u^*)>0$. Consequently the preceding lower bound, after change of numeraire, is multiplied by the fixed positive factor $\Zbar_e^0(u^*)/\Zbar_0(u^*)$ and still diverges. (ii) The floor inequality (6) between $u^*$ and $T_\star$ gives $1=\Zbar_t(T_\star)\le e^{-r_{\min}(T_\star-u^*)}\Zbar_t(u^*)$, hence $1/\Zbar_t(u^*)\le e^{|r_{\min}|(T_\star-u^*)}$ and $\Zbar^{u^*}_t(d)=\Zbar_t(d)/\Zbar_t(u^*) \le e^{|r_{\min}|(T_\star-u^*)}\,\Zbar_t(d)$ pathwise. Apply Theorem 4.3 to the rescaled domination; its maturity, carry, and finite-ask hypotheses are exactly those imposed in (ii). **Remark 5.6 (Anchor choice relocates the tail direction).** The escape of Definition 3.1 lives in $(T_m,T_\star)$, a segment whose location is fixed by the choice of anchor. If the retained book is truncated so that its last calendar and payment date coincide, $T=T_m$, then choosing the boundary anchor $T_\star=T_m$ empties the invisible segment. This is a different boundary configuration from the standing convention $T0)>0$.∎ ### 6.2 Account-convexity claims and the single-settlement budget The chaining bound is useful only if its right-hand side is itself quoted. The following family supplies exactly that, at one date. **Definition 6.3 (Account-convexity claim).** For $i=1,\ldots,N$ and contractual accrual factor $\tau_i>0$ for $[t_i,T_\star]$, the claim $\mathrm{AC}(t_i)$ pays at $T$ the currency amount $$ \begin{equation} X^{(i)}_T :=\frac{C_{i,N}\bigl(C_{i,N}\,\zeta_T-1\bigr)}{\tau_i} =\frac{(V^{(i)}_T)^2-V^{(i)}_T}{\tau_i\,\zeta_T}, \label{eq:pc1-ac-payoff}\tag{24} \end{equation} $$ a Borel function of the published tape and the date-$T$ curve. Under Assumption 2.2, $\zeta_T\ge1/C_P$, so the quadratic $C(C\zeta_T-1)$ is minimized over $C>0$ at value $\ge-C_P/4$ and $X^{(i)}_T\ge-C_P/(4\tau_i)$ is bounded below. Its anchor expectation is therefore well defined as an extended value, and a finite ask constraint forces finiteness. For $i=N$, $C_{N,N}=1$ and $\mathrm{AC}(T)=\mathrm{IA}(T)$: the spot claim of Definition 4.1 is the degenerate member of the family. **Lemma 6.4 (Account second-moment identity).** The anchor payoff of $\mathrm{AC}(t_i)$ is $\zeta_TX^{(i)}_T=\bigl((V^{(i)}_T)^2-V^{(i)}_T\bigr)/\tau_i$, and for every $Q\in\cM$, as an identity in $(-\infty,\infty]$ whose negative part is bounded by $1/(4\tau_i)$, $$ \begin{equation} \E_Q\bigl[(V^{(i)}_T)^2\bigr] =\tau_i\,\E_Q\bigl[\zeta_TX^{(i)}_T\bigr]+\Zbar_0(t_i). \label{eq:pc1-ac-identity}\tag{25} \end{equation} $$ Hence an ask constraint $\E_Q[\zeta_TX^{(i)}_T]\le\pi^{\rm AC}_i<\infty$ gives every calibrated law the budget $\E_Q[(V^{(i)}_T)^2]\le\tau_i\pi^{\rm AC}_i+\Zbar_0(t_i)$, and with Theorem 6.2, $\E_Q[\zeta_{t_i}^2]\le\tau_i\pi^{\rm AC}_i+\Zbar_0(t_i)$. *Proof.* Multiply (24) by $\zeta_T$. The variable $V^2-V\ge-\tfrac14$ is bounded below, so its extended expectation is defined. If $\E_Q[V^{(i)}_T]<\infty$, then Lemma 6.1 gives $\E_Q[V^{(i)}_T]=\E_Q[\zeta_{t_i}]=\Zbar_0(t_i)$ and the identity follows by subtraction. If $\E_Q[V^{(i)}_T]=\infty$, then both $\E_Q[(V^{(i)}_T)^2]$ and $\E_Q[(V^{(i)}_T)^2-V^{(i)}_T]$ are infinite, so the same equality holds in the extended sense. Finally, $V^2-V\ge\tfrac12V^2-\tfrac12$; hence a finite ask forces $V^{(i)}_T\in L^2$, and the finite budget follows from the first case.∎ **Corollary 6.5 (Single-settlement gauge budget).** Replace Assumption 4.5 by: the book contains the ask constraints of $\mathrm{AC}(t_i)$, $i=1,\ldots,N$, all settling at the single date $T$. Then Theorems 4.6 and 4.7 hold verbatim with $M_{\rm quoted}=1+\sum_i(\tau_i\pi^{\rm AC}_i+\Zbar_0(t_i))$, and the class cut is by the ask constraints (25), which are weakly closed by the same Fatou/monotone-truncation argument since $(V^{(i)}_T)^2$ is a continuous nonnegative function of the coordinates on $\Omega$. No spot in-advance quote at any intermediate date is needed, and no state beyond the existing curve and tape coordinates is introduced. *Proof.* By Lemma 6.4 and Theorem 6.2, every calibrated law satisfies $\E_Q[\zeta_{t_i}^2]\le\tau_i\pi^{\rm AC}_i+\Zbar_0(t_i)$ for every $i$, which is the only property of the strip the proofs of Theorems 4.6 and 4.7 use. Continuity of $(V^{(i)}_T)^2$ on $\Omega$ holds because $1/P_{t_j}(t_{j+1})=\zeta_{t_j}/\Zbar_{t_j}(t_{j+1})$ is a ratio of coordinates with positive denominator on $\Omega$.∎ The chaining is consistent with Theorem 3.6: $\mathrm{AC}(t_i)$ references $\zeta_T$, the inverse price of the full segment $[T,T_\star]$, and pays at $T$, the *start* of that segment—a payment-timing mismatch outside the standard class, as it must be. ### 6.3 Margined implementation and the plain-futures obstruction It remains to put these claims in the form desks actually trade—a margined settlement—and to ask whether the contracts already listed on the same tape would do instead. **Lemma 6.6 (Discrete-margin futures pricing).** Consider a contract with resettlement at the calendar dates, entered at $0$ at futures level $G_0$, whose margin flows $G_{t_{j+1}}-G_{t_j}$ are held in the rolled account $V^{(0)}$, with terminal settlement level $G_T=S$ for an $\cF_T$-measurable index $S$. Let $G$ be adapted and assume the value products are integrable, $V^{(0)}_{t_j}G_{t_j}\in L^1(Q)$ for every $j$ (in particular $V^{(0)}_TS\in L^1$). Suppose each unit margin flow has zero anchor value at its preceding date, $$ \begin{equation} \E_Q\!\left[\zeta_{t_{j+1}} (G_{t_{j+1}}-G_{t_j})\mid\cF_{t_j}\right]=0, \qquad j=0,\ldots,N-1. \label{eq:pc1-zero-margin-value}\tag{26} \end{equation} $$ Then $(V^{(0)}_{t_j}G_{t_j})_j$ is a $Q$-martingale and $$ \begin{equation} G_0=\frac{\E_Q\bigl[V^{(0)}_T\,S\bigr]}{\zeta_0}. \label{eq:pc1-futures-price}\tag{27} \end{equation} $$ *Proof.* Scale (26) over $(t_j,t_{j+1}]$ by $C_{0,j+1}$, an $\cF_{t_j}$-measurable quantity since $R_j$ is fixed at $t_j$. The margin then received at $t_{j+1}$ is $C_{0,j+1}(G_{t_{j+1}}-G_{t_j})$ currency, of anchor value $C_{0,j+1}\zeta_{t_{j+1}}(G_{t_{j+1}}-G_{t_j}) =V^{(0)}_{t_{j+1}}(G_{t_{j+1}}-G_{t_j})$. Zero conditional value means the last variable has conditional mean zero. All terms below are integrable: since $G_{t_j}$ is $\cF_{t_j}$-measurable and $V^{(0)}$ is positive, $\E[V^{(0)}_{t_{j+1}}|G_{t_j}|] =\E[V^{(0)}_{t_j}|G_{t_j}|]<\infty$. Therefore $\E_{t_j}[V^{(0)}_{t_{j+1}}(G_{t_{j+1}}-G_{t_j})]=0$, i.e. $\E_{t_j}[V^{(0)}_{t_{j+1}}G_{t_{j+1}}] =G_{t_j}\E_{t_j}[V^{(0)}_{t_{j+1}}]=G_{t_j}V^{(0)}_{t_j}$ by Lemma 6.1. Iterating from $G_T=S$ and $V^{(0)}_0=\zeta_0$ gives (27). This is the paper’s idealized discrete-resettlement form of the classical forward–futures pricing mechanism (Cox et al., 1981), stated under the explicit margin and integrability hypotheses above and in anchor units.∎ **Corollary 6.7 (Designed settlement prices the chaining moments exactly).** For $i=1,\ldots,N$, consider the tape-and-curve settlement index $$ \begin{equation} S_i:=\frac{C_{i,N}}{C_{0,i}\,P_T(T_\star)}, \qquad\text{which satisfies}\qquad V^{(0)}_T\,S_i=(V^{(i)}_T)^2 , \label{eq:pc1-designed-settlement}\tag{28} \end{equation} $$ the margined contract of Lemma 6.6 has $\zeta_0\,G_0^{(i)}=\E_Q[(V^{(i)}_T)^2]$ exactly, under every candidate law with finite budget $\E_Q[(V^{(i)}_T)^2]<\infty$. Indeed, the unique integrable reset process is determined backwards by $$ \begin{equation} V^{(0)}_{t_j}G^{(i)}_{t_j} =\E_Q\!\left[(V^{(i)}_T)^2\mid\cF_{t_j}\right], \label{eq:pc1-designed-reset}\tag{29} \end{equation} $$ so all integrability and zero-margin hypotheses of Lemma 6.6 hold automatically, with no sign or correlation condition. A quoted family $(G_0^{(i)})_{1\le i\le N}$ therefore supplies the entire single-settlement budget of Corollary 6.5 in margined form. *Proof.* Using $C_{0,N}=C_{0,i}C_{i,N}$ and $\zeta_T=1/P_T(T_\star)$, ∎ $$ V^{(0)}_TS_i =C_{0,i}C_{i,N}\zeta_T\cdot\frac{C_{i,N}\zeta_T}{C_{0,i}} =C_{i,N}^2\zeta_T^2=(V^{(i)}_T)^2 ; $$ conditional expectations in (29) are integrable. Writing their martingale as $M_j=V^{(0)}_{t_j}G^{(i)}_{t_j}$, both $M_{j+1}$ and $V^{(0)}_{t_{j+1}}G^{(i)}_{t_j}$ are in $L^1$: indeed, using the martingale property of $V^{(0)}$, $$ \E_Q\!\left[V^{(0)}_{t_{j+1}}|G^{(i)}_{t_j}| \mid\cF_{t_j}\right] =|G^{(i)}_{t_j}|\E_Q[V^{(0)}_{t_{j+1}}\mid\cF_{t_j}] =V^{(0)}_{t_j}|G^{(i)}_{t_j}|=|M_j|. $$ Taking expectations proves the required integrability. Hence $V^{(0)}_{t_{j+1}}(G^{(i)}_{t_{j+1}}-G^{(i)}_{t_j})$ is integrable and has conditional mean zero. Moreover $C_{0,j+1}\ge C_P^{-(j+1)}$ by Assumption 2.2, so division by the strictly positive, $\cF_{t_j}$-measurable factor $C_{0,j+1}$ preserves integrability and gives (26). Now apply Lemma 6.6. The pathwise identity and the futures identity are checked in the regression (Section 7). **Proposition 6.8 (Plain rate futures quotes carry no upper budget).** A margined contract settling on the compounded factor alone, $S=C_{0,N}$, an $\cF_{t_{N-1}}$-measurable index, prices $\zeta_0G_0=\E_Q[V^{(0)}_TC_{0,N}]=\E_Q[C_{0,N}^2\zeta_T]$, whereas the budget is $\E_Q[(V^{(0)}_T)^2]=\E_Q[(C_{0,N}^2\zeta_T)\cdot\zeta_T]$. For a finite standard book $\cB$, let $c(Q)$ collect its exact anchor values, fix $Q^*\in\cM(\cB)$ with $\E_{Q^*}[C_{0,N}^2\zeta_T]<\infty$, and put $$ \begin{align*} g(Q)&:=\E_Q[C_{0,N}^2\zeta_T], \\ \cM^{\rm aug}_{\rm f}:= &\{Q\in\cM:\ c(Q)\text{ and }g(Q) \text{ are absolutely finite}\}, \\ \mathcal J(Q)&:=(c(Q),g(Q)). \end{align*} $$ - 1. Let $\kappa_T(u):=(u-T)^+/(T_\star-T)$ be the terminal ramp profile, well defined since $TT_m$ with in-advance accruals $\tau_1=1.5$, $\tau_2=1.0$; deterministic continuation after $e$, so $\zeta_{q_j}=\Zbar_e(q_j)$ and the conditional Jensen step of Theorem 4.3 holds with equality. Forty base curve scenarios; one escape state obtained by applying Definition 3.1 literally to base scenario 3—identical swap-segment currency prices, terminal bond price divided by $\Lambda$, equality of the swap-segment observables asserted programmatically—with $\Lambda=10^6$ and mass $p=\Lambda^{-5/4}=3.162\times10^{-8}$. Law A is the uniform reference law on the base states. Law B carries the escape mass, with the forty base weights re-solved by exact linear programming so that every book functional matches law A: the LP residual over all nine rows is $2.2\times10^{-16}$ and all weights are nonnegative. The book comprises two physical payer swaptions ($K=3.0\%,4.0\%$), two caplets ($K=3.0\%,4.5\%$), one CMS caplet ($K=3.2\%$), one cash-settled swaption ($K=3.2\%$, ten-period cash annuity), and the three martingale rows $\Zbar(q_1),\Zbar(q_2),\zeta_e$. | | law A (no escape) | law B (escape) | |---|---|---| | swaption \(K{=}3.0\%\) | 0.056924940 | 0.056924940 | | swaption \(K{=}4.0\%\) | 0.045462099 | 0.045462099 | | caplet \(K{=}3.0\%\) | 0.023994297 | 0.023994297 | | caplet \(K{=}4.5\%\) | 0.011468444 | 0.011468444 | | CMS \(K{=}3.2\%\) | 0.055325291 | 0.055325291 | | cash-settled \(K{=}3.2\%\) | 0.226374272 | 0.226374272 | | \(\E [\Abar _e]\) (martingale-pinned) | 1.188674 | 1.188674 | | \(\E [\Abar _e^{\,2}]\) | 1.4194 | \(4.4779\times 10^{4}\) | | strip price \(\pi _1\) | 0.167592 | \(3.0520\times 10^{4}\) | | strip price \(\pi _2\) | 0.210580 | \(4.3788\times 10^{4}\) | The two laws are indistinguishable on the standard book to ten digits while their second annuity moments differ by the factor $3.155\times10^{4}$; the in-advance strip separates them by the same order. The identity (13) holds with zero recorded defect under both laws at both strip dates, and the certificate (16) holds with relative slack $1.2\times10^{-4}$: $\E_B[\Abar_e^2]=4.47795\times10^4 \le4.47850\times10^4$. ### 7.2 Chaining, designed settlement, and plain-futures invariance The chaining layer of Section 6 is checked three times. On the fixture, whose frozen continuation makes the rolled account trivial ($V_T=\Zbar_e(q_1)$ pathwise, so $\mathrm{AC}(q_1)$ degenerates to $\mathrm{IA}(q_1)$ and Theorem 6.2 holds with equality), the account identity (25) holds with zero recorded defect under both laws. An independent two-state toy with a stochastic roll exhibits the strict submartingale gap, $\E[\zeta_s^2]=2.84766<2.94531=\E[V_T^2]$, with the account identity again exact. A third, two-period toy (three nodes at $u$, mean-one terminal leaves) checks the margined layer: the designed settlement satisfies $V^{(0)}_wS_1=(V^{(1)}_w)^2$ pathwise to relative defect $7.8\times10^{-16}$; the check has discriminating power, since perturbing the settlement by an extra factor $C_{1,2}$ produces a pathwise defect of at least $1.5\times10^{-2}$. The $VG$-martingale recursion of Lemma 6.6 returns $\zeta_0G_0=\E[(V^{(1)}_w)^2]$ to $1.6\times10^{-16}$; and a terminal mean-preserving surgery ($\Lambda=10^4$, $p=10^{-6}$, $q=\tfrac12$) leaves every martingale row and the plain-futures quote fixed to $1.9\times10^{-16}$ while the budget and the designed quote both grow by the factor $50.99$, equal to $1-q+qp\Lambda^2+q(1-p\Lambda)^2/(1-p)$ to machine precision. A companion three-node, three-leaf toy then exercises the *correction* layer of Proposition 6.8 (i): a book-only Slater mixture restores a two-quote book to machine precision but shifts the plain-futures functional by $5.2\times10^{-6}$ (relative)—the algebraic obstruction that forces the augmented panel. A stronger same-fibre fixture restores both book and futures quote to machine precision while the budget remains inflated by the factor $50.4$. Finally, a genuinely augmented affine-hull correction uses individual correcting laws whose futures values move by as much as $4.1\times10^{-4}$, yet cancels the whole book–futures vector to machine precision while retaining a budget factor of $29.7$. A separate deterministic check records the carry-closure counterexample of Lemma 2.5: with a maturity retained at $e$ and $d$ but dropped at an intervening date, a law with $\Zbar_e(d)=100$, $\zeta_d=1$ satisfies every printed adjacent row vacuously while the certificate would assert $10^4\le1$; imposing the carry chain excludes the law. Finally, a reserve-interval check prices the annuity-tail exotic $((\zeta_e-1.2)^+)^2$ over the exactly calibrated class: its price interval is $[4.9\times10^{-3},\,1.2\times10^{5}]$ without the strip and $[4.9\times10^{-3},\,2.6\times10^{-2}]$ with the two strip asks imposed at $1.05$ times their reference levels—a width collapse by the factor $5.8\times10^{6}$, quantifying the reserve discussion of Section A. ### 7.3 The multi-date surgery The surgery of Theorem 3.16 has its own fixture (`pc1_multidate_regression.py`, seed 23): two stochastic periods, seven $t_1$-nodes with nine conditional states each on a four-coordinate state carrying one numeraire-segment maturity ($\kappa=\tfrac12$), and a four-claim book (a $t_1$-fixing caplet-type claim, an $s$-fixing physical swaption and CMS caplet, and a far-segment caplet). At $\Lambda=10^{8}$, $q=10^{-2}$, $p\Lambda=10^{-3}$: all unconditional and per-node conditional martingale rows hold to $8.9\times10^{-16}$ after the surgery; the book deviates by $6.0\times10^{-6}$ (predicted order $qp\Lambda=10^{-5}$) and is restored to $2.8\times10^{-17}$ by the closed-form fixed-mixture correction; $\E[\Abar_s^2]$ grows by the factor $907$, tracking the scheduled $1+qp\Lambda^2=10^{3}$ reduced by the correction weight; and an account-convexity quote at $s$ moves by the factor $4.1\times10^{3}$, confirming that augmented books block the construction. The run also exhibits Remark 3.18 (iii): with all maturities in the swap segment the book is preserved to machine precision with no correction at all. Two further blocks probe the structure at its boundaries. A carry-chain check confirms the unconditional date-$s$ means are pinned through the $t_1$ layer to $2.2\times10^{-16}$. A stopped-coupon block gives a matured coupon a historical value: the full-annuity identity $\Abar_s(D_\Lambda x)=\Lambda\Abar_s(x)$ fails by $46\%$, as it must—historical components are copied, not scaled—while the live-tail identity $\Abar^{\rm live}_s(D_\Lambda x)=\Lambda\Abar^{\rm live}_s(x)$ holds with zero recorded defect and the full-annuity second moment still grows by the factor $246$, the numerical content of Step 5 of Theorem 3.16. ## 8 Boundaries, discussion, and open problems The fixtures of Section 7 close the constructive story at machine precision; what remains is to say where it stops. This final section draws the sharp boundaries—support scope, calendar and exercise scope, carry closure, and what is and is not claimed about instruments—then reads the results against desk practice, records four open problems, and concludes. The through-line is that everything the theorems deliver is conditional on declarations made explicit here, and everything they cannot deliver is localized to one quoted family whose live availability is an empirical question. ### 8.1 Sharp boundaries - 1. *Support scope.* Theorem 3.6 requires the escape family inside the declared support (Assumption 3.3). A hard support capping $\zeta$—a finite tree over bounded states, or any model class with bounded rates—evades it by construction; the negative theorem then says only that the cap itself is the tail assumption. The two regimes are complementary: on compact declared supports, exact joint calibration is a compact-class feasibility problem with support duality (finite-dimensional only after an explicit tree or grid reduction), and everything in this paper concerns what happens when the support is not so truncated. The hypotheses are not vacuous or engineered: the rate-floor host of Proposition 2.12 satisfies the support and richness requirements at every base path and the multi-date branch requirements on every interior-margin surgery region. The quote-Slater condition remains explicit: Lemma 2.13 shows only that relative-interior targets are dense in a chosen finite-dimensional attainable image, not that the observed target is interior. Finally, Theorem 5.5 shows that no retained normalization maturity strictly beyond the last annuity payment removes the obstruction in the fixed-$T_\star$ formulation, and gives the literal change-of-numeraire conclusion for whole-calendar anchors $u^*>T$; the boundary normalization is the separate regime of Remark 5.6. - 2. *Single collateral, fixed calendar, one exercise date.* Collateral optionality, cross-collateral weights, moving calendars, and genuine multi-period *exercise* are outside this paper. The chaining question is settled by Section 6, and the negative theorem is extended to multi-date books by Theorem 3.16; the remaining multi-date scope items are Bermudan-style books, collateral kernels, and, for uncapped swap-segment claims fixing strictly after the escape date, the frozen-payoff integrability condition of Lemma 3.15, for which a declared settlement-domain bound is sufficient but not necessary. - 3. *Instruments.* The exact carriers of the budget—the $\mathrm{AC}$ claims or the designed margined settlement (28)—are constructible from the published tape and one terminal bond fixing. We do not identify them with any currently listed product, and make no exhaustive listing or liquidity claim. Related in-arrears and timing-mismatched OTC structures motivate a market-data search, but whether a convention-complete, liquid quote panel exists is an empirical question outside the theorem. Plain compounded-rate futures quotes are exactly invariant under a terminal escape surgery, so they cannot supply the one-sided budget, alone or—given Slater room for the panel augmented by the futures quotes themselves—jointly with any finite standard book (Proposition 6.8). - 4. *Absent a quoted carrier.* The correct output of an exact joint calibration remains *relative*: feasibility and duality hold on a declared envelope; an entropic selection may be posed only after choosing a reference law and verifying a finite-KL feasible law. The declaration should be governed and reported as a modeling input. This paper identifies one sufficient quoted tail-budget field: a finite dated price family on the compounded tape, not a model parameter. - 5. *Carry closure.* Every conditional-mean and pinned-mean statement requires the maturities involved to be retained at every intervening calendar date (Lemma 2.5); without this the second-moment certificate admits an explicit counterexample, so carry closure is part of the market declaration, on a par with the support itself. - 6. *Declared support closedness.* Theorem 4.6 requires the support to be closed in the ambient Euclidean product (Assumption 2.1): the price-only gauge controls coordinates from above, so a support accumulating at zero-price states at bounded levels would defeat inf-compactness. Degeneracy control is thus a support declaration here, playing the role inverse-price gauge terms play in expectation-based formulations. - 7. *Bid sides and higher moments.* Only strip asks enter the budget. Bid-side constraints on the strip need not be weakly closed, and continuity of $Q\mapsto\E_Q[\zeta^2]$—which would follow, for example, from a fourth-moment budget—is never used. - 8. *No priority claim on the building blocks.* The hinge decomposition and the forward–futures economic mechanism are classical (Breeden and Litzenberger, 1978; Carr and Madan, 2001; Cox et al., 1981); the in-advance identity used here is an elementary pathwise calculation under the printed anchor convention (Bang and Daboussi, 2022, Corollary 3.1.1). The contribution claimed here is their exact placement as a *sufficient quoted fixed-envelope carrier* for well-posed joint calibration—observable properness on the unbudgeted class $\cM$ is not thereby verified; on escape-rich supports it is proved *false*—together with the negative theorem showing that, within its escape-rich and quote-Slater scope, no finite standard-timing book can substitute for them. The *technique* of the negative theorem is likewise elementary once the anchor coordinates are fixed: in those coordinates the escape is a scaling direction along which every standard-timing anchor payoff grows at most linearly. Choosing $p_\Lambda$ so that $p_\Lambda\Lambda\to0$ but $p_\Lambda\Phi(\Lambda\Abar_e^0)\to\infty$ settles the one-date case. What that reduction does not supply is what the printed proofs do: the classification that *every* standard-timing claim has at most degree-one anchor growth—including numeraire-segment natural-timing claims, where the rate’s gain and the discount factor’s loss cancel exactly (Proposition 3.5). Moving the payment date can break that pattern; explicit superlinear dependence on far-segment inverse bond prices is the other route identified in Remark 3.12. Nor does the reduction supply the multi-date extension, where atom-mixing is unavailable and the conditional-mean-preserving branch surgery of Theorem 3.16 replaces it. ### 8.2 Relation to practice The theorems formalize, and sharpen into class-level statements, two pieces of practitioner knowledge. First, the terminal-swap-rate folklore: CMS pricing requires an annuity mapping that vanilla quotes do not fully identify (Hagan, 2003; Andersen and Piterbarg, 2010). Theorem 3.6 shows the unidentified direction is not a nuisance parameter but an unbounded superlinear moment, immune to any finite selection of CMS and cash-settled quotes at arbitrary strikes (Corollary 3.11). Second, any finite-strike implementation requires a wing extrapolation or another tail assumption. By Propositions 5.2 and 5.3, that choice is the tail envelope of the joint calibration within the declared escape-rich far-segment setting. The in-advance quote, or its designed margined relative, would replace that modeling input by a price. In the meantime, the practical reading of Theorems 4.6 and 4.7 is conditional: any desk willing to declare a budget $M$ obtains the closed image and the exact dual on $\{\E_Q\chi\le M\}$, and the results here identify both the relevant tail direction and the quote that would replace the declaration. Support values can be recomputed as the governed budget $M$ varies; differentiability or a dual sensitivity formula is not claimed here. The positioning of the completing instruments against adjacent traded classes is itself largely theorem-governed: Table 1 records, for each class, why it does not substitute for the budget carrier, with the governing statement; the numerical certificate of Section 7 makes the first three rows concrete. Table 1. Positioning against adjacent instrument classes. | Adjacent class | Why it does not substitute | |---|---| | Finite swaption-smile panels at arbitrary strikes and tenors | Rate-space convexity with numeraire-linear anchor weight (Proposition 3.5); blind to the escape (Theorem 3.6). | | CMS swaps, caps, floors; cash-settled swaptions | Covered for arbitrary finite panels (Corollary 3.11); their convexity correction is a model output, which the carrier replaces by a price. | | Plain compounded-rate futures | Exactly invariant under a terminal escape surgery: no one-sided budget, alone or jointly with any finite standard book (Proposition 6.8); the designed settlement (28) is one exact sufficient moment carrier and could be quoted as a basis. | | In-arrears and timing-mismatch structures | With accrual inside the swap segment they are class (S2): the paid rate is frozen by the escape, so they fail with the CMS panel. Early payment of a far-segment-reaching rate is precisely the in-advance family (Definition 4.1); bespoke structures of that kind motivate the empirical search recorded among the boundaries above. | | Finite long-bond-option panels at arbitrary strikes | Class (S1) (Definition 3.4): put anchor payoffs are tail-linear, call anchor payoffs bounded; see (20). | | Equity variance template | Different logical role: all-strike index options span realized variance, so a variance quote is a convenience; here the far-segment strike continuum is idealized and its availability is not established, while an integrable ask surface would complete the unspanned direction (Theorem 3.6 and Proposition 5.2). | ### 8.3 Open problems Four directions seem to us both natural and tractable. (i) *Relaxed formulation.* The escape is a recession direction of the calibrated set; a compactified formulation—laws plus an explicit boundary defect measure, in the spirit of concentration-compactness—should represent the closure of the weighted image as the image plus an explicitly parametrized recession cone, with genuine feasibility characterized by a recession-corrected dual. The defect coordinate would be exactly what the in-advance quote pins. (ii) *Robust bounds and the hedgeable cone.* Proposition 3.5 gives a necessary recession-direction test for finite model-free bounds over the vanilla-calibrated class; a full pricing–hedging duality in the semistatic idiom would determine when that test is also sufficient and make it an operational tool. (iii) *Multi-period exercise and collateral.* Bermudan books introduce weights measurable at stopping times; collateral optionality prices maxima of numeraire ratios and may already carry second-moment information of the required type. (iv) *Selection and the anchor.* On the compact strip-calibrated class, an entropic selection can be imposed when the fibre contains a law of finite relative entropy with respect to the chosen reference; strict convexity then gives uniqueness of the primal minimizer. Existence of dual potentials and a numerical scheme are not proved here. Relative entropy is not numeraire-invariant (Remark 5.6); quantifying the anchor dependence of the selected law is open. ### 8.4 Conclusion On supports rich enough to contain a one-parameter terminal-segment deformation family, and for quote targets with the printed relative-interior room, no finite book of caplets, floorlets, physical or cash-settled swaptions, or CMS products—at any strikes, fixing and paying anywhere on the calendar under the stated envelopes—bounds any superlinear moment of the annuity pricing weight; and, when the Slater correctors have finite budget for the gauge under test, no admissible coercive gauge is observably proper. Exact joint calibration on the declared untruncated class therefore fails this paper’s unbudgeted weight-tail and observable-properness criteria on such standard data. That failure is invisible to the quoted panel, as the numerical certificate makes concrete. One exact sufficient completion is a family of payment-timing convexity quotes: in-advance claims, their account-convexity relatives on the compounded tape settling at a single date, or a designed margined settlement—all explicit and elementary in form, without any liquidity or availability claim here. With finite asks for the required dated family, the second moment of every declared linearly dominated regular pricing weight is bounded by exact identities, the regular weighted-smile image is compact convex and closed, and feasibility has an exact support-function dual: the joint calibration problem becomes well posed with its tail budget quoted rather than declared. Within the printed support, carry, regularity, and nonemptiness hypotheses, one sufficient market-structural closure input is therefore a finite dated convexity-price family, whose live availability must be checked empirically. **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. ## A Adoption economics and contract design of the budget carrier This appendix discusses why the completing instruments of Sections 4 and 6 could plausibly be quoted and traded, and which contract-design choices the mathematics leaves free.[^1] The instruments in question are the in-advance claim (Definition 4.1), its account-convexity relatives on the published compounded tape settling at a single date (Definition 6.3), and the designed margined settlement (28); each quoted ask is affinely one $p$-th moment of an inverse terminal-bond price (Lemma 4.2 and Proposition 4.9), a market-implied uniform-integrability budget. The theorems say the market needs the quote; this appendix addresses why firms would make it. *The design-freedom principle.* Closure of the weighted-smile image requires only that some quoted ask dominate some superlinear moment of the inverse terminal-bond price, up to constants: the certificate machinery never uses the exact functional form (Proposition 4.9), tolerates caps with a governed residual (Proposition 4.10), and needs no tight two-way market (Remark 4.8). The contract can therefore be optimized for tradability without touching the theory. The paragraphs below walk the natural holders of each side of the market, attaching to each the design refinements, tagged (D1)–(D8), that spend this freedom on their behalf. **The natural buyers already hold the risk, and already pay for it.** Every CMS and exotics desk warehouses annuity-direction convexity that its vanilla hedges provably cannot span (Theorem 3.6 and Corollary 3.11). Today that residual lives inside the annuity-mapping model choice and the smile-wing extrapolation (Remark 5.4), and it is paid for through model-risk reserves, prudent-valuation adjustments, and independent price-verification haircuts. A traded budget carrier converts the reserve into a hedgeable position: buy the claim, release the reserve, mark against a price instead of a committee. Reserve release is realized profit and loss, and it carries an accounting counterpart—a model-dependent fair-value mark migrating toward an observable input, and a model-risk adjustment shrinking into an ordinary hedged market risk. Negative-convexity books (mortgage servicing, callable issuance hedging) are likewise structurally short the high-rate tail states in which the claim pays. The regression of Section 7 quantifies the stake (D8): a representative annuity-tail exotic, $((\zeta_e-1.2)^+)^2$, priced over the exactly calibrated class of the fixture, ranges over $[4.9\times10^{-3},\ 1.2\times10^{5}]$ without the strip—a model-risk interval carried entirely by the unquoted tail—while adding the two strip asks at $1.05$ times their reference levels collapses the interval to $[4.9\times10^{-3},\ 2.6\times10^{-2}]$, a width reduction by the factor $5.8\times10^{6}$. This is the impossibility theorem as a reserve number: the difference between the two intervals is what one ask per date is worth to the holder of such a book. **The natural market-makers can manufacture it from existing inventory.** By Proposition 5.2, the in-advance claim is a natural-timing FRA plus an all-strike cap/floor strip: a dealer hedges the body from the listed smile and warehouses only the wing, so quoting the instrument is not a new business but a monetization of the wing extrapolation the desk already maintains. The natural quoting format is a spread (D4): the designed settlement minus a multiple of the plain compounded settlement isolates the pure convexity leg—Corollary 6.7 and Proposition 6.8 prove the two legs differ by exactly the budget content—with smaller notional risk and a natural home on the futures desks that already trade the futures–swap convexity basis; a second spread family runs against swaption wings. **The natural sellers exist.** The claim’s seller collects premium for a tail paying in high-rate states—the classic variance-seller position. Institutions with structural offsets can run it: liability-driven and insurance books whose liabilities deflate in exactly those states, and systematic volatility sellers extending an existing franchise from the rate direction to the numeraire direction. Two-way structural interest is what separates completing products that trade from those that do not, and what makes the seller side feasible is tail governance. By Proposition 4.9, a $p=1.25$ contract carries the identical well-posedness content—Minkowski replacing Cauchy–Schwarz—with dramatically smaller tail exposure for the seller (D2). Unbounded payoffs are unpopular with desks and clearing houses, but Proposition 4.10 (i) shows a capped claim alone cannot carry the budget; the honest compromise (D3) is Proposition 4.10 (iii)–(iv): cap at a deterministic far-out level $K$ and decompose the budget as quoted-below-the-cap plus declared-beyond-the-cap. The declared field does not vanish, but it collapses from the entire wing extrapolation to a single far-out parameter with a computable sensitivity in $K$: a governance dial instead of a model. **The venue logic is engineered into the mathematics.** Three design facts matter to a listing venue. First, by Theorem 6.2 and Corollary 6.5, one settlement family at a single date carries every calendar date’s budget, so the listed footprint is a handful of standardized contracts rather than a strip of illiquid line items (D7), with settlement dates matched to the listed futures cycle for cross-margining, the anchor $T_\star$ fixed at the longest liquid curve point—which also shrinks the invisible numeraire segment (Remark 5.6)—and confirmations reduced to the published compounded index plus one discount-factor fixing. Second, the settlement is a Borel function of the published compounded overnight tape plus that one fixing, riding the post-2020 benchmark infrastructure; the fixing is also the one manipulable point in the baseline design, and the repair (D1) is to define the settlement discount factor off a clearing house’s end-of-day settlement curve—a regulated fixing that already margins large volumes—optionally averaged over a short window, date-averaging perturbing the budget only by constant factors, which the design-freedom principle makes free. Third, the designed contract is margined futures-style by construction (Lemma 6.6 and Corollary 6.7), slotting into existing clearing. **The index-first template.** The historical template is the volatility index: an index existed for a decade and became a market only after its redesign as a replicable variance functional of listed options, at which point it completed the equity volatility market. The designed settlement starts at the post-redesign stage, and the corresponding staging move (D5) is to publish the index before listing the contract: a terminal discount volatility index, quoted in volatility units from the second moment and computable daily from the replication formula of Proposition 5.2 with a published, transparent wing rule, is free to produce, converts the currently invisible wing assumption into a public number, builds the demand narrative, and fixes legibility. **Why it has not listed already, and what the theorems change.** Two reasons. First, the settlement infrastructure—public compounded-in-arrears tapes with legal standing—is a post-2020 artifact of the benchmark transition. Second, the demand was inarticulate: the terminal-swap-rate folklore lets every desk believe its CMS convexity is spanned by the swaption smile up to a modeling choice, so no unspanned risk was perceived. Theorem 3.6 and Corollary 3.11 convert the modeling choice into a provably unspanned exposure, and the dual multiplier of the declared-envelope program (Section 8) gives each risk function a number for what the missing quote is worth to its own book. Aggregated willingness-to-pay is what a listing case is made of. **Honest counterweights and a staged activation path.** Completing products fail more often than they succeed: flow can turn one-way in a crisis; the terminal bond fixing is the manipulable point and needs the robust fixing design (D1); early liquidity is circular. The realistic path is staged, in increasing activation energy: dealer consensus marks on in-arrears-style structures, which already make the tail budget market-implied in the sense of this paper; bilateral quoting, as an extension of the timing-mismatch business that has historically existed; and a listed margined contract once flow justifies it. The mathematics is indifferent among the three, because only the ask side is ever needed (D6): the budget uses no bid (Remark 4.8), an executable-in-size ask alone closes the calibration problem, and a risk system can consume the ask directly—wide early markets are fully useful, and few new contracts are informative at day-one liquidity, while this one provably is. Even the weakest stage replaces the declared tail envelope by a price. **Capital mechanics of the reserve release.** Two regulatory channels give the willingness-to-pay analysis above an institutional denomination; both are recorded as market-structure analysis, not as a reading of any particular rulebook. First, valuation adjustments for model risk and market-price uncertainty are capital deductions under prudent-valuation regimes, and independent price verification consumes observable inputs: an executable ask on the budget carrier moves the annuity-tail reserve from a committee mark toward a hedged position with an observable input, so the reserve-interval collapse of Section 7 (fixture R11) prices a capital line as well as a profit-and-loss one. Second, trading-book capital frameworks separate modellable from non-modellable risk factors by the existence of real price observations, with committed executable quotes qualifying; the terminal-discount convexity direction generates no such observations today, and the committed-quote stage of the staged activation path above would create them for the contributing institutions. Both channels reward the quoting side as well as the buying side, which is the incentive structure a completing quote requires. ### Summary The instrument does not create a new risk to trade; it prices a risk firms already hold and already pay for through reserves, capital, and model governance. Buyers get reserve release and observable marks; market-makers monetize an extrapolation they currently give away; sellers collect premium on a tail they structurally offset; venues get a futures-complex-adjacent contract with built-in spread families and a minimal footprint. The refinements (D1)–(D8) spend the theory’s built-in slack—constants, powers, averaging, one-sidedness—to buy fixing robustness, thin tails, margin efficiency, spread-form quoting, and governance dials; by Propositions 4.9 and 4.10 none of them weakens the well-posedness content. The instrument the desks would actually trade is, by the design-freedom principle, the same instrument the theorems demand. Its positioning against every adjacent instrument class is theorem-governed and collected in Table 1. ## B Deferred proofs This appendix collects the three longest proofs of the paper; all notation, standing conventions, and displayed objects are those of the sections in which the corresponding statements appear. *Proof of Theorem 3.16 .* *Step 1 (truncation).* The date-$s$ state alone is not enough here: bounding it need not bound the stopped history retained by a frozen branch. Let $H_s=(x^{t_0},\ldots,x^s)$ be the full date-$s$ prefix and define the finite Borel measure $$ \eta_s(E):=\E_{Q^*}\bigl[\Abar^{\rm live}_s \one_{\{H_s\in E\}}\bigr] $$ on the ambient finite-dimensional prefix space. This measure is Radon (Kallenberg, 2021), and $\eta_s(\R^{k_{t_0}+\cdots+k_s})=\Abar_0^{\rm live}(s)$ by Lemma 2.5. Inner regularity therefore supplies a compact prefix set $C_0$ with $\eta_s(C_0)\ge\Abar_0^{\rm live}(s)/2$. Since $Q^*(x^s\in S_s)=1$, the Borel set $C:=C_0\cap\operatorname{pr}_s^{-1}(S_s)$ carries the same $\eta_s$-mass, and $B:=\operatorname{pr}_s(C)$ is a bounded subset of the surgery region $S_s$ (it need not be closed); write $\varepsilon_0:=\varepsilon_0(B)$ for its slack from Assumption 3.13. We trigger the surgery on $\{H_s\in C\}$. This avoids the invalid inference that a projection of the closed path space must itself be closed. *Step 2 (surgery).* Disintegrate $Q^*$ at $s$ into the pre-$s$ law, the state kernel $\mu_h$, and the continuation kernels. For $q\in(0,1)$ and $p$ with $p\Lambda\le\varepsilon_0\wedge\tfrac12$, define $\widetilde Q$ by sampling the pre-$s$ history, the state $x$, and the continuation as under $Q^*$. On $\{H_s\in C\}$ (and only there), with probability $qp$ replace the path from $s$ by $F_s(D_\Lambda x)$, and with probability $q(1-p)$ by $F_s(S_{\Lambda,p}x)$; otherwise keep the original path. The modified kernel is Borel by Assumption 3.13 and because $C$ is Borel. *Step 3 (*$\widetilde Q\in\cM$*).* Nothing before $s$ changes. The row into $s$ holds because, pointwise in $x$, $(1-q\one_C)x+q\one_C\bigl[(1-p)S_{\Lambda,p}x+pD_\Lambda x\bigr]=x$ on live coordinates by (10), and stopped coordinates are copied; integrating against $\mu_h$ preserves every conditional mean exactly. Rows after $s$ hold on the original branch because $Q^*$’s kernels are untouched, and pathwise on the frozen branches. Coordinates on modified branches are bounded by $\Lambda\times$ integrable base coordinates, so true martingality is retained. Support membership is Assumption 3.13. *Step 4 (book deviation).* Claims paying before $s$ are unaffected. For a claim paying at $b\ge s$ with envelope $\varphi_i$, using $\varphi_i(h,S_{\Lambda,p}x)\le\varphi_i(h,x)$ and $\varphi_i(h,D_\Lambda x)\le\Lambda\varphi_i(h,x)$, $$ |c_i(\widetilde Q)-c_i^*| \le q\Bigl(\E_{Q^*}|X_i| +(1+p\Lambda)\,\E_{Q^*}[\varphi_i(h,x^s)]\Bigr) \le C_\cB\,q . $$ *Step 5 (correction and conclusion).* Fix, by the Slater condition, laws $Q_i^{\pm}\in\cM_{\rm f}$ with $c(Q_i^{\pm})=c^*\pm\rho e_i$ along an orthonormal basis $(e_i)_{i=1}^r$ of the direction space ($r=0$ being the empty-sum case). Expand $d:=c(\widetilde Q)-c^*=\sum_i d_i e_i$, put $\sigma_i=-\operatorname{sign}(d_i)$ when $d_i\ne0$ (arbitrary when $d_i=0$), and read $Q_i^{+1}=Q_i^+$, $Q_i^{-1}=Q_i^-$. With $\nu_i=|d_i|/\rho$, $\nu=\sum_i\nu_i=O(q)$, and $\mu_i=\nu_i/(1+\nu)$, the convex mixture ∎ $$ Q_\Lambda:=\Bigl(1-\sum_i\mu_i\Bigr)\widetilde Q +\sum_i\mu_iQ_i^{\sigma_i} $$ has total mass one, lies in $\cM_{\rm f}$, and satisfies $$ c(Q_\Lambda)-c^* =\frac{d}{1+\nu} -\sum_i\frac{|d_i|}{\rho(1+\nu)}\rho\operatorname{sign}(d_i)e_i=0. $$ Finally, the surgery scales the live coordinates only, so on the deformed branch $\Abar_s(h,D_\Lambda x) =\Abar_s^{\rm hist}(h)+\Lambda\Abar_s^{\rm live}(x) \ge\Lambda\Abar_s^{\rm live}(x)$, and by convexity $\psi(t):=(\Phi(t)-\Phi(0))/t$ is nondecreasing on $(0,\infty)$ with $\psi(t)\to\infty$ by superlinearity. With $a:=\Abar_0^{\rm live}(s)/4$ and $\Phi(t)\ge\psi(t)\,t$, $$ \begin{aligned} \E_{Q^*}\bigl[\one_C\,\Phi\bigl(\Abar_s(h,D_\Lambda x)\bigr)\bigr] &\ge\E_{Q^*}\bigl[\one_{C\cap\{\Abar^{\rm live}_s\ge a\}} \psi\bigl(\Lambda\Abar^{\rm live}_s\bigr) \Lambda\Abar^{\rm live}_s\bigr] \\ &\ge\psi(\Lambda a)\,\Lambda\, \E_{Q^*}\bigl[\Abar^{\rm live}_s \one_{C\cap\{\Abar^{\rm live}_s\ge a\}}\bigr] \ge\psi(\Lambda a)\,\Lambda a, \end{aligned} $$ because $\E[\Abar^{\rm live}\one_{C\cap\{\Abar^{\rm live}\ge a\}}] \ge\Abar_0^{\rm live}/2-a=a$. Hence $\E_{Q_\Lambda}[\Phi(\Abar_s)] \ge(1-\sum_i\mu_i)\,qp\,\psi(\Lambda a)\,\Lambda a$. With $g_\Lambda:=\psi(\Lambda a)\to\infty$, choose $p:=(\Lambda g_\Lambda^{1/3})^{-1}$ and $q:=g_\Lambda^{-1/3}$: then $p\Lambda=g_\Lambda^{-1/3}\to0$ meets the richness window, $q\to0$ makes the correction weights vanish, and $qp\,\psi(\Lambda a)\,\Lambda a=a\,g_\Lambda^{1/3}\to\infty$. For $s\le q_1$ the weight is the full annuity; for $q_1c)\le M_{\rm quoted}/c$ with $\{\chi\le c\}$ compact: the class is tight, and Prokhorov’s theorem applies. For closedness, let $Q_n\Rightarrow Q$ in $\cM_{\rm str}$. Since $\chi$ is nonnegative and continuous, the Portmanteau theorem gives $\E_Q\chi\le\liminf_n\E_{Q_n}\chi\le M_{\rm quoted}$; the fixed-initial-state condition also passes to $Q$ because its cylinder is closed, and the currency-price bound makes both endpoints of every martingale row integrable under $Q$. First, for any continuous $f$ with $|f|\le C_\varepsilon+\varepsilon\chi$ and any $R>0$, choose a continuous cutoff $\rho_R$ equal to one on $\{\chi\le R\}$ and vanishing off $\{\chiR\}}\le2\varepsilon\chi$, so $$ \limsup_n\bigl|\E_{Q_n}f-\E_Qf\bigr| \le0+4\varepsilon M_{\rm quoted}, $$ and $\varepsilon\downarrow0$ shows every $\chi$-sublinear expectation passes to the limit. Applied to the martingale tests with *continuous* $\phi$ this closes (3) for such $\phi$; bounded Borel $\phi$ then follow because the continuous-$\phi$ rows say that the two finite signed measures the tests induce on the history space agree against all bounded continuous functions, hence coincide. Thus $Q\in\cM$; the carried rows then give $\E_Q[\zeta_{t_i}]=\Zbar_0(t_i)$. For the strip constraints, the continuous function $g_i=\zeta_{t_i}^2$ is nonnegative, so monotone truncation gives $\E_Q[g_i\wedge R]=\lim_n\E_{Q_n}[g_i\wedge R] \le\tau_i\pi_i+\Zbar_0(t_i)$ for every $R$, and monotone convergence preserves the ask bound. (The bid side of a strip quote is not used and need not be weakly closed.) Convexity is clear. Hence $\cM_{\rm str}$ is compact. *(ii) Continuity and the image.* For bounded continuous $\varphi$, $W_a\varphi(Y_a)$ is $\chi$-sublinear by Theorem 4.6, and taking $\varphi(y)=|y|$ handles the first absolute moments, since $W_a(1+|Y_a|)$ is linearly dominated; the displayed truncation argument then gives convergence of $\int\varphi\,\dd\Gamma_a(Q_n)$ and of the first moments along any weakly convergent sequence in the class; continuity of the integrands themselves is Assumption 2.3. Thus $\Gamma$ is continuous; it is affine by construction. A continuous affine image of a compact convex set is compact convex, and compact sets are closed in the Hausdorff output space. *(iii) Separation.* On the positive cone, weak convergence together with convergence of first absolute moments is equivalent to convergence against every continuous function of at most linear growth (the usual $1$-Wasserstein topology after adjoining the mass coordinate). Thus the observation metric is the restriction of the locally convex weak topology used next. Embed the output product in the locally convex space of tuples of finite signed measures with finite first absolute moment, in duality with tuples of continuous functions of at most linear growth; the topology of (ii) is the weak topology of this pairing. If $\gamma\notin\Gamma(\cM_{\rm str})$, the Hahn–Banach strict separation theorem for a point and a compact convex set in a Hausdorff locally convex space furnishes a continuous linear functional, i.e. a tuple $(\varphi_a)_a$ in the pairing class, with $\langle\varphi,\gamma\rangle> \sup_{Q}\langle\varphi,\Gamma(Q)\rangle=\sup_Q\E_Q[\sum_aW_a\varphi_a(Y_a)]$, violating the displayed inequality; conversely, membership makes every inequality an instance of monotonicity of suprema.∎ *Proof of Proposition 6.8 .* The pricing statements are Lemma 6.6 with the two settlement indices, and $V^{(0)}_TC_{0,N}=C_{0,N}^2\zeta_T$. For (i): the settlement $C_{0,N}$ is a Borel function of the rates fixed at $t_0,\ldots,t_{N-1}$ and is therefore copied by both terminal surgery branches. Choose a compact positive-probability set $B_0$ of terminal price states under $Q^*$; since $Q^*$ loads the terminal surgery region fully, the Borel set $B:=B_0\cap S_T$ has the same positive probability and is bounded, and we write $\varepsilon_0:=\varepsilon_0(B)$ for its slack. Perform Steps 2–4 of Theorem 3.16 at $s=T$, with $\kappa_T$ in place of $\kappa$, triggering on $B$. The asserted terminal richness gives support membership and the price bound, while the frozen envelopes give absolute integrability and an $O(q)$ deviation of the finite book. Pointwise in the sampled history and terminal state, $$ C_{0,N}^2\bigl((1-q\one_B)x_T +q\one_B[(1-p)(S^T_{\Lambda,p}x)_T +p(D^T_\Lambda x)_T]\bigr) =C_{0,N}^2x_T, \qquad x_T=\zeta_T, $$ by (10). Hence the raw law $\widetilde Q$ belongs to $\cM^{\rm aug}_{\rm f}$ and preserves $g$ exactly. It remains to make the finite correction without assuming more than (30). Set $v^*:=\mathcal J(Q^*)$, let $L:=\operatorname{span}(\mathcal J(\cM^{\rm aug}_{\rm f})-v^*)$, and choose an orthonormal basis $b_1,\ldots,b_r$ of $L$. Relative interiority supplies $\rho>0$ and fixed laws $Q_i^\pm\in\cM^{\rm aug}_{\rm f}$ satisfying $\mathcal J(Q_i^\pm)=v^*\pm\rho b_i$. Write $d:=\mathcal J(\widetilde Q)-v^*=\sum_i d_i b_i$; its futures coordinate is zero and $\lVert d\rVert=O(q)$. Put $$ \nu_i:=|d_i|/\rho,\qquad\nu:=\sum_i\nu_i, \qquad\mu_i:=\frac{\nu_i}{1+\nu},\qquad\sigma_i:=-\operatorname{sign}(d_i), \qquad Q_\Lambda:=\frac1{1+\nu}\widetilde Q +\sum_i\mu_iQ_i^{\sigma_i}, $$ omitting zero coordinates. Direct substitution gives $\mathcal J(Q_\Lambda)=v^*$ exactly. Notice that this argument uses directions in the affine hull of the *whole* augmented image; the individual $Q_i^\pm$ need not themselves have futures quote $g(Q^*)$. On the deformed branch $\zeta_T\mapsto\Lambda\zeta_T$ because $\kappa_T(T)=0$, so with $B$ as chosen above the nonnegative budget obeys ∎ $$ \E_{Q_\Lambda}[C_{0,N}^2\zeta_T^2] \ge\frac{qp\Lambda^2}{1+\nu}\, \E_{Q^*}[\one_BC_{0,N}^2\zeta_T^2]. $$ The bracket is strictly positive and finite: $\zeta_T$ is bounded on the bounded set $B$, while $\E_{Q^*}[C_{0,N}^2]\le C_P\,\E_{Q^*}[C_{0,N}^2\zeta_T] =C_P\,\zeta_0G_0<\infty$ by Assumption 2.2 and the finiteness of the quote. Take $q>0$ fixed and small enough that $\nu=O(q)$ is bounded, and set $p=(\varepsilon_0\wedge\tfrac12)/(2\Lambda)$. The right-hand side then tends to infinity linearly in $\Lambda$, proving (31). The same pointwise calculation works coordinatewise for the stated finite family of past-measurable plain settlements. For (ii), whenever the displayed products are integrable, both inequalities are the standard association bounds for nonnegative variables, applied with the stated sign. **Acknowledgments and reproducibility.** This paper was revised through iterative machine-assisted hostile review and exact-source verification; the tools used are recorded in the AI-use disclosure closing Section 8. That process is not a substitute for independent human peer review. 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