---
title: "Common-Budget Certification of Rolling SPX/VIX Markets: Finite Witnesses, Exact Continuous Realization, and the Clock Boundary"
authors:
  - name: "Shaosai Huang"
    affiliation: "Kspectra Research Inc., Toronto, Canada"
date: "2026-09"
status: "Working paper"
url: https://kspectra.ai/papers/common-budget-certification-spx-vix/
doi: 10.2139/ssrn.7399598
ssrn: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7399598
---

# Common-Budget Certification of Rolling SPX/VIX Markets: Finite Witnesses, Exact Continuous Realization, and the Clock Boundary

Shaosai Huang — Working paper, version of September 2026.

Links: [Web page](https://kspectra.ai/papers/common-budget-certification-spx-vix/) · [SSRN](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=7399598)

> 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{huang2026common,
  author = {Huang, Shaosai},
  title  = {{Common-Budget Certification of Rolling SPX/VIX Markets: Finite Witnesses, Exact Continuous Realization, and the Clock Boundary}},
  year   = {2026},
  month  = sep,
  note   = {Working paper, version of September 2026},
  doi    = {10.2139/ssrn.7399598},
  url    = {https://kspectra.ai/papers/common-budget-certification-spx-vix/}
}
```

## Macros

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

```latex
\newcommand{\E}{\mathbb E}
\newcommand{\R}{\mathbb R}
\newcommand{\Q}{\mathbb Q}
\newcommand{\cA}{\mathcal A}
\newcommand{\cC}{\mathcal C}
\newcommand{\cF}{\mathcal F}
\newcommand{\cG}{\mathcal G}
\newcommand{\cH}{\mathcal H}
\newcommand{\cP}{\mathcal P}
\newcommand{\cW}{\mathcal W}
\newcommand{\1}{\mathbf 1}
\newcommand{\VIX}{\operatorname{VIX}}
\newcommand{\Law}{\operatorname{Law}}
```

## Abstract

Fix a positive VIX tenor and a coercive continuous-path class. We ask when an all-strike SPX call surface and a continuously rolling VIX call surface are generated by one risk-neutral filtered law. Constructive joint calibration is formulated where consecutive windows abut, and there window-by-window solutions suffice; listed weekly windows instead share twenty-three of their thirty days. This paper proposes a coherence theory for the structure that gluing ignores. The state carries log spot, the quadratic variation $A$ of its stochastic logarithm, and one nonnegative additive conditional budget field

\[ B(t;u,v)=\E [A_v-A_u\mid \cF _t],\qquad t\leq u\leq v, \]

so every overlapping window is a projection of one tower, and the tower yields an explicit overlap inequality between consecutive VIX smiles that complete vanilla call surfaces alone can violate. Certification is then a dichotomy. On a finite grid the quote image is compact and convex, the alternative returns either a calibrating law or a strict pathwise separator, and every calibrating finite law has an exact positive continuous conditional-Bass realization preserving its skeleton and all window budgets. Inside one fixed compact class, a calibrated continuum law exists if and only if every determining finite quote restriction is feasible within the structural class, so failure is witnessed by finitely many quotes and structural rows together with the class tail bound; every such calibrated law is already a stochastic-clock model, and absolute continuity of the clock is exactly the additional condition for a calendar-time Brownian representation. The analysis is filtration-sensitive throughout: what a model reveals, and when, is part of its identity, which yields a payoff-typed early-revelation diagnostic and closedness of the feasible surface set in the enhanced topology. All statements are relative to one declared class, and examples show that this cannot be relaxed.

**Keywords:** SPX options; VIX options; martingale transport; conditional variance budgets; feasibility certificates; Bass martingales; stochastic clocks.

**MSC 2020:** 91G20 (primary); 60G44, 60G42, 49N15, 90C25.

**JEL:** G13, G12, C02, C61.

## 1 Introduction

The traded VIX complex is a *rolling* market. With weekly VIX expiries, listed contracts reference thirty-day forward-variance windows whose start dates are only days apart, so the quoted windows overlap heavily. In the continuous positive-martingale idealization, a VIX fixing at time $t$ with tenor $\Delta$ is the conditional future log-variance budget

$$
\begin{equation}\label{eq:intro-vix}\tag{1} (V_t^\Delta)^2 =\frac1\Delta\E\!\left[A_{t+\Delta}-A_t\mid\cF_t\right] =-\frac2\Delta\E\!\left[\log\frac{S_{t+\Delta}}{S_t} \Bigm|\cF_t\right], \end{equation}
$$

where $A=\langle M\rangle$ is the quadratic variation, or bracket, of the stochastic logarithm $M=\int dS/S$. Overlapping windows therefore price increments of one bracket, and their quotes are linked across fixing dates by the tower property of conditional expectation, not only by static convex order.

For SPX alone the static side is classical: complete call prices identify the one-time marginals, and convex order characterizes whether some martingale carries them (Breeden and Litzenberger, 1978; Strassen, 1965; Kellerer, 1972). VIX admits no such static reduction: validity of every SPX and VIX marginal separately does not settle joint generability, because the VIX readout is conditional and filtration-sensitive.

Constructive joint calibration is formulated where the windows tile. Guyon (2024) closes the one-window problem exactly, with entropic selection, and continuous-time, dual, and computational counterparts follow (Guo et al., 2022; Guyon, 2025; Zitridis, 2025); the multi-maturity extensions place consecutive maturities one tenor apart, so that the windows abut (Bourgey and Guyon, 2024; Dong et al., 2025; Zhang, 2025). In that exact tiled regime, provided adjacent blocks use the same prescribed intermediate SPX marginal, local and global feasibility coincide, and a global law exists whenever every local block does (Acharya et al., 2026). The listed weekly complex is not in that regime: consecutive thirty-day windows share twenty-three days, and an extension to overlapping spans requires a model of the variance allocated to the overlap together with martingale rows at every adjacent event time (Acharya et al., 2026). That object is the subject of this paper.

The object every overlapping window projects from is one budget field

$$
\begin{equation}\label{eq:intro-budget}\tag{2} B(t;u,v):=\E[A_v-A_u\mid\cF_t], \qquad0\le t\le u\le v\le H, \end{equation}
$$

nonnegative, additive in $(u,v)$, and a martingale in its valuation time. The question of this paper is whether a complete SPX call surface and the entire rolling fixed-tenor VIX call surface are generated by *one* filtered risk-neutral law carrying this field, and, when they are not, how failure is witnessed. *This paper proposes a coherence theory for the structure that gluing ignores.*

The hypotheses are sharp, and Section 4 shows it. Example 4.1 gives a fully specified four-date panel of SPX smiles, VIX smiles at three rolling fixings, and three pairwise dispersion claims: deleting any one pairwise claim leaves a market exactly calibratable by a positive continuous martingale, yet no law matches the whole panel, and the failure carries an explicit finite witness with gap one half. Neither the individual smiles nor any leave-one-cross-claim-out test detects it. Its windows abut and its cross-window claims are not vanilla options, so the theory below is stated both for declared panels that may carry such joint claims (Section 2) and for vanilla surfaces (Section 3); Example 4.2 exhibits the same failure with complete vanilla smiles alone on genuinely overlapping windows, through an explicit overlap inequality (Proposition 3.5) that the mean rows and every one-window test miss. Ordinary marginal-hypergraph combinatorics does not supply the failure: rolling windows are intervals, so the hypergraph they form is acyclic and the classical marginal problem (Vorob’ev, 1962) would grant extension rather than obstruct it (Remark 4.3).

### 1.1 Main results

Three themes run through the results. *Certification is a dichotomy*: every alternative below returns either a calibrating law or a strict separator, and both are explicit objects rather than existence statements; “certificate” is used in both senses, but every realization statement below concerns only a feasible law certificate. *The analysis is filtration-sensitive throughout*: the budget field is a prediction coordinate, and what a model reveals, and when, is part of its identity (Section 2.3, Theorem 3.11, Remarks 3.12 and 4.3). *Everything is relative to one declared coercive class*, and the examples show that this cannot be relaxed.

*The first result is finite-dimensional but architecture-free.* For a finite grid and a finite continuous raw-payoff vector, conditional martingality, diagonal log-contract identities, and every budget tower are affine rows (Lemma 2.1). Under a common $q$-moment envelope their feasible quote image is compact and convex. Membership is equivalent to all support inequalities, and failure admits a strict pathwise structural separator (Theorem 2.2). A conditional Bass construction then turns every feasible discrete law—including boundary and atomic ones—into a positive continuous model without changing the grid skeleton or any declared VIX budget (Lemma 2.8 and Theorem 2.9). For a finite payoff panel, exact recursive cubature can first replace the law by a finite nonrecombining skeleton while preserving the quote vector and coercive moment (Lemma A.3).

*The second and headline result concerns the complete surface.* We augment path space by the bracket and the full jointly continuous budget field and fix one lower-semicontinuous coercive envelope. Within that single compact class, the following are equivalent (Theorem 3.6):

- (i) one law calibrates all SPX and rolling fixed-tenor VIX calls;
- (ii) every finite subset of one countable determining quote panel is feasible in the same class;
- (iii) no finite strict support-functional witness exists; and
- (iv) one positive continuous stochastic-clock model generates both surfaces and its own VIX budgets.

The finiteness is on the quote side: each restriction in (ii) is still tested against the full structural class, so (i)$\Leftrightarrow$(ii) is a compact finite-intersection statement rather than a finite-dimensional market test; the genuinely finite object is the witness of failure in (iii).

*The third result locates the model boundary* (Theorem 3.9). If the clock has density $a_t$, the model can be written $dS_t/S_t=\sqrt{a_t}\,dW_t$ on an extension, and absolute continuity of the clock is exactly what decides this: a singular clock gives an exact continuous model but no calendar-time volatility density. This separates weak realization from smooth, elliptic, or low-dimensional diffusion realization; Examples 4.5 and 4.6 inhabit the two sides.

Further results support these. The tower has a market-facing consequence: an explicit overlap inequality between consecutive VIX smiles and the SPX log strip over their non-overlapping part (Proposition 3.5), void without overlap, not implied by the mean rows, evaluable from the two complete VIX strips and the intervening complete SPX log-strip increment, and violated by a vanilla-only all-strike panel on overlapping windows (Example 4.2). The finite alternative has a band form covering bid–ask and confidence panels, in which a witness is a separator that clears the quoted spread (Proposition 2.5). The feasible surface set is closed in the enhanced topology carrying spot, quadratic variation, budget, and filtration information (Theorem 3.11). The no-early-revelation step of the realization is isolated as a payoff-typed diagnostic: the revelation defect vanishes row by row exactly under conditional mean-independence (Proposition 2.11), spot martingality can survive a premature mark while the budget row fails (Example 2.12), and panel-safety is the exact checkable requirement for preserving conditional values (Definition 2.13).

The proofs combine classical tools—compact separation, conditional randomization, Bass interpolation, continuous-martingale time change—with the one common rolling budget, full-history testing, finite obstruction, and the exact feasible-law-to-model boundary.

### 1.2 Relation to adjacent results

The problem descends from model-free SPX/VIX bounds (De Marco and Henry-Labordère, 2015; Guyon et al., 2017; Papanicolaou, 2018) and from Guyon (2024), who proves the exact one-window feasibility-and-arbitrage alternative with entropic selection. Its continuous-time, dual, and computational successors (Guo et al., 2022; Guyon, 2025; Zitridis, 2025) are formulated for one window, with further maturities noted as possible at the cost of one state per maturity; multi-maturity constructions iterate or chain consecutive maturities, the gluing argument for which places those maturities exactly one tenor apart, so that the windows abut (Bourgey and Guyon, 2024; Dong et al., 2025; Zhang, 2025); in that exact tiled setting with prescribed common seam marginals, local and global nonemptiness coincide, while the exact overlapping-window reference construction is left open (Acharya et al., 2026); term-structure no-arbitrage between VIX futures and equity derivatives is expressed by Badran and Goldys (2015) as drift restrictions on model coefficients; and term-structure consistency inside a fixed architecture is studied by Papanicolaou (2022) and Forde and Smith (2023). Closest in spirit, Zaugg and Grzelak (2026) build a parametric VIX-first model on the rolling-window definition, with single-window consistency enforced by a backward Feynman–Kac equation; positivity of the recovered variance and general existence are left open, and no feasibility characterization is given. None of the works reviewed here proves a nonparametric, complete-surface, common-class alternative for the rolling pair with overlapping windows, which is what is established below.

The key methods are those of martingale-transport duality (Beiglböck et al., 2013), the Bass embedding (Bass, 1983), adapted topologies for filtration-sensitive quantities (Backhoff-Veraguas et al., 2020), and Tchakaloff cubature (Bayer and Teichmann, 2006). The comparison is feature-specific and is not a global priority claim.

### 1.3 Scope

All prices are discounted and deterministic carry is absorbed in the forward numeraire. VIX means the idealized conditional log contract in (1); exchange discretization, jumps, rates, dividends, and settlement conventions are outside the theorem. We fix one $\Delta>0$ and do not take $\Delta\downarrow0$. Existence is always relative to one stated compact class. A separator is a model-class certificate; it is an executable arbitrage only when every structural and coercive term is traded at the assumed cost. No uniqueness, physical law, statistical identification, or hedging conclusion is claimed. The mathematical theorems are not empirical claims: they neither assert that historical SPX/VIX data satisfy the maintained conditions nor select a forecasting law.

Throughout, $R$ denotes a candidate risk-neutral law and $A$ denotes the bracket clock. This convention is maintained in both the finite and continuous formulations below.

The paper follows the three results. Section 2 treats the finite grid: the structural rows (Lemma 2.1), the quote-space alternative and its band form (Theorem 2.2, Proposition 2.5), the conditional-Bass realization of every structural finite law (Theorem 2.9), and the early-revelation diagnostic that the realization isolates (Section 2.3). Section 3 passes to complete surfaces: the certificate class and its identification (Definition 3.2, Lemma 3.3), the overlap inequality between consecutive VIX smiles (Proposition 3.5), the finite-intersection alternative (Theorem 3.6), the clock boundary (Theorem 3.9), and closedness in the enhanced topology (Theorem 3.11). Section 4 sharpens each hypothesis by example, Section 5 records what the theorems change, what is checkable from market input, and the limitations, and Section 6 concludes. Appendices A and B supply the finite cubature reduction and the details of the Bass realization.

## 2 The finite common-budget market

On a finite grid every structural row is affine and the alternative is finite-dimensional; this section proves it, realizes each structural law as a continuous model, and isolates the information step that the realization depends on. Section 3 lifts the same architecture to complete surfaces.

Fix a grid $0=t_0<\cdots<t_J$ and write $\delta_j=t_{j+1}-t_j$. At date $t_j$ store

$$
Z_j=\left(X_j,(B_{j,h})_{h=j}^{J-1}\right), \qquad S_j=e^{X_j},
$$

where $B_{j,h}$ is the date-$j$ conditional budget rate for cell $[t_h,t_{h+1}]$. The root $Z_0=z_0$ is deterministic. Let $H_j=(Z_0,\ldots,Z_j)$ and $\cF_j=\sigma(H_j)$ on the corresponding Polish path space $\Omega_J$. Define

$$
\begin{align} R_j^S&:=S_{j+1}-S_j,\label{eq:resS}\tag{3} \\
R_j^D&:=2(X_{j+1}-X_j)+\delta_jB_{j,j},\label{eq:resD}\tag{4} \\
R_{j,h}^B&:=B_{j+1,h}-B_{j,h},\qquad j<h.\label{eq:resB}\tag{5} \end{align}
$$

Let $\cW_J$ be the linear span of these residuals multiplied by bounded continuous functions of $H_j$ at their conditioning date.

Put

$$
\begin{equation}\label{eq:finite-envelope}\tag{6} \Gamma=1+\sum_{j=0}^J(|X_j|+e^{X_j}) +\sum_{0\le j\le h<J}B_{j,h}. \end{equation}
$$

For $q>1$ and $M>\min\Gamma^q$, define

$$
\begin{align*} \cP_M^J&:=\{R\in\cP(\Omega_J):\E_R\Gamma^q\le M\}, \\
\cA_M^J&:=\{R\in\cP_M^J:\E_RW=0\text{ for every }W\in\cW_J\}. \end{align*}
$$

Only nonnegativity of the marks is imposed; no strictly positive lower bound on budgets is needed anywhere below. Nonnegativity is not a feasibility test: Remark 2.3 shows that the structural class can be empty with nonnegative root budgets.

**Lemma 2.1 (Structural rows are the common budget).**  For $R\in\cP_M^J$, membership in $\cA_M^J$ is equivalent to

$$
\begin{align} \E[R_j^S\mid\cF_j]&=0,& \E[R_j^D\mid\cF_j]&=0,& \E[R_{j,h}^B\mid\cF_j]&=0.\label{eq:finite-conditional}\tag{7} \end{align}
$$

Consequently $S$ is a positive discrete martingale and, for every $0\le i<k\le J$,

$$
\begin{equation}\label{eq:finite-window}\tag{8} A_{i,k}:=\sum_{h=i}^{k-1}\delta_hB_{i,h} =-2\E[X_k-X_i\mid\cF_i]. \end{equation}
$$

Thus $V_{i,k}=\sqrt{A_{i,k}/(t_k-t_i)}$ is the model’s own conditional log-contract volatility for every grid-aligned window, simultaneously.

*Proof.* The conditional equations imply annihilation after conditioning. Conversely, fix one residual $R$ at date $j$ and define the finite signed measure $\nu(A)=\E[R\1_{\{H_j\in A\}}]$ on the Polish history space. Its finiteness follows from $|R|\le C\Gamma$ and $q>1$. Annihilation against every bounded continuous history function says that $\nu$ integrates every such function to zero. These functions determine finite signed Borel measures, hence $\nu=0$ and $\E[R\mid\cF_j]=0$. Apply this to (3)–(5).

Iterating the tower row gives $B_{i,h}=\E[B_{h,h}\mid\cF_i]$. Condition the diagonal identity at date $h$ back to date $i$, multiply by $\delta_h$, sum, and telescope the log returns. This proves (8).∎

### 2.1 Finite-panel feasibility and strict separation

Let $\Phi=(\Phi_1,\ldots,\Phi_m)$ be continuous on $\Omega_J$ and satisfy $\|\Phi\|\le C_\Phi\Gamma$. It may contain raw SPX calls, VIX futures and calls, or continuous joint grid payoffs. Define

$$
\cC_M^J:=\{\E_R\Phi:R\in\cA_M^J\},\qquad h_M^J(\lambda):=\sup_{R\in\cA_M^J}\E_R[\lambda^\top\Phi],
$$

with the convention $\sup\varnothing=-\infty$; when $\cA_M^J\ne\varnothing$ the supremum is a maximum, by the compactness proved next.

**Theorem 2.2 (Finite quote-space alternative).**  Exactly one of the following two cases holds.

- (a) $\cA_M^J\ne\varnothing$. Then $\cA_M^J$ and $\cC_M^J$ are compact and convex, and $$
\begin{equation}\label{eq:support-test}\tag{9} c\in\cC_M^J \quad\Longleftrightarrow\quad\lambda^\top c\le h_M^J(\lambda) \quad\text{for every }\lambda\in\R^m. \end{equation}
$$ If $c\notin\cC_M^J$, there are $\lambda$, a finite structural gain $W\in\cW_J$, $\eta\ge0$, and $a\in\R$ such that $$
\begin{equation}\label{eq:finite-pathwise-separator}\tag{10} a+\eta\Gamma^q+W\ge\lambda^\top\Phi\quad\text{on }\Omega_J, \qquad a+\eta M<\lambda^\top c. \end{equation}
$$ Conversely, (10) excludes every calibrating law in $\cA_M^J$.
- (b) $\cA_M^J=\varnothing$. Then $h_M^J\equiv-\infty$, and for every $\lambda$ the pathwise form (10) holds with $a+\eta M$ arbitrarily negative, already at $\lambda=0$, where the certificate reads $a+\eta\Gamma^q+W\ge0$ with $a+\eta M<0$: emptiness is certified by finitely many structural rows together with the moment-envelope constraint, without using quote rows.

*Proof.* Case (a) first. The sublevel sets of $\Gamma$ are compact. Markov’s inequality gives tightness of $\cP_M^J$, and lower semicontinuity of $\Gamma^q$ makes it closed, hence compact. Moreover,

$$
\sup_{R\in\cP_M^J}\E_R[\Gamma\1_{\{\Gamma>K\}}] \le MK^{1-q}.
$$

Therefore expectations of continuous functions dominated by $C\Gamma$ are continuous on $\cP_M^J$. The structural rows define a closed affine subset, and the quote map is continuous and affine. This proves compactness and convexity. Formula (9) is finite-dimensional separation.

For completeness, let $G=\lambda^\top\Phi$. Sion’s minimax theorem (Sion, 1958) on the compact convex set $\cP_M^J$ and the linear space $\cW_J$ gives ∎

$$
\begin{equation}\label{eq:finite-minimax}\tag{11} \sup_{R\in\cA_M^J}\E_RG =\inf_{W\in\cW_J}\max_{R\in\cP_M^J}\E_R(G-W), \end{equation}
$$

where the right-hand maximum exists because $\cP_M^J$ is compact and nonempty. Indeed the inner infimum in the reverse order equals $\E_RG$ on $\cA_M^J$ and $-\infty$ off it, by scaling a violated row. For each fixed $W$, the concave value function

$$
r\longmapsto\max\{\E_R(G-W):\E_R\Gamma^q\le r\}
$$

has a nonnegative supergradient at the interior point $M$. Applying the supergradient inequality to Dirac masses yields

$$
\max_{R\in\cP_M^J}\E_R(G-W) =\inf_{\eta\ge0}\left\{\eta M+ \sup_{\omega}(G-W-\eta\Gamma^q)(\omega)\right\}.
$$

Writing the pointwise supremum as the least cash $a$ gives the pathwise dual. Strict separation of $c$ from $\cC_M^J$ supplies a positive gap; choosing a dual tuple within that gap proves (10). Integration under a calibrating law proves the converse. For case (b), if $\cA_M^J=\varnothing$, the inner infimum $\inf_{W\in\cW_J}\E_R(G-W)$ of the reverse order is $-\infty$ at every $R\in\cP_M^J$, so (11) holds in $[-\infty,\infty)$ and the minimax value is $-\infty$: for every $K$ some $W$ has $\max_{\cP_M^J}\E_R(G-W)<-K$, and the one-moment dual for that $W$ supplies $(a,\eta)$ with $a+\eta\Gamma^q+W\ge G$ and $a+\eta M<-K$.

**Remark 2.3 (A checkable nonemptiness criterion).**  The standing condition $M>\min\Gamma^q$ does not by itself make $\cA_M^J$ nonempty. With $J=1$, $\delta_0=1$, $X_0=0$, $B_{0,0}=2$, $q=2$, and $M=30$ one has $\min\Gamma^2=25<30$, yet the spot and diagonal rows force $\E e^{X_1}=1$ and $\E X_1=-1$ for every structural law, hence $\E\Gamma\ge6$ and $\E\Gamma^2\ge36>M$. A sufficient condition is explicit: the reference law $R^{\mathrm{ref}}$ with independent lognormal cell returns $X_{j+1}-X_j\sim N(-\tfrac12\delta_jB_{0,j},\delta_jB_{0,j})$ and frozen marks $B_{j,h}=B_{0,h}$ satisfies every structural row, so $\cA_M^J\ne\varnothing$ whenever $M\ge\E_{R^{\mathrm{ref}}}\Gamma^q$, a finite lognormal-moment computation from the root budgets. In the example, $\E_{R^{\mathrm{ref}}}\Gamma^2\approx48.4$; an explicit emptiness certificate is $\eta=1$, $W=-12R_0^S+6R_0^D$, $a=-36$, and none with $\eta=0$ exists, since $R^{\mathrm{ref}}$ satisfies the rows.

**Remark 2.4 (Structural witness versus traded arbitrage).** The $R^S$ portion of $W$ is an ordinary discounted SPX trading gain. The diagonal and tower portions are Lagrange multipliers for the model’s log-budget identities. They are traded gains only if the corresponding forward log contracts and budget claims are available. Likewise $\eta\Gamma^q$ is a tail certificate unless a dominating payoff can be bought for at most $M$. The theorem always supplies a strict mathematical witness relative to the class; literal execution needs these extra instruments.

**Proposition 2.5 (Band and noisy panels).**  Let $c^-\le c^+$ componentwise and $\mathcal B=\prod_i[c_i^-,c_i^+]$. No nonemptiness is assumed. Exactly one of the following holds:

- (i) some $R\in\cA_M^J$ has $\E_R\Phi\in\mathcal B$; or
- (ii) there is $\lambda\in\R^m$ with $$
\begin{equation}\label{eq:band-witness}\tag{12} h_M^J(\lambda)<\underline c(\lambda) :=\sum_i\bigl(\lambda_i^+c_i^--\lambda_i^-c_i^+\bigr) =\min_{c\in\mathcal B}\lambda^\top c . \end{equation}
$$

In case (ii) the witness upgrades to the pathwise form (10) with $a+\eta M<\underline c(\lambda)$.

*Proof.* If $\cA_M^J=\varnothing$ then (i) fails and (ii) holds with $h_M^J\equiv-\infty$, with the pathwise upgrade of the empty case of Theorem 2.2. Otherwise $\cC_M^J$ and $\mathcal B$ are compact and convex, so they are disjoint if and only if some linear functional strictly separates them, which is (ii). The pathwise upgrade is Theorem 2.2 applied to $G=\lambda^\top\Phi$ inside the gap $\underline c(\lambda)-h_M^J(\lambda)$.∎

**Remark 2.6 (Transaction costs and statistical bands).** $\underline c(\lambda)$ is the revenue of assembling $\lambda$ at the quoted spread: positions sold at the bid and bought at the ask. A band witness is therefore a separator that clears the spread. Statistical confidence bands give the same statement conditionally on coverage: whenever the bands jointly cover the true prices, band infeasibility certifies that no law in the class matches them; constructing such bands from data is outside this paper. The reproducibility package implements exactly this band form, in elastic-programming shape, on the four-date panel of Example 4.1 and on listed SPX/VIX quotes.

**Remark 2.7 (Root budgets as declared inputs).**  The root state $z_0$, including every root mark $B_{0,h}$, is a declared input, matching the continuum situation where complete SPX smiles force the root strip through (31); a finite panel need not determine it. A panel without declared root budgets is handled in the same framework by dropping the deterministic-root restriction on the marks: the structural rows remain affine, $\cA_M^J$ remains compact and convex, and Theorem 2.2 and Proposition 2.5 apply verbatim, with the same extended-real convention if the relaxed class is empty. Nonemptiness of the structural class with a random root implies nonemptiness with some deterministic root: testing the rows against a countable separating class and conditioning on $Z_0$ yields, outside one null set, deterministic-root conditional laws satisfying every structural row, while $\E[\Gamma^q]=\E\bigl[\E[\Gamma^q\mid Z_0]\bigr]\le M$ enforces the moment cap on a set of root values of positive measure. Quoted prices, by contrast, are matched only on average across roots, so calibrated feasibility is genuinely weaker with a random root than with a declared one and is decided directly in the relaxed class, which is the correct primitive when root budgets are unquoted.

### 2.2 Every finite structural law has a continuous realization

The finite alternative does not yet say that a structural law is a continuous model. We now supply that implication. The construction uses a conditional version of the Bass embedding (Bass, 1983); unlike an endpoint interpolation performed after calibration, it also preserves the successor budget marks and does not reveal their randomizer at the left edge.

Let $E$ be standard Borel and let $\kappa(e,ds)$ be a probability kernel on $(0,\infty)$ satisfying

$$
\begin{equation}\label{eq:bass-moments}\tag{13} \int s\,\kappa(e,ds)=m(e),\qquad\int(s^q+|\log s|)\,\kappa(e,ds)<\infty. \end{equation}
$$

Let $q(e,u)$ be its generalized quantile, $\Phi_N$ the standard-normal CDF, and

$$
\begin{equation}\label{eq:bass-heat}\tag{14} f(e,z)=q(e,\Phi_N(z)),\qquad u(r,e,x)=\int_\R f(e,x+\sqrt{1-r}\,z)\varphi(z)\,dz. \end{equation}
$$

**Lemma 2.8 (Conditional Bass edge and log budget).**  For a Brownian motion $W$, conditional on $e$ the process

$$
Y_r=u(r,e,W_r),\quad0\le r<1, \qquad Y_1=f(e,W_1),
$$

is a continuous strictly positive true martingale from $m(e)$ to terminal law $\kappa(e,\cdot)$. It is jointly measurable in $(e,r,W_r)$ and, on $r<1$,

$$
dY_r=u_x(r,e,W_r)\,dW_r.
$$

Moreover,

$$
\begin{equation}\label{eq:bass-entropy}\tag{15} \E\!\left[\int_0^1\left(\frac{u_x(r,e,W_r)} {u(r,e,W_r)}\right)^2dr \Bigm|e\right] =-2\int\log\frac{s}{m(e)}\,\kappa(e,ds). \end{equation}
$$

*Proof.* Measurability of the parameterized quantile follows from

$$
\{(e,v):q(e,v)<a\} =\bigcup_{r\in\Q,\,0<r<a}\{(e,v):\kappa(e,(0,r])\ge v\}.
$$

The generalized-inverse identity gives the terminal law. Conditional on $W_r=x$, the terminal Brownian value is $x+\sqrt{1-r}Z$, so $Y_r=\E[f(e,W_1)\mid\cF_r^W,e]$. The $q>1$ moment closes the martingale and gives a continuous version up to $r=1$. A nonnegative martingale with a strictly positive terminal value cannot hit zero, proving strict positivity. Gaussian convolution and Hölder’s inequality justify differentiation under the integral on compact subsets of $[0,1)\times\R$, and Itô’s formula gives the displayed SDE.

Write $dY_r/Y_r=\theta_r\,dW_r$, localize $\Lambda_r=\int_0^r\theta_s^2ds$, and apply Itô’s formula: ∎

$$
\log Y_{\tau_n}=\log m(e)+\int_0^{\tau_n}\theta_s\,dW_s -\frac12\Lambda_{\tau_n}.
$$

The stopped logarithms are uniformly integrable. Indeed $\log^+Y_\tau\le Y_\tau$, while conditional Jensen gives $\log^-Y_\tau\le\E[\log^-Y_1\mid\cF_\tau,e]$. Letting $n\to\infty$ and using monotone convergence proves (15). This is a complete proof of the Bass fact needed below; no optimality property of Bass martingales is used.

**Theorem 2.9 (Exact finite common-budget realization).**  Let $R\in\cA_M^J$. There exists a filtered model with a continuous positive true martingale $S$ and a continuous finite-dimensional Gaussian memory factor such that:

- (i) its full grid skeleton $\bigl(X_j,(B_{j,h})_{h=j}^{J-1}\bigr)_{j=0}^J$ has law exactly $R$;
- (ii) conditional on the continuous filtration at $t_j$, the successor state has the same kernel as under $R$ conditional on $H_j$;
- (iii) on each open grid cell, $$
\begin{equation}\label{eq:piecewise-lsv}\tag{16} \frac{dS_t}{S_t}=\bm\sigma(t,S_t,Y_t)^\top d\bm W_t \end{equation}
$$ for a Borel time-inhomogeneous coefficient; and
- (iv) every grid-aligned window is exact: $$
\begin{equation}\label{eq:finite-realized-window}\tag{17} \E\!\left[\int_{t_i}^{t_k}\|\bm\sigma_t\|^2dt \Bigm|\cF_{t_i}\right] =A_{i,k} =-2\E\!\left[\log\frac{S_{t_k}}{S_{t_i}}\Bigm|\cF_{t_i}\right]. \end{equation}
$$

Consequently every finite feasible raw quote panel has an exact continuous history-state realization. If the panel contains finitely many raw payoffs, the finite-cubature reduction in Lemma A.3 may first be applied, yielding a realization with a finite nonrecombining skeleton. No relative-interior, density, or uniform ellipticity condition is required.

*Proof.* Disintegrate $R$ into Borel successor kernels $K_j(h_j,dz_{j+1})$. Parameterized disintegration separates each kernel into its successor-spot marginal $\kappa_j(h_j,ds)$ and a conditional kernel on the fiber having that spot. The standard randomization lemma for probability kernels on standard Borel spaces realizes the fiber kernel by a Borel map of one independent uniform variable.

For each grid cell take two fresh Brownian coordinates. Normalize their increments over the cell to processes $W^{j,1},W^{j,2}$ running from $0$ to a standard normal endpoint. Interpolate the successor-spot marginal with Lemma 2.8 using $W^{j,1}$. At $t_{j+1}$ use $\Phi_N(W^{j,2}_1)$ to draw the remaining successor mark conditionally on the realized spot. Freeze both coordinates after the cell. Recursing over $j$ produces a continuous Gaussian memory factor of dimension at most $2J$ and the exact skeleton law $R$.

The active endpoint randomizers are independent of the past at $t_j$; therefore

$$
\begin{equation}\label{eq:no-early-revelation}\tag{18} \Law(Z_{j+1}\mid\cF_{t_j})=K_j(H_j,\cdot). \end{equation}
$$

This is the no-early-revelation step. The Bass conditional-expectation identity inside each cell and the conditional mean row across boundaries make $S$ a global true martingale. On cell $j$, with $r=(t-t_j)/\delta_j$, Itô’s formula gives

$$
\frac{dS_t}{S_t} =\frac1{\sqrt{\delta_j}} \frac{u_{j,x}(r,H_j,W_r^{j,1})} {u_j(r,H_j,W_r^{j,1})}\,dW_t^{j,1},
$$

which is (16) after reading $H_j$ from the frozen memory.

By (15), the conditional integrated variance on cell $j$ is ∎

$$
-2\E[X_{j+1}-X_j\mid\cF_{t_j}]=\delta_jB_{j,j}.
$$

Condition back to $t_i$, use the common tower for $B_{i,h}$, and sum over $h=i,\ldots,k-1$. Lemma 2.1 then yields (17). All grid payoffs retain their prices because the complete skeleton law is unchanged.

**Remark 2.10 (Exact level of realization).** The coefficient may switch at grid boundaries and degenerate near an atomic endpoint. Time plus the Gaussian memory is a finite-dimensional Markov state, but no small factor count or smooth uniformly elliptic coefficient follows. The conclusion is exact continuous weak realization, not a classical regular diffusion theorem.

### 2.3 Early revelation: a payoff-typed diagnostic

The construction above consumes its randomizers at the left edge of each cell, and (18) records the consequence. Because a premature mark can preserve some declared rows while destroying others, we state the failure mode in the form in which an implementation can be tested against it.

Fix a grid date $j$ and a mark $U$ on the same space, and enlarge the history to $\cG_j=\cF_j\vee\sigma(U)$. For an integrable payoff $\psi$ define the *revelation defect*

$$
\begin{equation}\label{eq:revelation-defect}\tag{19} D_j(\psi):=\E[\psi\mid\cG_j]-\E[\psi\mid\cF_j]. \end{equation}
$$

**Proposition 2.11 (Revelation defect).**  Let $\psi$ be integrable. Then

- (i) $\E[D_j(\psi)\mid\cF_j]=0$;
- (ii) if in addition $\psi\in L^2$, then $$
\begin{equation}\label{eq:revelation-variance}\tag{20} \E\bigl[D_j(\psi)^2\mid\cF_j\bigr] =\operatorname{Var}\bigl(\E[\psi\mid\cG_j]\bigm|\cF_j\bigr), \qquad\|D_j(\psi)\|_2^2 =\E\Bigl[\operatorname{Var}\bigl(\E[\psi\mid\cG_j]\bigm|\cF_j\bigr)\Bigr]; \end{equation}
$$
- (iii) $D_j(\psi)=0$ almost surely if and only if $\E[\psi\mid\cF_j,U]=\E[\psi\mid\cF_j]$, that is, conditional mean-independence of $U$ and $\psi$ given $\cF_j$.

*Proof.* Since $\cF_j\subseteq\cG_j$, the tower property gives $\E[\E[\psi\mid\cG_j]\mid\cF_j]=\E[\psi\mid\cF_j]$, which is (i). Given (i), $D_j(\psi)$ is conditionally centred, so its conditional second moment is the stated conditional variance, and the unconditional identity follows by taking expectations; the $L^2$ hypothesis is used only here, and is not implied by the standing $q>1$ envelope of Section 2. Part (iii) is the definition of the conditional expectations in (19) being equal.∎

Part (iii) is stated for one payoff, and that is exactly the point: mean independence is a property *of a row*, not of the mark.

**Example 2.12 (Spot survives, the budget does not).**  At a fixing date let $S=1$ and let a fair coin $U$ select between the two mean-one terminal laws

$$
\tfrac12\delta_{1/2}+\tfrac12\delta_{3/2}, \qquad\tfrac12\delta_{1/4}+\tfrac12\delta_{7/4}.
$$

Both have mean one, so $D_j(S_{j+1})=0$: revealing $U$ leaves the martingale row intact. Their log budgets are $-2\E\log S=\log(4/3)\approx0.2877$ and $\log(16/7)\approx0.8267$. With the coin hidden the squared VIX readout is the average $0.5572$; with the coin revealed it is $0.2877$ or $0.8267$ according to the branch, a ratio of $2.87$. Conditional mean-independence for spot therefore does not protect the diagonal row, and a model that still calibrates every SPX quote can carry the wrong VIX.

**Definition 2.13 (Panel-safe revelation).**  Let $\Phi$ be the declared quote vector and let $\mathcal R_j$ be the structural residuals conditioned at date $j$. The mark $U$ is panel-safe at $j$ if $D_j(\psi)=0$ for every $\psi\in\Phi\cup\mathcal R_j$.

Panel-safety is sufficient for the enlarged model to satisfy every declared structural row at date $j$ and to calibrate every unconditional quote, and it is necessary and sufficient for it to preserve every date-$j$ *conditional* value in $\Phi\cup\mathcal R_j$. It is not necessary for unconditional calibration alone: two branch laws with the same spot mean and the same log mean preserve the spot row, the diagonal row, and every unconditional price while changing a conditional call value, and an enlargement never alters an unconditional expectation. The conditional target is the relevant one here, because a VIX fixing is itself a conditional value. Panel-safety is strictly weaker than requiring $D_j(\psi)=0$ for all bounded $\psi$, which is conditional independence of $U$ from the future given $\cF_j$ — the immersion or $\mathcal H$-hypothesis of the enlargement-of-filtrations literature — and strictly stronger than mean-independence for the spot row alone, by Example 2.12. As a diagnostic it is finite and checkable: for each declared row separately, compare the two conditional expectations in (19), using (20) to report a defect magnitude when the row is square-integrable. Testing the spot row alone is not sufficient.

**Remark 2.14 (What the construction supplies).**  Theorem 2.9 does not merely achieve panel-safety. Its cell randomizers are independent of $\cF_{t_j}$, so (18) preserves the entire successor kernel, hence every row simultaneously and every undeclared payoff as well. The diagnostic above is therefore addressed to constructions that introduce auxiliary marks by other means, not to the one built here. The same sensitivity to what is revealed when reappears at the continuum level: Theorem 3.11 is stated in the enhanced topology for exactly this reason, and Remark 3.12 gives the continuous-time counterpart of Example 2.12.

## 3 The complete rolling market

We now pass to complete surfaces. The state is enlarged so that the quadratic variation and the entire budget field are path coordinates, making filtration information part of the path itself; the finite alternative then lifts through compactness of one declared class, and every structural continuum law is already a stochastic-clock model.

Fix a horizon $H<\infty$ and one tenor $\Delta\in(0,H)$. SPX calls are declared on $[0,H]\times[0,\infty)$ and VIX calls on $[0,H-\Delta]\times[0,\infty)$. The quoted surfaces $C^S,C^V$ are assumed continuous in maturity and strike, and every strike section is a valid call transform of an integrable nonnegative law. Equality on a dense set of strikes determines the corresponding marginal law because call transforms are $1$-Lipschitz and their right derivatives recover survival functions.

Put

$$
\Theta=\{(t,u,v):0\le t\le u\le v\le H\}.
$$

The canonical space $\Omega$ is the closed subset of

$$
C([0,H])\times C_{\uparrow,0}([0,H])\times C(\Theta)
$$

with coordinates $(X,A,B)$ satisfying a deterministic root $X_0=x_0$, $B(0;u,v)=b_0(u,v)$ and, pathwise,

$$
\begin{align} B(t;u,v)&\ge0,& B(t;u,u)&=0,\label{eq:Bpositive}\tag{21} \\
B(t;u,w)&=B(t;u,v)+B(t;v,w),&&t\le u\le v\le w.\label{eq:Badditive}\tag{22} \end{align}
$$

The root field is also continuous, nonnegative, and additive. Let $S=e^X$ and $M_t=X_t-x_0+A_t/2$. The raw filtration records $X,A$ and only budget valuations observed no later than the current time.

**Assumption 3.1 (One common coercive class).**  Choose $p>2$. There is a lower-semicontinuous $\Psi:\Omega\to[1,\infty]$ with compact sublevel sets such that

$$
\begin{equation}\label{eq:Psi-dominates}\tag{23} 1+\|X\|_\infty^p+A_H^p+\|B\|_\infty^p+\|e^X\|_\infty^p \le C_\Psi\Psi. \end{equation}
$$

Moreover $\Psi$ is the increasing supremum of bounded continuous cylinder functions on one countable dense coordinate set. Fix $M_\Psi>\inf_{\{\Psi<\infty\}}\Psi$ and set

$$
\cP_\Psi=\{R\in\cP(\Omega):\E_R\Psi\le M_\Psi\}.
$$

Every finite restriction in the continuum theorem uses this same $(\Psi,M_\Psi)$.

The cylinder presentation enters only through monotone convergence: $\E_R\Psi=\sup_n\E_R\psi_n$ along the defining supremum, so the moment cap is checkable on finite-dimensional marginals. Concrete sufficient choices append coercive moduli of continuity for $X,A,B$ to the left side of (23). The theorem does not infer such moduli from option prices.

For $t\le H$, let $\cH_t$ be the bounded continuous functions of finitely many canonical coordinates observed by $t$.

**Definition 3.2 (Common-budget continuum certificate).**  A law $R\in\cP_\Psi$ is structural if, for every admissible time tuple and conditioning multiplier $\phi$, it satisfies the following identities, with all expectations taken under $R$:

$$
\begin{align} \E[\phi(M_t-M_s)]&=0, &&s<t,\quad\phi\in\cH_s,\label{eq:Mrow}\tag{24} \\
\E[\phi\{(M_t^2-A_t)-(M_s^2-A_s)\}]&=0, &&s<t,\quad\phi\in\cH_s,\label{eq:Arow}\tag{25} \\
\E[\phi\{B(t;u,v)-B(s;u,v)\}]&=0, &&s<t\le u\le v,\quad\phi\in\cH_s,\label{eq:Brow}\tag{26} \\
\E[\phi\{B(u;u,v)+2(X_v-X_u)\}]&=0, &&u\le v,\quad\phi\in\cH_u.\label{eq:diagrow}\tag{27} \end{align}
$$

Write $\cA_\Psi$ for the set of structural laws. It is calibrated if, in addition,

$$
\begin{align} \E[(S_t-K)^+]&=C^S(t,K),\label{eq:SPXquote}\tag{28} \\
\E[(V_t^\Delta-K)^+]&=C^V(t,K),\qquad(V_t^\Delta)^2=\Delta^{-1}B(t;t,t+\Delta),\label{eq:VIXquote}\tag{29} \end{align}
$$

for every declared $(t,K)$.

**Lemma 3.3 (Identification of bracket and budget).**  For every $R\in\cA_\Psi$, in the usual augmented canonical filtration:

- (i) $M$ is a continuous square-integrable martingale and $A=\langle M\rangle$;
- (ii) $S=S_0\mathcal E(M)=e^X$ is a positive continuous true martingale; and
- (iii) for $t\le u\le v$, $$
\begin{equation}\label{eq:Bidentified}\tag{30} B(t;u,v)=\E[A_v-A_u\mid\cF_t] =-2\E[X_v-X_u\mid\cF_t]. \end{equation}
$$ In particular (29) is the model’s own rolling VIX.

*Proof.* It is enough to test (24)–(27) at a countable dense set of times against a countable separating algebra of history cylinders. A signed-measure and monotone-class argument, as in Lemma 2.1, upgrades those rows to conditional identities. Path continuity and the $p>2$ envelope then extend them to all real times. Martingality transfers to the usual augmentation by the downward martingale theorem along $\cF_{s'}^0\downarrow\cF_{s+}^0$, path continuity, and the same $L^1$ domination; for the diagonal row, whose window start moves with the conditioning date, the transfer also uses $\E|\xi_u-\xi_{u'}|\to0$ for $\xi_u=B(u;u,v)+2(X_v-X_u)$.

Rows (24) and (25) say that $M$ and $M^2-A$ are true martingales. Since $A$ is continuous increasing from zero, uniqueness in the quadratic-variation characterization gives $A=\langle M\rangle$. From the definition of $M$, $X_t=x_0+M_t-A_t/2$, so Itô’s formula gives $S=S_0\mathcal E(M)$. Localize this positive local martingale. The envelope controls $\sup_tS_t$ in $L^p$, so dominated convergence removes the localization and $S$ is true.

At time $u$, (27) and martingality of $M$ give ∎

$$
B(u;u,v)=-2\E[X_v-X_u\mid\cF_u] =\E[A_v-A_u\mid\cF_u].
$$

Condition back to $t$ and use (26); this proves (30).

**Corollary 3.4 (Fast necessary market identities).**  Complete SPX smiles force

$$
\begin{equation}\label{eq:root-forced}\tag{31} b_0(u,v)=-2\left(\int\log s\,\mu_v^S(ds) -\int\log s\,\mu_u^S(ds)\right), \end{equation}
$$

and

$$
\begin{equation}\label{eq:vix-second-moment}\tag{32} \int z^2\,\mu_t^V(dz) =-\frac2\Delta\left(\int\log s\,\mu_{t+\Delta}^S(ds) -\int\log s\,\mu_t^S(ds)\right). \end{equation}
$$

The right side of (31) must be finite, continuous, nonnegative, and additive. These conditions are necessary, not sufficient: they do not impose a common conditional tower.

*Proof.* Take expectations in (30) at time zero, then use the SPX and VIX marginal laws.∎

**Proposition 3.5 (Overlap inequality).**  Let $R\in\cA_\Psi$ and let $t<t'<t+\Delta$, with $t'\le H-\Delta$, be two fixings whose windows overlap. Then, almost surely,

$$
\begin{equation}\label{eq:overlap-pathwise}\tag{33} B(t;t,t')\;\ge\;\Delta\,(V_t^\Delta)^2-\Delta\,\E\bigl[(V_{t'}^\Delta)^2\bigm|\cF_t\bigr], \end{equation}
$$

and consequently, for every $K\ge0$, with $\ell(u):=-2\,\E[X_u-x_0]$,

$$
\begin{equation}\label{eq:overlap}\tag{34} \E\bigl[((V_t^\Delta)^2-K)^+\bigr]-\E\bigl[((V_{t'}^\Delta)^2-K)^+\bigr] \;\le\;\frac{\ell(t')-\ell(t)}{\Delta}. \end{equation}
$$

If, in addition, $R$ is calibrated, every term in (34) is determined by complete all-strike call strips: $\E[((V_u^\Delta)^2-K)^+]=2\sqrt K\,C^V(u,\sqrt K)+2\int_{\sqrt K}^\infty C^V(u,k)\,dk$, and $\ell(t')-\ell(t)$ is the SPX log-strip increment over the non-overlapping part $[t,t']$. This is an exact all-strike identity; evaluation from a finite grid of exchange-listed strikes additionally requires declared interpolation and tail bounds. At $K=0$ the inequality follows from the two mean identities of Corollary 3.4, with slack $(\ell(t'+\Delta)-\ell(t+\Delta))/\Delta\ge0$, the expected budget of the trailing cell $[t+\Delta,t'+\Delta]$ dropped in the proof; for $K>0$ it is not implied by the mean rows (Example 4.2); and for $t'\ge t+\Delta$ the windows share no budget coordinate and the argument yields nothing beyond the mean rows.

*Proof.* By additivity, $\Delta(V_t^\Delta)^2=B(t;t,t+\Delta)=B(t;t,t')+B(t;t',t+\Delta)$. By Lemma 3.3, additivity, and (21), $\Delta\,\E[(V_{t'}^\Delta)^2\mid\cF_t]=B(t;t',t'+\Delta) =B(t;t',t+\Delta)+B(t;t+\Delta,t'+\Delta)\ge B(t;t',t+\Delta)$. Subtracting gives (33). For $K\ge0$, ∎

$$
(V_t^\Delta)^2-K \le\frac{B(t;t,t')}{\Delta}+\E[(V_{t'}^\Delta)^2\mid\cF_t]-K \le\frac{B(t;t,t')}{\Delta}+\E\bigl[((V_{t'}^\Delta)^2-K)^+\bigm|\cF_t\bigr],
$$

and the right side is nonnegative, so it dominates $((V_t^\Delta)^2-K)^+$. Take expectations and use $\E B(t;t,t')=\E[A_{t'}-A_t]=-2\E[X_{t'}-X_t]$ from Lemma 3.3. The strip formula is Carr–Madan replication of the convex payoff $f(v)=(v^2-K)^+$, with $f(0)=f'(0)=0$, whose second-derivative measure is $2\,dk$ on $(\sqrt K,\infty)$ plus the atom $2\sqrt K\,\delta_{\sqrt K}$ carried by the jump of $f'$.

### 3.1 Finite determination and pathwise obstruction

Choose a countable dense set $D\subset[0,H]$ containing $0,H$ and closed under admissible $\Delta$-shifts, and rational nonnegative strikes. Continuity of $(X,A,B)$ and uniform integrability from Assumption 3.1 imply that structural rows and quote rows on this countable panel determine all real times and strikes. Denote its quote indices by $\Theta_\circ$ and the corresponding payoffs and prices by $(\Phi_\theta,c_\theta)$.

Enumerate a countable determining collection of homogeneous structural residuals from (24)–(27) as $\rho_1,\rho_2,\ldots$, and let $\mathscr W$ be their finite linear span. The envelope gives, for every $W\in\mathscr W$ and every finite quote combination $G$,

$$
\begin{equation}\label{eq:sublinear-growth}\tag{35} |W|+|G|\le C(1+\Psi^\gamma) \quad\text{on }\Omega_\Psi:=\{\Psi<\infty\} \end{equation}
$$

with $\gamma=2/p<1$; the quadratic row $M^2-A$ is where $p>2$ binds.

Here a stochastic-clock model means a positive continuous true martingale $S$ whose stochastic logarithm has continuous integrable bracket $A$, with the forward budget given by conditional bracket increments. The sharper calendar-time distinction is made in Section 3.2.

**Theorem 3.6 (Complete-market common-budget alternative).**  If $\cA_\Psi\ne\varnothing$, the following are equivalent:

- (a) there is a calibrated continuum certificate;
- (b) for every finite $F\subset\Theta_\circ$, there is $R_F\in\cA_\Psi$ satisfying $\E_{R_F}\Phi_\theta=c_\theta$ for all $\theta\in F$;
- (c) there is no finite set $F=\{\theta_1,\ldots,\theta_m\}$ and $\lambda\in\R^m$ such that $$
\begin{equation}\label{eq:finite-support-witness}\tag{36} \sup_{R\in\cA_\Psi}\E_R\!\left[\sum_{i=1}^m \lambda_i\Phi_{\theta_i}\right] <\sum_{i=1}^m\lambda_i c_{\theta_i}; \end{equation}
$$
- (d) there is a positive continuous stochastic-clock model in the class whose conditional future bracket has the jointly continuous version $B$ and whose SPX/VIX call surfaces are $(C^S,C^V)$.

In this nonempty-class case, if these equivalent conditions fail, the finite witness can be strengthened to a pathwise infeasibility certificate: for some such finite $F$, some $W\in\mathscr W$, $a\in\R$, and $\eta\ge0$,

$$
\begin{align} a+\eta\Psi+W&\ge\sum_{i=1}^m\lambda_i\Phi_{\theta_i} &&\text{on }\Omega_\Psi,\label{eq:continuum-pathwise}\tag{37} \\
a+\eta M_\Psi&<\sum_{i=1}^m\lambda_i c_{\theta_i}. \label{eq:continuum-gap}\tag{38} \end{align}
$$

If instead $\cA_\Psi=\varnothing$, there are $W\in\mathscr W$, $a\in\R$, and $\eta\ge0$ such that

$$
a+\eta\Psi+W\ge0\quad\text{on }\Omega_\Psi, \qquad a+\eta M_\Psi<0.
$$

Thus structural infeasibility is already witnessed by finitely many structural rows together with the class bound, without using quote rows.

*Proof.* We first record compactness. Markov’s inequality and compact sublevels of $\Psi$ make $\cP_\Psi$ tight; Portmanteau makes it closed. Hence it is compact. Every structural or quote payoff is continuous and is dominated by $C(1+\Psi^\gamma)$ for some $\gamma<1$. The bound

$$
\sup_{R\in\cP_\Psi}\E_R[\Psi^\gamma\1_{\{\Psi^\gamma>K\}}] \le M_\Psi K^{1-1/\gamma}
$$

gives uniform integrability, so its expectation is weakly continuous. Therefore $\cA_\Psi$ is compact and convex.

Assume first $\cA_\Psi\ne\varnothing$. For each $\theta\in\Theta_\circ$, let

$$
F_\theta=\{R\in\cA_\Psi:\E_R\Phi_\theta=c_\theta\}.
$$

These sets are closed in the compact space $\cA_\Psi$. Condition (b) is their finite-intersection property, so it is equivalent to a nonempty total intersection. A law in the intersection satisfies every dense quote row. Path continuity, surface continuity, the $1$-Lipschitz property in strike, and the same uniform-integrability estimate extend calibration to every time and strike. This proves (a)$\Leftrightarrow$(b).

If the total intersection is empty, compactness produces a finite empty subintersection. The finite quote image of $\cA_\Psi$ is compact and convex, so strict finite-dimensional separation gives (36). Conversely, a calibrated law contradicts that inequality. Thus (a)$\Leftrightarrow$(c).

For the pathwise strengthening, put $G=\sum_i\lambda_i\Phi_{\theta_i}$. Let

$$
\cA_{\Psi,n}=\{R\in\cP_\Psi:\E_R\rho_j=0,\ j\le n\}.
$$

These compact sets decrease to $\cA_\Psi$, so their support values at $G$ decrease to $\sup_{\cA_\Psi}\E G$: maximizers accumulate, along a weakly convergent subsequence, in $\bigcap_n\cA_{\Psi,n}=\cA_\Psi$, since each $\E\rho_j$ is weakly continuous and the intersection identity is the determination argument of Lemma 3.3. Finite-dimensional Lagrange separation gives

$$
\max_{R\in\cA_{\Psi,n}}\E_RG =\inf_{\vartheta\in\R^n} \max_{R\in\cP_\Psi}\E_R \left(G+\sum_{j=1}^n\vartheta_j\rho_j\right).
$$

For each finite structural combination, the concave moment-value argument used in Theorem 2.2 gives the exact one-moment dual with $\Psi$. Hence

$$
\begin{equation}\label{eq:continuum-no-gap}\tag{39} \sup_{R\in\cA_\Psi}\E_RG =\inf_{\substack{a\in\R,\eta\ge0,W\in\mathscr W \\
a+\eta\Psi+W\ge G\text{ on }\Omega_\Psi}} (a+\eta M_\Psi). \end{equation}
$$

The outer infimum need not be attained. The strict separation gap allows a feasible tuple within half the gap, proving (37)–(38).

Finally, Lemma 3.3 says that the calibrated law in (a) is itself the clock model in (d). Conversely, project an admissible clock model onto $(X,A,B)$. Its stochastic logarithm and squared-minus-bracket are martingales; the conditional budget has the tower and diagonal log identity; conditioning from any larger original filtration down to the canonical filtration preserves these rows. The jointly continuous version of the conditional bracket satisfies the pathwise nonnegativity and additivity constraints almost surely, per countable tuple and then by continuity. Membership in $\cP_\Psi$ and calibration then give the calibrated law in (a). This proves (a)$\Leftrightarrow$(d).

It remains to consider $\cA_\Psi=\varnothing$. The closed sets ∎

$$
D_j:=\{R\in\cP_\Psi:\E_R\rho_j=0\}
$$

have empty total intersection. Compactness of $\cP_\Psi$ therefore gives an $n$ for which $\bigcap_{j=1}^nD_j=\varnothing$. The compact convex residual image

$$
K_n:=\{(\E_R\rho_1,\ldots,\E_R\rho_n):R\in\cP_\Psi\}
$$

does not contain zero. Strict separation yields $\vartheta\in\R^n$ and $\epsilon>0$ such that, with $W=\sum_{j=1}^n\vartheta_j\rho_j$, $\E_RW\ge\epsilon$ for every $R\in\cP_\Psi$. Hence $\max_{R\in\cP_\Psi}\E_R(-W)\le-\epsilon$. Applying the exact one-moment dual with $\Psi$ to $-W$ and choosing a tuple within half this gap gives $a\in\R$ and $\eta\ge0$ such that $a+\eta\Psi+W\ge0$ on $\Omega_\Psi$ and $a+\eta M_\Psi<0$. This proves the empty-class branch.

**Remark 3.7 (Why the common class cannot vary).** Condition (b) is not the assertion that each finite panel can be fitted by some increasingly irregular model. Every $R_F$ already lies in the same compact $\cA_\Psi$. If one starts instead from genuinely finite arrays, a valid projective route must impose uniform integrability, full-history rows, and common tight moduli for spot, bracket, and the three-parameter budget field. Example 4.4 shows that omitting this uniformity invalidates the implication.

The finiteness in (b) and (c) is on the quote side only. Each $R_F$ ranges over $\cA_\Psi$, which is cut out by the complete countable family of structural rows together with the class, so (a)$\Leftrightarrow$(b) is a compact finite-intersection theorem and not a finite-dimensional market test. What is genuinely finite is the witness in (c) and its pathwise form: finitely many quotes, finitely many structural rows, and the class tail bound. Finite-dimensional tests are the grid programs of Theorem 2.2 and Section 5.2.

### 3.2 The clock/calendar-time boundary

**Definition 3.8 (Clock and calendar-time realizations).** A clock realization is a positive continuous true martingale $S$ whose stochastic logarithm $M=\int dS/S$ has continuous integrable quadratic variation $A=\langle M\rangle$. Its budget is $\E[A_v-A_u\mid\cF_t]$. It has a calendar-time Brownian representation if $A_t=\int_0^ta_s\,ds$ for a nonnegative predictable process $a$. This is the weakest sense in which the model is a local–stochastic volatility model: $a$ may be any predictable functional of the stopped history, and no Markov or finite-dimensional structure is implied.

**Theorem 3.9 (Exact threshold for a calendar-time Brownian representation).**  Every structural continuum law $R\in\cA_\Psi$ is an exact clock realization. It has a weak calendar-time representation

$$
\begin{equation}\label{eq:dt-lsv}\tag{40} \frac{dS_t}{S_t}=\sqrt{a_t}\,dW_t,\qquad A_t=\int_0^ta_sds, \end{equation}
$$

on a Brownian extension if and only if $A\ll dt$ almost surely. In that case $a_t$ has a predictable Borel version as a functional of time and the complete stopped canonical history. Smooth coefficients, uniform ellipticity, and a fixed finite factor dimension require additional assumptions.

*Proof.* The clock statement is Lemma 3.3. Suppose $A_t=\int_0^ta_sds$. Enlarge the space by an independent Brownian motion $W'$ and set

$$
W_t=\int_0^t\1_{\{a_s>0\}}a_s^{-1/2}\,dM_s +\int_0^t\1_{\{a_s=0\}}\,dW'_s.
$$

The summands are orthogonal continuous local martingales and $\langle W\rangle_t=t$; Lévy’s characterization makes $W$ Brownian. The integral of $dM$ over $\{a=0\}$ has zero bracket, hence vanishes, giving (40). Conversely, that SDE implies $A_t=\int_0^ta_sds$, so absolute continuity is necessary.

The completed raw canonical filtration is generated, modulo null sets, by the stopped histories of $X,A,B$. Concretely, $a_t=\limsup_{n\to\infty}n\,(A_t-A_{(t-1/n)^+})$ is raw-predictable and is a version of the density by Lebesgue differentiation; standard-Borel factorization makes it a Borel nonanticipative functional of the stopped history. This history is a Markov state in the tautological transition-kernel sense, but can be infinite-dimensional. Nothing in the argument supplies smoothness, nondegeneracy, or finite-dimensional sufficiency.∎

**Remark 3.10 (Identification inside the \(dt\) subclass).** A complementary gauge freedom holds inside the absolutely continuous subclass: in deterministic one-factor operational-time models the surface determines only the projected clock–variance product, not its factorization (Angstmann and Gebbie, 2026). The boundary above is orthogonal: it characterizes when a calendar-time density exists at all, and claims no identification.

### 3.3 Closedness and information-sensitive stability

Let $\mathcal Y$ be the product of the two continuous call-surface spaces, with locally uniform convergence in time and strike, and write

$$
\mathfrak C(R)=\left( (\E_R(S_t-K)^+)_{t,K}, (\E_R(V_t^\Delta-K)^+)_{t,K}\right).
$$

**Theorem 3.11 (Closed feasible surface set).**  The map $\mathfrak C:\cA_\Psi\to\mathcal Y$ is continuous. Its image is compact and convex. Hence locally uniform limits of feasible surface pairs are feasible, calibration fibers are compact with a closed graph, and $R_n\Rightarrow R$ in $\cA_\Psi$ implies weak convergence of the complete readout paths $(S,V^\Delta)$. This conclusion concerns the enhanced $(X,A,B)$ topology; ordinary weak convergence, and even equality in law, of the unaugmented $(S,A)$ paths does not by itself determine conditional VIX when the filtrations differ.

*Proof.* On each compact $\Psi$-sublevel the canonical coordinate family is bounded and equicontinuous. Moving-time SPX payoffs and ∎

$$
\sqrt{B(t;t,t+\Delta)/\Delta}
$$

therefore converge uniformly there. The sublinear envelope and uniform integrability remove the complement uniformly. This proves joint continuity of expectation in $(R,t,K)$, and compactness of the parameter sets gives locally uniform surface convergence. A continuous affine image of compact convex $\cA_\Psi$ is compact and convex; closed graph and compact fibers follow. Finally, the map

$$
(x,q,b)\longmapsto\left(e^x, \left(\sqrt{b(t;t,t+\Delta)/\Delta}\right)_{0\le t\le H-\Delta} \right)
$$

is continuous in the uniform topology, using $|\sqrt{x}-\sqrt y|^2\le|x-y|$. The continuous-mapping theorem proves the path statement.

**Remark 3.12 (Why the filtration coordinate cannot be dropped).**  Let a fair coin $H$ choose between two deterministic variance profiles $a_0,a_1$ that agree through $u+\eta$ but have different integrals over $(u+\eta,v]$, and set

$$
S_t=\exp\!\left(\int_0^t\sqrt{a_H(s)}\,dW_s -\frac12\int_0^t a_H(s)\,ds\right), \qquad A_t=\int_0^t a_H(s)\,ds.
$$

Give this same pair of paths either a filtration that reveals $H$ at $u$ or the natural filtration of $(S,A)$, which cannot yet distinguish the profiles. The full unaugmented $(S,A)$ path law is identical in the two models, but at $u$ the conditional increment $\E[A_v-A_u\mid\cF_u]$ is branch-specific in the first model and the mixture average in the second. Their conditional VIX laws therefore differ.

Theorem 3.11 avoids this loss by fixing the raw canonical filtration and carrying $B$, the relevant prediction coordinate, in the state, in the spirit of adapted topologies (Backhoff-Veraguas et al., 2020). The moment, uniform-integrability, and modulus controls encoded by Assumption 3.1 then make that enhanced state compact enough for closure. This is a qualitative closedness statement; quantitative marked-Bass, budget, and fixed-tenor VIX error rates are not asserted here.

## 4 Analytic examples and sharp boundaries

Each example isolates one hypothesis or boundary of the theory. Example 4.1 shows that pairwise consistency does not glue and exhibits the finite witness; Example 4.2 shows the same failure with complete vanilla smiles alone on genuinely overlapping windows, through the overlap inequality of Proposition 3.5; Remark 4.3 records why this is not a marginal-problem phenomenon; Example 4.4 shows that the common-class qualifier in Theorem 3.6(b) cannot be dropped; Examples 4.5 and 4.6 inhabit the two sides of the clock boundary; Example 4.7 computes a separator and shows it need not be a traded arbitrage; and Example 4.8 separates weak realization from positive-variance mixtures.

**Example 4.1 (A four-date panel: pairwise-consistent fixings, no common law).**  Take the grid $0<1<2<3<4$ with unit cells, tenor $\Delta=1$, spot frozen on $[0,1]$, and write $A_i=B(i;i,i+1)=(V_i^\Delta)^2$ for the fixings $i=1,2,3$. The panel consists of:

- (i) VIX calls $C^V(i,K)=\tfrac12(1-K)^+$ at all strikes $K\ge0$, $i=1,2,3$, so each $A_i$ is Bernoulli$(\tfrac12)$ on $\{0,1\}$;
- (ii) three pairwise dispersion claims with payoff $A_iA_j\wedge1$ and price zero, $i<j$: since $A_iA_j\ge0$ by (21), each claim forces $A_iA_j=0$ almost surely; and
- (iii) SPX calls at dates $1,2,4$ generated by $S_1\sim\delta_1$, $S_2\sim\tfrac12\delta_1+\tfrac12\operatorname{LN}(1)$, and $S_4\sim\tfrac12\operatorname{LN}(1)+\tfrac12\operatorname{LN}(2)$, where $\operatorname{LN}(v)$ is the unit-mean lognormal law with log-variance $v$.

Deleting any one claim in (ii) leaves an exactly calibratable market. Reveal a regime $R\sim\text{Bernoulli}(\tfrac12)$ on $[0,1]$ through the budget coordinates by a continuous $[0,1]$-valued martingale absorbed at $\{0,1\}$, independent of the spot driver; set $(A_1,A_2,A_3)=(R,1-R,R)$, $(R,1-R,1-R)$, or $(R,R,1-R)$ according to whether the deleted pair is $(1,3)$, $(2,3)$, or $(1,2)$; and on each cell run a unit lognormal edge if its budget is one, freezing the spot otherwise. Each model is a positive continuous martingale generating its own budgets and matching (i), (iii), and both retained claims. All three budgets are then $\cF_1$-measurable, which no quote or structural row forbids.

No law matches the whole panel: by (i) the events $\{A_i=1\}$ have probability $\tfrac12$ each, and by (ii) they are pairwise disjoint, so their probabilities would sum to $\tfrac32$. The witness is a finite portfolio. For $v\ge0$ put $q(v)=2(v-\tfrac12)^+-2(v-1)^+$, a call spread at strikes $\tfrac12$ and $1$; then $q(v)\le v^2\wedge1$ pathwise, since $v^2-(2v-1)=(v-1)^2$ on $[\tfrac12,1]$, and the quoted curve prices it at $2C^V(i,\tfrac12)-2C^V(i,1)=\tfrac12$. On the whole state space,

$$
\begin{equation}\label{eq:gluing-witness}\tag{41} \sum_i q(V_i)\;\le\;\sum_i\bigl(A_i\wedge1\bigr) \;\le\;1+\sum_{i<j}\bigl(A_iA_j\wedge1\bigr), \end{equation}
$$

where the second inequality is multilinear on $[0,1]^3$, hence holds by checking its vertices, and extends off the cube because $A_iA_j\wedge1\ge A_j\wedge1$ whenever $A_i\ge1$. The panel prices the left side at $\tfrac32$ and the right side at $1$: six calls, three pair claims, and cash $1$ realize the witness of Theorem 2.2 with $a=1$, $\eta=0$, $W=0$, and a strict gap of $\tfrac12$. The obstruction is joint: every smile is matched and each pairwise test passes in some model, yet the three pairwise tests cannot pass together. The three windows here are the abutting cells $[1,2]$, $[2,3]$, and $[3,4]$, and the contradiction uses the nonvanilla pairwise dispersion claims in (ii). Accordingly, the example does not establish failure of joint calibration for a panel consisting only of listed vanilla weekly SPX and VIX options.

**Example 4.2 (Overlapping vanilla windows with no common law).**  Take the grid $0<1<2<3<4$ with unit cells and tenor $\Delta=2$, so that the window $[1,3]$ of the fixing at $1$ and the window $[2,4]$ of the fixing at $2$ share the cell $[2,3]$. Quote only complete vanilla smiles: SPX smiles $S_1=S_2\sim\delta_1$ and $S_3=S_4\sim\tfrac12\operatorname{LN}(\Delta/2)+\tfrac12\operatorname{LN}(3\Delta/2)$, and VIX smiles $(V_1^\Delta)^2\sim\tfrac12\delta_{1/2}+\tfrac12\delta_{3/2}$ and $(V_2^\Delta)^2\sim\delta_1$.

Each sub-panel with one VIX smile deleted is calibratable. Without the fixing at $2$: reveal $(V_1^\Delta)^2$ through the budget coordinates on $[0,1]$ as in Example 4.1, freeze the spot to date $2$, run a lognormal edge of log-variance $\Delta(V_1^\Delta)^2$ on $[2,3]$, and freeze again. Without the fixing at $1$: freeze the spot to date $2$, reveal the mixing variable $R$ of the cell $[2,3]$ after the date-$2$ fixing—for some fixed $0<\varepsilon<\tfrac12$, continuously through the budget coordinates on $(2,2+\varepsilon]$ by the absorbed martingale of Example 4.1 started at $\tfrac12$, with bracket (variance) rate $dA_s/ds=2$ meanwhile and $dA_s/ds=(1+2R-2\varepsilon)/(1-\varepsilon)$ on $[2+\varepsilon,3]$, so that $B(2;2,3)=\Delta$ and $B(2;3,4)=0$—and freeze on $[3,4]$; an instantaneous draw at $2^{+}$ would be $\cF_2$-measurable under the usual augmentation and would break continuity of $B$ in valuation time. Both models satisfy every row and every retained smile. For the full panel the mean rows of Corollary 3.4 hold with $\ell=(0,0,\Delta,\Delta)$, and each window is feasible as a one-window problem.

No law matches the full panel. The frozen spot forces $B(1;1,2)=0$, and (33) at $t=1$, $t'=2$ reads $0\ge\Delta\bigl((V_1^\Delta)^2-1\bigr)$, which fails on $\{(V_1^\Delta)^2=\tfrac32\}$; equivalently (34) at $K=1$ reads $\tfrac14\le0$. The obstruction is conditional: the spot is frozen on $[1,2]$ and, since $S_3\overset{d}{=}S_4$ and $S$ is a martingale, on $[3,4]$ as well, so the tower gives $(V_1^\Delta)^2=\E[(V_2^\Delta)^2\mid\cF_1]=1$ almost surely—a convex-order violation between the two squared-VIX laws, invisible to the mean rows and to each window separately. The frozen spot is a normalization: with both VIX smiles held fixed, (34) at $K=1$ shows that any SPX smiles with $\ell(2)-\ell(1)<\Delta/4$ produce the same contradiction, and there the positivity of $B(1;3,4)$ carries the argument.

**Remark 4.3 (Why ordinary marginal combinatorics does not decide the problem).**  It is worth recording why the classical marginal problem does not decide this question. Write the quoted intervals as $I_i=[\ell_i,r_i]$ and order them by nondecreasing left endpoint. For $i>1$, if $b_i:=\max_{j<i}r_j$, then, whenever the intersection is nonempty,

$$
I_i\cap\bigcup_{j<i}I_j =[\ell_i,\min(r_i,b_i)] =I_i\cap I_{j^*}, \qquad j^*\in\arg\max_{j<i}r_j.
$$

Thus the interval hypergraph has the running intersection property and is acyclic, so Vorob’ev’s theorem (Vorob’ev, 1962) would grant extension were the data joint laws of the cell budgets. The cited theorem is stated for finite coordinate spaces; for real-valued budgets the positive direction follows by disintegrating each hyperedge law along its running-intersection separator and adjoining the new coordinates through the resulting regular conditional kernels in running-intersection order, which preserves every earlier marginal. They are not: a VIX quote fixes the law of their sum, a pushforward that neither determines the joint law nor restricts canonically to an intersection, so neither direction of the dichotomy transfers. Moreover, admissibility here includes the martingale tower (26), positivity, the diagonal log-budget identity, and any joint structural or quoted rows declared by the panel; these constraints have no counterpart in the ordinary marginal-extension theorem. Failure of that theorem to transfer is therefore not a claim that the obstruction is purely conditional. Example 4.1 shows both: its windows abut without overlapping, and its bite comes from joint dispersion claims that are not vanilla; Example 4.2 shows that the conditional route bites with complete vanilla smiles alone once the windows overlap. Whether some Vorob’ev-type dichotomy holds for sum-functional data under a martingale tower, and what the right acyclicity notion for such data would be, we leave open.

**Example 4.4 (Finite feasibility with a changing class).**  Fix $t_0\in(0,H)$ and $a\in(0,1)$. Prescribe SPX marginals

$$
\mu_t=\delta_1\quad(t<t_0),\qquad\mu_t=\tfrac12\delta_{1-a}+\tfrac12\delta_{1+a}\quad(t\ge t_0).
$$

Every finite ordered subpanel has a positive continuous martingale realization: keep spot at one until the last declared date before $t_0$, run a time-scaled Bass martingale to the two-point law before the first declared date after $t_0$, and keep it constant thereafter. But no continuous process has the entire marginal family. Continuity and $t\uparrow t_0$ force $S_{t_0}=1$ almost surely, contradicting the nondegenerate prescribed law. The finite realizations compress variation into shrinking intervals and cannot satisfy a common compact modulus. Thus “all finite panels fit” is false without the same-class qualifier in Theorem 3.6.

**Example 4.5 (An explicit compatible rolling market with a stochastic clock).**  Fix $\kappa,\theta,v_0>0$ and $\sigma>0$ with $\kappa^2\ge6\sigma^2$, independent Brownian motions $W,Z$, and the CIR variance $dv_t=\kappa(\theta-v_t)\,dt+\sigma\sqrt{v_t}\,dZ_t$. Let $A_t=\int_0^tv_s\,ds$, $X_t=x_0+\int_0^t\sqrt{v_s}\,dW_s-\tfrac12A_t$, $S=e^X$, and

$$
B(t;u,v)=\theta(v-u) +\frac{v_t-\theta}{\kappa} \bigl(e^{-\kappa(u-t)}-e^{-\kappa(v-t)}\bigr).
$$

Then $B(t;u,v)=\int_u^v\E[v_s\mid\cF_t]\,ds$: it is jointly continuous, nonnegative (each $\E[v_s\mid\cF_t]$ is a convex combination of $v_t\ge0$ and $\theta$), additive, and a martingale in the valuation time because the mean flow $\E[v_t\mid\cF_s]=\theta+(v_s-\theta)e^{-\kappa(t-s)}$ composes; Fubini gives $B(t;u,v)=\E[A_v-A_u\mid\cF_t]$, so (26) holds. Conditionally on the variance path $S$ is a Gaussian exponential, hence a positive continuous true martingale, and (24)–(27) follow. The rolling VIX is affine in the variance state,

$$
(V_t^\Delta)^2=\theta(1-\varphi)+\varphi\,v_t, \qquad\varphi=\frac{1-e^{-\kappa\Delta}}{\kappa\Delta}\in(0,1),
$$

a nondegenerate noncentral-$\chi^2$ smile at every fixing $t>0$, while SPX smiles are the corresponding integrated-variance lognormal mixtures.

For the class take $p=3$ and

$$
\Psi=1+\|X\|_\infty^3+A_H^3+\|B\|_\infty^3+\|e^X\|_\infty^3 +[X]_{1/8}^3+[A]_{1/8}^3+[B]_{1/8}^3,
$$

with $1/8$-Hölder seminorms taken over rational coordinates, so $\Psi$ is an increasing supremum of bounded continuous cylinder functions (finite rational maxima truncated at level $n$) and its sublevels are compact by Arzelà–Ascoli. Every polynomial coordinate has finite expectation for all parameters, by Burkholder–Davis–Gundy bounds and the moment form of the Kolmogorov–Chentsov criterion, reducing to polynomial moments of $\sup_tv_t$; in particular $\|X\|_\infty^3\le C(|x_0|^3+\sup_t|M_t|^3+A_H^3)$ is covered. The exponential coordinate is controlled by conditioning on the variance path: $\E S_H^3=e^{3x_0}\,\E e^{3A_H}$, which is finite for every horizon precisely when $3\le\kappa^2/(2\sigma^2)$, the Riccati explosion threshold for integrated CIR exponential moments (Andersen and Piterbarg, 2007); Doob’s $L^3$ maximal inequality applied to $S$ then bounds $\|e^X\|_\infty^3$. Hence the law lies in $\cA_\Psi$ for any $M_\Psi\ge\E\Psi$: the structural class contains an explicit market with genuinely stochastic budgets, nondegenerate rolling VIX smiles, and the absolutely continuous clock $a_t=v_t$. Together with Example 4.6 below, both sides of the clock boundary of Theorem 3.9 are inhabited. Nonzero spot–variance correlation is possible under the corresponding moment-explosion conditions and is not needed here.

**Example 4.6 (A strict clock model without a \(dt\) volatility).**  Let $C:[0,1]\to[0,1]$ be the Cantor function and $W$ Brownian motion. Set

$$
S_t=\exp\{W_{C(t)}-\tfrac12C(t)\}, \qquad A_t=C(t),
$$

with filtration $\cF_t=\cF_{C(t)}^W$. Then $S$ is a positive continuous true martingale with moments of every order, and

$$
B(t;u,v)=C(v)-C(u)
$$

is a continuous deterministic additive budget. Hence the model produces a complete compatible SPX/VIX surface and is an exact clock realization. The measure $dC$ is singular with respect to Lebesgue measure, so no predictable $a$ can satisfy $A_t=\int_0^ta_sds$. The $dt$ subclass is therefore strictly smaller than the clock class.

**Example 4.7 (A computed finite structural separator).**  Take one period of length one, $S_0=1$, and restrict the carrier to $S_1\in[1/2,3/2]$, enforced by zero-priced calls above $3/2$ and puts below $1/2$ as in Example 4.1, or by declaring the compact spot range in $\Omega_J$. Let $B$ denote the one-cell budget. On this interval, Taylor’s theorem and $(d^2/ds^2)(-2\log s+2(s-1))=2/s^2\le8$ give

$$
\begin{equation}\label{eq:separator-ineq}\tag{42} -2\log s+2(s-1)\le4(s-1)^2. \end{equation}
$$

Therefore, pathwise,

$$
\begin{equation}\label{eq:separator-pathwise}\tag{43} B-4(S_1-1)^2 \le\underbrace{(B+2\log S_1)-2(S_1-1)}_{W}. \end{equation}
$$

The first term in $W$ is the diagonal structural residual and the second is a constant-position SPX gain. Every admissible law has $\E W=0$, so

$$
\E B\le4\E(S_1-1)^2.
$$

A quoted pair $\E B=1$ and $\E(S_1-1)^2=0$ has strict separation margin one. If the squared-return payoff is synthesized from calls and $B$ is traded as a variance/log-contract claim, the certificate is executable. Without the budget instrument it remains an exact class-relative obstruction, not a claim of traded arbitrage.

**Example 4.8 (Weak realization versus positive-variance mixtures).**  Let $S_0=1$ and $R(S_1=1/2)=R(S_1=3/2)=1/2$. This is a nonconstant positive martingale with budget

$$
B=-2\E\log S_1=-\log(3/4)>0.
$$

The strike-$2$ call has price zero. Theorem 2.9 gives an exact continuous Bass realization. Every finite mixture of lognormal laws with strictly positive component variances has full support on $(0,\infty)$ and prices that call strictly positively. Thus an exact weak realization exists while an exact positive-variance finite-mixture realization does not. Relative-interior hypotheses for stronger regular architectures are substantive.

## 5 Implications and limitations

### 5.1 From marginal calibration to common-coupling certification

The practical distinction is between fitting marginal surfaces and certifying one common filtered coupling. In the complete-surface idealization, SPX calls determine each law $\operatorname{Law}(S_T)$ and VIX calls determine each law $\operatorname{Law}(V_t^\Delta)$; finitely many strikes with bid–ask bands only constrain those laws. Neither observation determines their joint law, the filtration, or the allocation of variance across overlapping windows.

Fix one ambient carrier and declared class. For each fixing $t$, let $\mathcal L_t$ be the one-window laws that fit the corresponding quote block and local martingale and diagonal rows, and let $\pi_t$ denote projection from a full law to that window. Separate calibration establishes only $\mathcal L_t\ne\varnothing$ for every $t$. Let $\mathcal G$ be the full laws on that carrier satisfying every full-history structural row: the spot martingale, diagonal, positivity, additivity, and budget towers, including in particular the shared-cell and cross-window conditions. Rolling calibration requires

$$
\mathcal G\cap\bigcap_t\pi_t^{-1}(\mathcal L_t)\ne\varnothing,
$$

so that every window budget is the projection of one nonnegative additive field

$$
B(t;u,v)=\E[A_v-A_u\mid\cF_t].
$$

This is the common-coupling problem; it is stronger than selecting an arbitrary copula between the VIX marginals. A parametric full-process model chooses a coupling by construction, while the architecture-free question is whether any such coupling exists. Example 4.2 shows that every one-window problem may be feasible while the rolling problem is not.

Theorems 2.2 and 3.6 therefore change the validation question from whether a selected calibrator fits chosen panels to whether *any* law in the declared class generates the entire rolling market. A rejection is localized to finitely many coordinates.

### 5.2 Practitioner coupling audit

The logical implications are asymmetric. Failure of a marginal/root row in Corollary 3.4, or exact violation of the overlap inequality in Proposition 3.5, rules out every common law satisfying the maintained idealization. Passing either test says only that the corresponding necessary obstruction is absent; it does not construct a coupling. At finite resolution, a validated primal common-law witness certifies existence for the stated panel, carrier, and class. A verified strict separator certifies nonexistence there. Bare numerical infeasibility of an inner discretization is not such a certificate unless it is supported by a valid dual separator or by a sound outer relaxation.

Table 1 separates data admissibility, marginal implications, local feasibility, a cross-window projection screen, finite common coupling, and the continuum lift. These stages are not interchangeable notions of “fit.”

Table 1. Practitioner coupling audit. Vanilla prices constrain marginal information; the later gates ask whether those marginals admit one filtered coupling. Passing P–O does not certify G.

| Code | Question | Evidence or test | If it fails | If it passes |
|---|---|---|---|---|
| **P** Data | Is the quoted panel a valid input? | Bid–ask, static-arbitrage, settlement, tenor, interpolation, and tail checks. | Repair or reject the panel; failure does not specifically diagnose coupling. | The inputs are usable; no mathematical feasibility conclusion. |
| **M** Marginal | Does one root-budget curve fit all marginals? | SPX convex order (the spot-martingale row), plus log-strip/root and VIX-square mean identities (Corollary 3.4). | No common rolling law can fit the panel. | Necessary unconditional rows only; no conditional coupling. |
| **L** Local | Does every window admit a one-window law? | Separate feasibility for \((S_t,V_t^\Delta ,S_{t+\Delta })\) on the declared carrier (the one-window problem; finite formulation via Theorem 2.2 and Remark 2.7). | No common rolling law can fit the panel in that class. | Possibly different laws fit different windows; no gluing conclusion. |
| **O** Screen | Does the observable overlap projection hold? | Complete VIX strips at \(t,t'\), the SPX log strip on \([t,t']\), and (34) for all \(K\) (Proposition 3.5). | No common filtered coupling exists. | This obstruction is absent only; there is no existence witness. |
| **G** Common | Does one finite rolling coupling exist? | One \(R\in \cA _M^J\) satisfies every quote, martingale, diagonal, positivity, additivity, and tower row (Theorem 2.2; Proposition 2.5). | A verified strict separator excludes every law in the declared finite class. | A common finite-panel law exists; Theorem 2.9 gives an exact positive continuous realization. |
| **C** Cont. | Does one continuum common law exist? | Every determining finite quote restriction is feasible in the same fixed \((\Psi ,M_\Psi )\) class (Theorem 3.6). | A finite quote, structural, or class witness excludes the continuum law. | One common-budget stochastic-clock law exists; uniqueness is not implied. |

Stages P, M, and O use observable marginal information. Stages L and G introduce latent joint variables and structural rows. Stage C requires complete continuous quote surfaces as a market idealization or interpolation assumption and, separately, the compactness, moment bound, and path/budget regularity supplied by the declared class. The continuum alternative is therefore not itself a finite-dimensional price test: each finite quote restriction remains tested against the same complete structural class.

### 5.3 What the current computations establish

The calculations concern different projections of Table 1; none identifies the market’s actual copula or filtration. On the committed 27 August 2026 delayed-quote snapshot, the date-collapsed transformed-band surrogate for Stage M is feasible and has a maximum common inward margin of $0.518$ basis points of total variance. Its Stage-O screen reports no flag on ten date-overlapping consecutive pairs; four use same-calendar-date SPX expiries as timing proxies and six use maturity brackets. Because those calculations collapse AM/PM settlement times, hard-code a thirty-day tenor, and use declared tail, parity, and quadrature conventions, neither the inward margin nor a nonpositive overlap residual is a settlement-aware feasibility certificate.

A separate settlement-aware, quote-constrained experiment on four proprietary ORATS panels tests a projected budget/tower subsystem. On 16 March 2020, 8 April 2025, and 30 June 2026 it constructs respectively 31-, 39-, and 19-atom joint VIX/budget witnesses fitting every retained raw SPX and VIX call/put band and the common-forward mean rows. A fail-closed checker independent of the generating program reconstructs every retained row; the largest equality, quote-band, or nonnegativity residual is $1.82\times10^{-12}$. These are floating-point witnesses at tolerance $10^{-8}$, not exact-rational proofs. On 5 August 2024 both selected VIX smiles fail butterfly convexity inside their raw snapshot bands. Exact calibration to that unmodified panel is impossible, but the Stage-P failure does not diagnose a specifically cross-window coupling defect.

The positive witnesses establish nonemptiness only of the projected budget/tower subsystem on the declared finite supports. The experiment has no exact AM SPX marginal at both VIX fixing times and does not impose the joint $(S_t,V_t^\Delta,S_{t+\Delta})$ laws, conditional spot-martingale rows, or conditional diagonal log rows. Only the 2026 panel has equal window tenors and directly matches the paper’s fixed-$\Delta$ setting; the 2020 and 2025 panels are unequal-tenor extensions of the same reduced subsystem. Thus these are not Stage-G certificates, and three positive witnesses are not a historical-frequency estimate. They show that the coupling question can be posed against actual quote bands and that no reduced budget–tower obstruction appears on those panels and supports; they do not establish the full SPX/VIX filtered coupling or the prevalence of market inconsistency.

### 5.4 Missing instruments and post-feasibility model checks

The principal observational gap is the conditional law. Listed VIX options give the marginal law of each fixing separately and never the $\cF_t$-conditional law of the next one. Further instruments narrow this gap; none closes it. A *mid-curve VIX option*, expiring at fixing $t$ on the VIX future for fixing $t'>t$, observes the law of the scalar $\E[V_{t'}^\Delta\mid\cF_t]$, which by conditional Jensen satisfies $\Delta\,(\E[V_{t'}^\Delta\mid\cF_t])^2 \le B(t;t',t'+\Delta)$ pathwise; an option on the forward variance level, that is on the VIX-squared future, observes the law of $B(t;t',t'+\Delta)/\Delta$, and hence the law of the budget up to the known tenor factor. Only the latter adds a quote row on the *marginal* law of a forward budget; the mid-curve adds the law of a conditional mean and the Jensen bound, and no more. Two models can share complete VIX and mid-curve strips while their forward-budget laws differ: with $\Delta=1$ and $V\sim\tfrac14\delta_0+\tfrac12\delta_1+\tfrac14\delta_2$, revealing nothing at $t$ gives $\E[V\mid\cF_t]=1$ and forward budget $\tfrac32$, whereas revealing at $t$ a fair coin that selects $\{V=1\}$ or $\{V\in\{0,2\}\}$ gives conditional mean $1$ on both branches and forward budget $1$ or $2$. The tower row is a conditional identity between that budget and the later fixing; its verification would require the joint law with the conditioning history, so it remains a structural row imposed in the program. VIX calendar-spread options likewise observe the law of the scalar spread between two futures, not the pairwise joint law of the fixings; the dispersion claims of Example 4.1 are a different object. Realized-variance options on $[t,t+\Delta]$ paired with VIX options at $t$ price the unrevealed variance of the budget, $\E[(A_{t+\Delta}-A_t)^2]-\Delta^2\E(V_t^\Delta)^4 =\E\bigl[\operatorname{Var}(A_{t+\Delta}-A_t\mid\cF_t)\bigr]$, which bounds the revelation defect of any mark for the budget row. Making every history-conditioned row observable would require a spanning family of path-dependent claims that no listed market supplies. These instruments therefore enrich the quote panel against which the structural rows are tested; they do not convert the structural rows into quote rows, and the continuum class remains a declaration in every case.

Panel-safety (Definition 2.13) and the early-revelation diagnostic apply after a candidate construction has been specified. They test whether marks introduced by an algorithm preserve the intended conditional rows; they are not additional market coupling tests. The clock/Brownian distinction is later still. Once a common continuum law exists, Theorem 3.9 asks whether its bracket is absolutely continuous and hence admits a calendar-time volatility density. Passing the coupling audit does not imply that regularity, low-dimensionality, or parsimony.

### 5.5 Limitations

The class-relative nature is unavoidable, and Section 1.3 records what that excludes. What option prices leave undetermined is specific: they supply neither path tightness, nor budget continuity, nor logarithmic uniform integrability, nor a volatility density, so each is a declaration rather than a quote row. The calendar-time representation is likewise an existence category and not a parsimony statement: a regular low-dimensional diffusion requires in addition a sufficient finite-dimensional state and a regular factorization of the bracket density through that state.

## 6 Conclusion

Within one fixed coercive continuous-path class, a complete SPX surface and a rolling fixed-tenor VIX surface are compatible precisely when their dense finite restrictions share one common martingale–bracket–budget law. Failure has a finite strict witness. At finite resolution every feasible structural law admits an exact positive continuous conditional-Bass realization; at continuum resolution every calibrated structural law is itself a stochastic-clock model. Absolute continuity of the clock is the exact additional gate to a calendar-time Brownian representation. The examples show why none of common-window gluing, common compactness, filtration augmentation, or clock regularity can be suppressed. The CIR and Cantor-clock examples explicitly inhabit the absolutely continuous and singular sides of the clock boundary. Throughout, the decisive coordinate is informational: what a model reveals, and when, is part of its identity, which is why the state carries the budget field and why closedness is stated in the enhanced topology. The main question left open is whether a Vorob’ev-type dichotomy exists for sum-functional data under a martingale tower (Remark 4.3). The overlap inequality shows that the tower bites on complete vanilla smiles (Example 4.2); whether it and its conditional refinements exhaust the vanilla obstructions is open.

**Code and data availability.** The public reproducibility package contains the linear-programming scripts, the 27 August 2026 CBOE delayed-quote snapshot, and its outputs. Archived separately with the working paper are the reduced-coupling program, aggregate output, compressed numerical witnesses, frozen hashes, and a checker independent of the generating program. The underlying ORATS chains are proprietary and are not redistributed; reproducing those calculations requires local access to the same raw files. The documentation records the settlement, support, tail, and tolerance conventions behind the numerical statements in Section 5.3.

**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 Exact finite structural cubature

This appendix supplies the finite-tree reduction invoked after Theorem 2.9. It preserves a chosen finite payoff panel and the coercive moment exactly; it makes no positive-variance mixture claim.

**Lemma A.1 (Finite barycentric cubature).**  Let $\mu$ be a probability measure on a measurable space $E$, let $F:E\to\R^d$ be integrable, and let $D\subset E$ have full $\mu$-measure. There are $N\le d+1$, points $x_1,\ldots,x_N\in D$, and positive weights $p_i$ summing to one such that

$$
\int_EF\,d\mu=\sum_{i=1}^Np_iF(x_i).
$$

*Proof.* Put $m=\int F\,d\mu$ and $K=\overline{\operatorname{conv}}F(D)$. Separation gives $m\in K$. If $m$ is on the relative boundary of $K$, a supporting affine functional through $m$ has a nonnegative gap on $F(D)$ with zero expectation. The gap therefore vanishes almost surely. Restrict $D$ to that full-measure hyperplane section and repeat. After at most $d$ steps, $m$ lies in the relative interior of the closed convex hull of the remaining image. A convex set and its closure have the same relative interior, so $m$ lies in the convex hull itself. Carathéodory’s theorem then gives the representation with at most $d+1$ points; zero weights are discarded. This is the finite-dimensional cubature argument used in Bayer and Teichmann (2006).∎

For $h=0,\ldots,J-1$, define the stopped budget coordinate

$$
C_j^h= \begin{cases} B_{j,h},&j\le h, \\
B_{h,h},&j>h, \end{cases} \qquad N_0=0,\qquad N_j=\sum_{a=0}^{j-1} \{2(X_{a+1}-X_a)+\delta_aB_{a,a}\},
$$

and the adapted vector

$$
\begin{equation}\label{eq:augmented-martingale}\tag{44} \mathbf M_j=(S_j,N_j,C_j^0,\ldots,C_j^{J-1}). \end{equation}
$$

**Lemma A.2 (The structural rows form one vector martingale).**  For a law with finite first $\Gamma$-moment, the conditional identities in (7) hold if and only if $\mathbf M$ in (44) is a vector martingale in the full-history filtration.

*Proof.* The first coordinate gives the spot row. The increment of $N$ is $R_j^D$. For fixed $h$, the increment of $C^h$ is $R_{j,h}^B$ when $j<h$ and zero afterward. These are exactly the three residual families.∎

**Lemma A.3 (Finite structural cubature).**  Let $R\in\cA_M^J$ and let $\Phi$ be a finite $m$-dimensional payoff vector with the growth imposed in Section 2. There is a finite nonrecombining atomic tree law $R^{\rm at}\in\cA_M^J$, supported on paths from an $R$-full carrier, such that

$$
\E_{R^{\rm at}}\Phi=\E_R\Phi, \qquad\E_{R^{\rm at}}\Gamma^q=\E_R\Gamma^q.
$$

Each selected node has at most $J+m+4$ successors.

*Proof.* Set $Y=(\Phi,\Gamma^q)$ and choose Borel versions $G_j(H_j)=\E_R[Y\mid\cF_j]$. Regular conditional successor laws exist because all history spaces are standard Borel. Starting from a Borel full-measure carrier $D_J$ on which $G_J=Y$, choose carriers $D_j$ backward so that the conditional successor law at every $h\in D_j$ is concentrated on prefix extensions in $D_{j+1}$ and

$$
\E_R[\mathbf M_{j+1}\mid H_j=h]=\mathbf M_j(h), \qquad\E_R[G_{j+1}\mid H_j=h]=G_j(h).
$$

The finitely many conditional identities and the conditional concentration property hold almost surely; intersecting their Borel full-measure sets at each date gives these carriers. The deterministic root is retained in $D_0$.

At a selected history $h\in D_j$, apply Lemma A.1 to its conditional successor law on $D_{j+1}$ and the $J+2+m+1$ coordinates $(\mathbf M_{j+1},G_{j+1})$. At most $J+m+4$ successors reproduce both displayed conditional means. Recurse at every selected successor through date $J$. The resulting finite tree makes $\mathbf M$ a martingale, so all structural rows hold by Lemma A.2. Backward induction on the tree preserves $G_j$; at the root this preserves both $\E_R\Phi$ and $\E_R\Gamma^q$. The latter identity retains the moment cap. This is the marked full-history form of the martingale cubature reduction in Beiglböck and Nutz (2014).∎

The reduction preserves the selected finite panel, not an entire marginal law, a continuum of strikes, or the original conditional kernels.

## B Details for the conditional Bass realization

This appendix expands the four compressed steps behind Lemma 2.8 and Theorem 2.9. The statements are not repeated; notation is that of Section 2, and $f_j,u_j$ denote the objects of (14) built from the successor-spot kernel $\kappa_j$ below.

### (a) Parameterized disintegration and randomization

Two classical facts about standard Borel spaces are used, in the following exact forms. *Measurable disintegration* (Kallenberg, 2002, Theorem 6.4): if $K(h,dz)$ is a probability kernel between standard Borel spaces and $\pi$ is a Borel map on the target, there exist a kernel $\kappa(h,ds)$, the $\pi$-image of $K(h,\cdot)$, and a kernel $L((h,s),dz)$ concentrated on the fiber $\{\pi=s\}$, with $K(h,dz)=\int\kappa(h,ds)\,L((h,s),dz)$. *Randomization* (Kallenberg, 2002, Lemma 3.22): for a probability kernel $L$ between standard Borel spaces there is a Borel map $g$ such that $g(x,U)\sim L(x,\cdot)$ for *every* $x$ whenever $U$ is uniform on $[0,1]$.

Identify the successor state with $(s,\beta)\in(0,\infty)\times[0,\infty)^{J-j-1}$ through $(x,\beta)\leftrightarrow(e^x,\beta)$, and apply both facts with $h$ the date-$j$ history and $\pi(x,\beta)=e^x$: this yields the successor-spot kernel $\kappa_j(h,ds)$ and a mark kernel $L_j^{\mathrm{mark}}((h,s),d\beta)$ on $[0,\infty)^{J-j-1}$, realized by a Borel map $g_j^{\mathrm{mark}}(h,s,u)$ with $g_j^{\mathrm{mark}}(h,s,U)\sim L_j^{\mathrm{mark}}((h,s),\cdot)$ for uniform $U$. By Lemma 2.1 and $\E_R\Gamma^q\le M$, for $R$-a.e. $h$ the kernel $\kappa_j(h,\cdot)$ has mean equal to the current spot $s_j$ and satisfies (13). On the exceptional $R$-null set redefine $\kappa_j(h,\cdot)$ as the law of $s_j\exp(\sigma N-\sigma^2/2)$ with $N$ standard normal and $\sigma^2=\delta_j b_{j,j}(h)$ read from $h$, and let $g_j^{\mathrm{mark}}(h,s,\cdot)$ freeze the marks, $b_{j+1,\ell}:=b_{j,\ell}$. The patched kernels satisfy (13) at *every* history, are jointly Borel, and agree with the disintegration of $R$ almost everywhere, so the skeleton law of the construction is unchanged.

### (b) Measurable quantile and convolution regularity

The set identity displayed in the proof of Lemma 2.8 exhibits $\{q<a\}$ as a countable union of measurable sets, so $(e,v)\mapsto q(e,v)$ is jointly measurable, and is nondecreasing in $v$; hence $f(e,z)=q(e,\Phi_N(z))$ is jointly measurable, nondecreasing in $z$, and strictly positive for a.e. $z$, and $u$ in (14) is jointly measurable. Write $u(r,e,x)=\int f(e,y)\,\varphi_{1-r}(y-x)\,dy$ with $\varphi_t$ the $N(0,t)$ density. Fix $\epsilon\in(0,1)$, $K>0$, $k\ge0$. For $(r,x)\in[0,1-\epsilon]\times[-K,K]$, since $1-r\in[\epsilon,1]$,

$$
\bigl|\partial_x^k\varphi_{1-r}(y-x)\bigr| \le C(\epsilon,K,k)\,(1+|y|^{k})\,e^{K|y|}\varphi(y),
$$

and the $r$-derivative reduces to two $x$-derivatives through the heat equation for $\varphi_t$. Hölder’s inequality against $\varphi(y)\,dy$ with exponents $(q,q')$ and $\int f(e,y)^q\varphi(y)\,dy=\int s^q\kappa(e,ds)<\infty$ dominate every derivative of the integrand, uniformly on compacts of $[0,1)\times\R$, by one integrable function. Hence $u$ is $C^{1,2}$ there, strictly positive, and solves $u_r+\tfrac12u_{xx}=0$; Itô’s formula on $[0,1-\epsilon]$ gives $dY_r=u_x(r,e,W_r)\,dW_r$.

### (c) The no-early-revelation identity

On a space carrying $2J$ independent Brownian motions set, for cell $j$, $W^{j,a}_r=\delta_j^{-1/2}(B^{j,a}_{t_j+r\delta_j}-B^{j,a}_{t_j})$, $a=1,2$, $r\in[0,1]$, frozen after the cell. The model filtration $(\cF_t)$ is generated by the memory factor, i.e., by all cell coordinates run up to time $t$ (the root is deterministic). Let $\cG_j=\sigma(W^{i,a}:i<j,\ a=1,2)$. The recursion makes $H_j$ a Borel function of the frozen endpoints, so $H_j$ is $\cG_j$-measurable and $\cF_{t_j}\subseteq\cG_j$, while $(W^{j,1},W^{j,2})$ is independent of $\cG_j$. The successor state is $Z_{j+1}=\Xi_j(H_j,W^{j,1}_1,W^{j,2}_1)$ with

$$
\Xi_j(h,w_1,w_2) =\bigl(\log f_j(h,w_1),\ g_j^{\mathrm{mark}}(h,f_j(h,w_1),\Phi_N(w_2))\bigr) \in\R\times[0,\infty)^{J-j-1},
$$

a successor state in the coordinates $(X_{j+1},(B_{j+1,h})_{h>j})$. For bounded Borel $\psi$, the freezing lemma for conditional expectations under independence gives $\E[\psi(Z_{j+1})\mid\cG_j]=G_j(H_j)$ with $G_j(h)=\E[\psi(\Xi_j(h,N_1,N_2))]$ for independent standard normals $N_1,N_2$. Since $\Phi_N(N_1)$ is uniform, $f_j(h,N_1)\sim\kappa_j(h,\cdot)$; since $\Phi_N(N_2)$ is an independent uniform, the randomization property of $g_j^{\mathrm{mark}}$ and the product structure give

$$
G_j(h)=\iint\psi(\log s,\beta)\,L_j^{\mathrm{mark}}((h,s),d\beta)\, \kappa_j(h,ds) =\int\psi\,dK_j(h,\cdot),
$$

the last equality being the disintegration of $K_j$ along $\pi$. Conditioning from $\cG_j$ down to $\cF_{t_j}\supseteq\sigma(H_j)$ yields (18): no functional of the past drivers beyond $H_j$ biases the successor kernel. The same independence gives cell martingality in the full filtration: for $t_j\le s\le t\le t_{j+1}$ the increments of $W^{j,1}$ after $r(s)$ are independent of $\cF_s$, which adds to $\cG_j$ only the stopped $W^{j,1}$ and the $W^{j,2}$-path, itself independent of $W^{j,1}$; hence $\E[u_j(r(t),H_j,W^{j,1}_{r(t)})\mid\cF_s]=u_j(r(s),H_j,W^{j,1}_{r(s)})$. With the boundary identity $u_{j+1}(0,H_{j+1},0)=\int s\,\kappa_{j+1}(H_{j+1},ds)=S_{t_{j+1}}$ and the tower property across the finitely many cells, $S$ is a true martingale on $[0,t_J]$.

### (d) Closure and uniform integrability for (13)

Fix $e$. Since $Y_1=f(e,W_1)\in L^q$, $q>1$, the martingale $Y_r=\E[Y_1\mid\cF^W_r]$ is uniformly integrable and closed at $r=1$; Lévy’s upward theorem gives $Y_r\to Y_1$ a.s., so the version $u(r,e,W_r)$ is a.s. continuous on all of $[0,1]$. As $\kappa(e,\cdot)$ charges $(0,\infty)$, $Y_1>0$ a.s.; a nonnegative martingale is absorbed at zero, so $\inf_{[0,1]}Y>0$ a.s. With $\Lambda_r=\int_0^r(u_x/u)^2(s,e,W_s)\,ds$, put $\tau_n=\inf\{r:|W_r|\ge n\ \text{or}\ \Lambda_r\ge n\}\wedge(1-\tfrac1n)$; these are nested and increase to $1$ a.s. because $\Lambda$ is finite on $[0,1)$. On $[0,\tau_n]$ the integrand $\theta=u_x/u$ is bounded, so the stopped Itô expansion of $\log Y$ is valid, the stochastic integral has zero mean, and $\E[\log Y_{\tau_n}]=\log m(e)-\tfrac12\E[\Lambda_{\tau_n}]$. Uniform integrability of $(\log Y_{\tau_n})_n$ holds by splitting: optional sampling gives $Y_{\tau_n}=\E[Y_1\mid\cF_{\tau_n}]$, so $\log^+Y_{\tau_n}\le Y_{\tau_n}$ is dominated by the uniformly integrable family $\{\E[Y_1\mid\cF_{\tau_n}]\}_n$; and since $y\mapsto(-\log y)^+$ is convex, conditional Jensen gives $\log^-Y_{\tau_n}\le\E[\log^-Y_1\mid\cF_{\tau_n}]$, uniformly integrable because $\log^-Y_1\in L^1$ by (13). A family dominated in absolute value by uniformly integrable families is uniformly integrable; with the a.s. convergence $\log Y_{\tau_n}\to\log Y_1$ this gives $\E[\log Y_{\tau_n}]\to\E[\log Y_1]=\int\log s\,\kappa(e,ds)$, finite by (13). Monotone convergence along the nested $\tau_n$ gives $\E[\Lambda_{\tau_n}]\uparrow\E[\Lambda_{1-}]=\E\int_0^1\theta_r^2dr$. Combining the two limits proves (15), with both sides finite; in particular $\Lambda_{1-}<\infty$ a.s. Taking $e=H_j$, the diagonal row of (7) transported through (18) gives the cell identity, and summation with the tower rows gives (17).

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