A Gaussian shock applied at an active constraint will leave the no-arbitrage set with probability tending to 1/2 as the time step shrinks. The paper sets its own pre-registered daily-step floor at 25% of active-margin steps. That rate rises toward 0.5 when both the margin and the step contract. Drift cannot change either figure. A soft no-arbitrage penalty in the training loss cannot change them either. This result should alter desk practice before the data show up.
The scaling argument is brief. Write implied total variance as w(k,tau) = tau sigma^2(k,tau) over a compact rectangle of forward log-moneyness and maturity, then treat the whole surface as one state in a weighted Sobolev space. Three inequalities define static no-arbitrage on the window: the variance floor w >= m, calendar monotonicity d_tau w >= 0, and the Gatheral-Jacquier butterfly functional g[w] >= 0. The last is a slice-wise inequality ensuring that the implied risk-neutral density stays nonnegative. Denote the resulting set by K_m.
Now apply one Euler step of a Gaussian field model, effectively surface PCA in infinite dimensions: w_Delta = w_0 + Delta mu + sqrt(Delta) xi. Its shock has order sqrt(Delta), while its drift has order Delta. At an active constraint, provided the noise is nondegenerate in the constraint-normal direction, the chance of leaving the set tends to 1/2 as Delta shrinks (Theorem 7.3). The conclusion holds for every fixed drift.
Corollary 7.4 gives the form a practitioner can standardize. Violation probability converges to Nbar(Phi/(sqrt(Delta) sigma_Phi)), where Phi measures the constraint margin through a smoothed window functional. Increment variance along the constraint normal is sigma_Phi^2 = <l, C l>. Convergence is uniform for states whose standardized margins remain in a bounded range. Uniform controls must also hold on a common neighbourhood: ||DPhi|| + ||D^2Phi||_op <= L and sigma_Phi >= sigma_0. Under those curvature and non-degeneracy bounds, the formula guides the Delta goes to zero limit; it is no exact daily-step rate. Setting r = 0 gives one half.
Continuous time turns the result into a trichotomy (Corollary 7.5). Any K_m-invariant Itô model must suppress noise along the constraint normal at first contact, introduce a reflection term, or remain on a submanifold within K_m. Learned one-step generators face the same argument. When the fitted diffusion has Sigma_theta(w_0)* l!= 0 at an active constraint, the half-law applies regardless of the penalty used during training (Corollary 12.2). Feasibility therefore belongs in the architecture.
We could not test these claims. The paper is theoretical and contains no performance figures, leaving nothing to reproduce. Its proposed protocol uses SPX index options. Our options data instead covers end-of-day listed US equity and ETF contracts. The nearest buildable exercise would use SPY options as a proxy for the same index surface, making the result a construction inspired by the framework rather than a replication. End-of-day quotes are all we have, which rules out intraday surface dynamics and execution-sensitive claims.
The mathematics on offer
Noguer i Alonso presents this as mathematics. No numerical results appear in this version, and each formal statement receives one of three tiers: [Proved], [Conditional] or [Conjectural]. The paper builds several objects used by a surface desk around its boundary theorem.
Dupire local variance is exactly the ratio between the two constraint functionals, a = d_tau w / g[w] (Theorem 5.1). This requires a smooth surface, w in H^s with s >= 3, with w and its derivatives in k, kk and tau continuous. Uniform margins are also assumed: w >= m > 0 and g[w] >= gamma > 0, keeping the butterfly functional away from zero. Within {g > 0}, zeros of local variance coincide with the calendar-active set. Poles occupy the butterfly-active stratum. Positive local variance alone cannot certify static arbitrage freedom because numerator and denominator may both be negative. That loses one sign bit, recoverable only through price-chart convexity.
The Karhunen-Loève factors of the increment covariance operator have an exact mean-square tail, sum_{i>q} lambda_i. No linear rank-p decoder can improve on sum_{j>p} lambda_j. Portfolio sensitivity to the surface is collected in the vega field, the function whose inner product with a surface bump produces the book's P&L. Formally, it is the Riesz representer of the portfolio value derivative.
For n traded instruments, the minimum-variance hedge has the closed form alpha = (H C H)^{-1} H* C nu. Residual variance equals <nu, C nu> minus the projection term. It reaches zero only if C^{1/2} nu belongs to the span of the C^{1/2} instrument fields.
Section 15 fixes an empirical design and leaves it unexecuted. The intended sample uses end-of-day SPX quotes from January 2005 to December 2025, with tau in [7/365, 2] years and k in [-1.5, 0.5]. Increments sit on a shared 41 x 21 grid. Headline results use Sobolev weight 1, and the design specifies two construction pipelines: Fengler smoothing and price-space arbitrage repair.
The protocol names nine models M1-M9, six metric families F1-F6 and seven hypotheses H1-H7. Its thresholds are design constants chosen before data contact, rather than estimates. Under H1, the unconstrained Gaussian KL model M1 violates constraints on more than 25% of active-margin steps, with the rate rising toward 0.5. Models M2-M9 remain below 0.01. H3 requires five modes to explain at least 90% of increment variance. H4 requires the field hedge to reduce out-of-sample vega P&L variance by at least 10% relative to bucket regression. The sample window ends eight months before the manuscript date of August 7, 2026. Here, pre-registration means the author fixed the thresholds before touching data that had already existed for eight months.
Nothing has been run.
Two feasible surfaces, one infeasible midpoint
Proposition 3.3(ii) supplies a machine-checked certificate. Over k in [-2,2], with m = 10^-3, two slices in the SVI family are certified feasible. SVI and its surface extension SSVI are the standard parametrizations of a total-variance slice. The bounds are inf g[w1] >= 0.0037007 and inf g[w2] >= 0.0029721. Yet their midpoint reaches g <= -0.5871470 at k = 1.75010.
The computation uses directed-rounding interval arithmetic, 50-digit decimals and 25,000 subintervals. Total-variance admissibility is therefore nonconvex: averaging two arbitrage-free volatility surfaces can create an arbitrageable surface. A metric projection onto K_m exists, though uniqueness is not guaranteed. Variance-coordinate calibration consequently lacks a single answer unless one moves to the price chart. There, the banded set K_{c,m,M} is closed, convex and weakly closed.
The certified failure occurs at k = 1.75010, beyond the paper's quote window. A separate pre-registered certificate search requires the violating point to fall within k in [-1.5, 0.5].
Where does static feasibility stop?
A simulated surface that belongs to K_m at every date has static feasibility. Dynamic absence of arbitrage additionally requires the Musiela roll-down identity and a separate drift restriction. The discounted fixed-contract call price must be a local martingale jointly with the forward and discount-factor dynamics (Proposition 6.1).
Remark 6.2 says directly that heads, projections, reflections and flow maps provide static feasibility alone. The empirical program accordingly treats them as physical-measure scenario generators whenever the restriction is left unimposed. Sellers of an "arbitrage-free" generative surface model should be pressed on which meaning they intend.
Three available doors
The trichotomy doubles as a design brief, with different proof status for each route. Construction I works in convex price coordinates and adds reflection. Function-space well-posedness for this particular set remains [Conditional] (Proposition 11.1). Once the model is placed on a fixed quote grid, the constraint set becomes a compact convex polytope. The reflected SDE then has a unique strong solution, and projected Euler converges (Theorem 11.2). This proved grid construction is the version a desk could actually run.
Construction II stays in variance coordinates. Each Karhunen-Loève innovation is projected onto the linearized feasible cone through one small convex QP per step, a result proved on a finite grid. Its weak limit in total-variance coordinates is [Conjectural]. The author identifies this as the paper's main open problem.
Construction III restricts the dynamics to SSVI-type parameter manifolds. It is exact on the image, rigid away from it, and assumes innovations of exactly rank p.
The paper attaches two caveats to its boundary results. First, the half-law is an H^2 statement observed through smoothed window functionals. An isolated point of boundary contact disappears under the L^2 order structure. Activity over a set of positive measure resolves the problem. So does a stronger state space with s > 3, or the finite-grid viability test in Theorem 7.6.
The second caveat concerns the interior survival bound, exp(-(delta - M)^2 / (2 sigma_bar^2)), for joint margin delta above M = E||xi||_{C^2}. This bound requires C^2 sample paths for the noise. The paper states that regularity as a hypothesis on the model class because membership of the noise in the state space does not imply it.
A closed-form hedge awaits evidence
Two deductions arrive immediately. Reducing the number of Karhunen-Loève modes lowers M_q and cuts violation probability exponentially. Truncation rank therefore serves as a safety parameter, exchanged against the exact tail sum_{i>q} lambda_i.
The optimal field hedge alpha* and its residual variance are invariant to the Sobolev weight, although that choice transforms the vega field, kernel sections and covariance operator (Lemma 10.3). Vega-weighting or liquidity-weighting the inner product changes the representation used for hedging. Bucketed vegas equal point samples of the vega field only when bump shapes are kernel sections for the selected inner product. With arbitrary triangular bumps, the reported vector consists of Gram coordinates, beta = G_B a.
The proposed hedge is frictionless, first-order and one-period. It also requires estimation of a trace-class covariance operator, bringing every large-covariance pathology with it; the author refers to his own repair work. Flow-map likelihoods are exact only on finite grids or finite-rank manifolds. Genuine function-space likelihoods require Ramer-type quasi-invariance, a property coordinatewise neural flows need not satisfy. Such flows may send equivalent Gaussian measures to mutually singular laws. The scope section acknowledges all of these limits.
As a specification document, the paper is good work. Evidence has yet to arrive, a fact the author states in the tiering section rather than leaving readers to infer it. We have previously covered a study that was precise about estimation and silent about the model one would trade (our note on EGARCH asymmetry).
Measured H1 and H4 results would change my view. M1's violation frequency should follow Nbar(r) among active-margin states. The field hedge should also outperform bucket regression by that 10% out of sample during the 2008-09 and 2020 windows. For now, the boundary-layer formula is the paper's concrete deliverable. Everything else remains a promise attached to a 41 x 21 grid.