Compact operator representations for option pricing and risk

# Compact operator representations for option pricing and risk

## Executive assessment

Operator-derived representations can support a small, accurate European pricing kernel and accelerate selected repeated model-change workloads. A standalone CPU implementation prices a 65-contract layered-volatility chain, with spot delta and right-hand spot curvature, in 67-99 microseconds including preparation. It retains 14-47 KB of numeric state and reaches approximately 1.8 MiB peak serving-process memory. All 24 layered cases pass independently qualified synthetic checks under that right-sided convention. A final audit finds material left/right curvature differences at the spot coefficient interface, so this is not a validated smooth-gamma hedge service. An ordinary prefactored full solver also meets the one-millisecond target.[^1][^27]

This is a demonstrated implementation capability, not a demonstrated pricing-theory or commercial breakthrough. Exact cell elimination, inverse-Laplace pricing, adjoint readout, low-rank updates and analytic source integration have substantial prior art. No calibrated market surface, executable profit, lower-cost device, energy saving, general American-option solver or general Heston solver has been validated. The practical contribution is narrower: an auditable pathway from continuous operator identities to compact price/risk services, together with explicit evidence of where those identities cease to deliver accuracy, admissibility or lower total cost.

Representation stability and cost across changing maturity operators became the most consequential limitation. Raw pole expansions fail near collisions; retaining every pair in an unexpanded confluent form restores accuracy but makes three-interval pricing costly. A subsequent selective compilation retains only close pairs in that stable form. It passes all twelve existing three-interval cases and reduces complete time from 63-117 ms to 4-12 ms, a 5.30-17.18-fold internal speedup. Its numeric-artifact budget fails and is preserved as a separate failed gate. This is a concrete exponential/Hermite-calculus improvement, not evidence of economical arbitrary-depth propagation or market calibration.[^2][^20][^25]

The final nine-window native validation retains useful observed latency headroom: both methods pass all 216 case-windows, with selected-row cold p99 between 71 and 138 microseconds. No request exceeds one millisecond among 864,000 measured cold/prepared requests. These repeated synthetic workloads establish neither a future SLA nor a weaker-device result. All 134 campaign tests pass, and the numerical source records remain reproducible under their explicit contracts.[^28]

## Application boundary and evidence standard

Fast evaluation of the closed-form European Black-Scholes formula is not the research target. The relevant workloads involve variable-coefficient diffusion, usable derivatives, repeated coefficient changes and coherent propagation across maturity intervals. These calculations must be distinguished from market inference and execution. A numerically accurate price under a specified model does not establish that the model is economically accurate or that a quoted opportunity is tradable.

The evidence consists of bounded synthetic studies with predeclared cases, methods, tolerances and resource limits. Reference qualification is separate from candidate accuracy. Prices, first derivatives, second derivatives and sampled admissibility are reported separately. Construction and retained state are charged where preparation is part of the requested service; prepared-query speed is not substituted for total cost. Failed configurations, unqualified references, abstentions and cases where ordinary controls win remain in the record.[^3]

Most later European comparisons use normalized tolerances of $10^{-6}$ for price, $10^{-5}$ for first derivative and $10^{-4}$ for curvature. In the fixed-spot physical chain these mean price divided by 100, spot delta and 100 times right-hand spot curvature. At a coefficient jump, that last quantity need not be a unique classical gamma. Earlier American, Heston and barrier studies have their own frozen definitions. Spot derivatives, strike derivatives, one-sided interface traces and their normalizations must remain distinct.

Timing measurements come from one-thread numerical workloads on an Apple M4 Max Mac with 16 CPU cores and 128 GiB installed RAM. This is a capable workstation, not a measured edge device; thermal state and unrelated system activity are not controlled. Repeated-call medians and model ranges describe those workloads, not confidence intervals for future market traffic. Numeric arrays, temporary work, interpreter overhead, compiler memory and serving-process RSS are different quantities. The native serving executable has no Python, BLAS or GPU runtime, but this does not imply that competing financial libraries require one.

## Algorithmic foundation and prior art

### Exact native-cell interfaces

For a cellwise equation $-u''+q u=0$, write $\lambda=\sqrt q$. Endpoint values determine the interior through $H_L(y)=\sinh(\lambda(h-y))/\sinh(\lambda h)$ and its reflected right basis. The cell maps endpoint values to outward derivatives through a symmetric two-by-two matrix with diagonal $D=\lambda\coth(\lambda h)$ and off diagonal $E=-\lambda\operatorname{csch}(\lambda h)$. Assembling these maps produces a tridiagonal interface system, with one unknown per native interface rather than many interior samples.

This elimination is exact for the declared cellwise equation, not for an arbitrary smooth volatility function approximated by those cells. Temporal inversion, finite boundaries and model specification remain separate approximations. Stable evaluation uses decaying exponentials and cancellation-aware elementary functions. Complex symmetry is bilinear; it is not Hermitian symmetry, and inserting conjugation changes the calculation.

For zero-carry log-space diffusion, $V_t=a(x)(V_{xx}-V_x)$ with $a=\sigma^2/2$, a payoff-aware gauge gives the time-value resolvent $[-D_x^2+1/4+s/a(x)]\widehat w=\delta_0/s$. Each inverse-contour frequency therefore requires a small exact-cell solve. Layered spatial solutions and Laplace-based local-volatility calibration are established in Lipton-Sepp and Itkin-Lipton. The published contour families used here come from Trefethen, Weideman and Schmelzer. The new numerical evidence must not be relabeled as the invention of those methods.[^4][^5][^6]

The connection to cardinal spline theory is conceptual and algorithmic, not literal identity of every basis. The cell poles vary with frequency and volatility interval; the financial boundaries are not periodic. There is no single globally shift-invariant exponential B-spline generator in this implementation. Cardinality and compact support alone would not justify a circulant inverse, guarantee positivity or make a time integrator exact.

### Inner products, projection and finite changes

Badoual, Schmitter and Unser's inner-product calculus demonstrates how continuous integrals and projections between represented functions become explicit coefficient matrices. Its periodic setting permits circulant structure. The transferable principle here is exact cross-basis integration, not transplantation of periodic boundary formulas into a bounded pricing domain.[^7]

For two cell parameters $q=\lambda^2$ and $r=\mu^2$, Green's identity gives $\int H_L(q)H_L(r)=(D(q)-D(r))/(q-r)$, with the analogous formula for the cross-endpoint product using $E$. Repeated poles use derivative limits. This makes the source generated by a previous maturity interval accessible through small structured loads. Ordinary quadrature remains a necessary comparator: exact represented integration need not minimize application-level cost.

A coefficient change in one cell changes only its two-by-two interface block. Classical finite-rank inverse identities then permit exact finite responses from selected inverse entries and a small correction system. This is different from linearizing the price in a volatility parameter. Selected inversion, Woodbury-type updates and Schur elimination are established numerical tools; their value here depends on the requested batch, outputs, update size and preparation cost.[^8]

The bounded local variance gamma experiments have a different interpretation from deterministic-time diffusion. In LVG, a resolvent itself defines the model price, whereas ordinary diffusion requires time evolution or inversion. Carr-Nadtochiy and Le Floc'h supply direct prior art for the resolvent model and exponential/tension-spline calibration. A fast LVG experiment cannot be silently presented as a fast general Black-Scholes PDE solver.[^9][^10]

## Results across the tested problem classes

| Problem class | Established result | Binding limitation |
| --- | --- | --- |
| European polynomial splines | Derivative recovery passes all 48 cases at the finest tested grid | Equal-grid finite differences can be faster |
| American exercise | Selected published prices can be matched | Off-grid obstacle and broader Greek/admissibility failures remain |
| Heston diffusion | Improved spatial treatment repairs selected price/Greek errors | Negative prices elsewhere prevent a general admissibility claim |
| Discrete barriers | Shared single-event response preparation benefits large batches | Small requests and simpler direct controls often win |
| Positive response compression | Tens or hundreds of atoms can represent selected responses | Smaller ordinary quadratures often minimize total cost |
| Layered European diffusion | Small exact-cell representations pass independent references | Advantage depends on reconstruction versus prepared-query workload |
| Multi-maturity propagation | Selective confluence cuts verified three-interval cost 5.30-17.18-fold | Internal comparison only; arbitrary-depth closure remains open |

### Correct derivatives are not sufficient admissibility

The initial European study exposed short-maturity gamma errors despite acceptable price and delta. Classical derivative recovery reduced a representative dense-grid scaled gamma error from 0.163 to 0.0064. A subsequent 48-case panel passed completely with recovered spline Greeks at the finest grid, while direct spline and finite-difference outputs passed fewer cases. Finite differences were faster at equal grid size, so this was an accuracy result rather than an accuracy-matched speed claim.[^3]

American options exposed a more important structural limitation. Enforcing nodal physical values above the payoff did not ensure the interpolated curve respected the exercise obstacle between nodes. Focused meshes could match selected BENCHOP prices, yet off-grid violations persisted; broader panels also exposed Greek and shape failures. Heston experiments similarly showed that accurate queried outputs could coexist with negative prices elsewhere. These configurations were closed without promoting their local successes to general solver claims. Published American-Heston ADI work and BENCHOP provide stronger comparison standards than a few convenient price samples.[^11][^12]

### Barrier response and compression

For a single changed monitoring barrier, forward and backward conditional factors yield an exact finite response integral for the represented discrete model. Shared preparation improved seven of eight 1,000-request workloads, by approximately 1.21-4.02 times total cost, but lost all 48-request comparisons. Subsequent response-polynomial, positive-quadrature and sparse-transition studies preserved several useful small-state representations without producing a broad total-cost advantage.[^3]

Continuous-recursion error bounds were more conservative than finite-parent agreement suggested. Most tested configurations failed their requested price/Greek budgets even when all observed inclusion checks passed. This distinction is essential: agreement with a finite quadrature parent is not a bound on the continuous problem, and an ordinary floating-point analytical bound is not an outward-rounded certificate. Fast Gaussian option-pricing methods, including Broadie-Yamamoto, are established comparators rather than an opportunity to assume that a compact spline response is uniquely efficient.[^13]

## Compact deterministic-time pricing and risk

### Accuracy-matched representation comparison

The exact-cell deterministic-time study passed nine constant-volatility and 24 layered cases against independent references. Its original uniform-grid controls did not meet the requested accuracy, so they could not support an equal-accuracy speed ratio. A separate focused-polynomial comparison repaired that gap: at least one conventional control qualified on every layered case.[^14]

On those 24 layered cases, exact cells retained 4,584 or 14,184 numeric bytes, 3.11-35.92 times less than the fastest qualified focused-polynomial control. Construction plus 501 queries was 1.36-4.27 times faster. However, a prepared polynomial curve answered 501 queries in about 0.046-0.063 ms, compared with 0.594-0.799 ms for the exact-cell representation. A largely static model with many repeated queries can therefore favor the polynomial curve. The observed gain belongs to reconstruction-heavy workloads, not universal evaluation throughput.

Spline projection and dynamic polynomial interpolation are substantial existing approaches to financial computation. Kirkby's projection framework includes spline bases, Gram/dual calculations, multi-strike outputs and FFT structure. Dynamic Chebyshev methods explicitly exploit conditional expectations and approximation reuse. Neither was reproduced in full as an optimized end-to-end comparator, so the local comparisons do not establish superiority over those method families.[^15][^16]

### Finite bucket risk

The deterministic-time three-call risk study tested 2,112 finite coefficient-change scenarios. Exact finite updates agreed with full solves throughout. For larger native books containing 68 scenarios, they improved total runtime by approximately 2.12-2.32 times. Full batched solves won every 20-scenario and one-request comparison. Prepared finite-update state was larger, not smaller, than the full comparator's state.[^17]

First-order sensitivities failed at least one practical tolerance in every complete model/shock panel. Their failure does not imply that all small-change approximations are useless; it establishes that the requested complete finite-shock service could not replace finite repricing with that linearization. In the earlier LVG sequence, direct solves also won every tested sequential scalar inversion workload. These results argue for workload-specific dispatch rather than a universal insistence on precomputed updates.

## Fixed-spot chains and deployment footprint

At a fixed spot, an arbitrary strike contributes loads to at most two native endpoints. Price and spot-readout calculations can then use two selected rows of the interface inverse, with the local inhomogeneous source correction retained. The physical model fixes a reference scale of 100 and uses a drift-normalized coordinate under deterministic rates and dividends. Holding that reference fixed during spot perturbations is necessary for the reported delta and one-sided curvature to refer to the same model.[^18]

This convention is explicit but restrictive. With nonzero carry, a coefficient field fixed in the normalized coordinate is a moving field in absolute physical spot. The implementation does not thereby solve every time-independent physical local-volatility specification. Four constant-volatility controls use physical Black-Scholes references; 24 fresh layered models use independently translated, high-precision single-contract calculations. All 265 unique strikes per model pass the requested price, delta and right-curvature checks, with no sampled bound, monotonicity or convexity violations.

The final convention audit materially narrows the risk interpretation. Exact-cell matching makes price and delta continuous at the spot interface, but the PDE implies $a_L\Gamma_L=a_R\Gamma_R$. All 24 layered models have unequal adjacent coefficients and nonzero forward-ATM curvature. Their left/right gamma differences are 1.26-53.07 percent relative to the right trace; the four constant controls have equal traces. The implementation and independent reference both select the right side, so their agreement does not establish existence of a unique second derivative. A future smooth-gamma service must qualify an appropriate coefficient model near spot, or explicitly offer directional/finite-bump risk instead. Neither smoothing nor relocating an interface has been tested as a remedy here.[^27]

The standalone C++ study gives both the full and selected-row methods cached tridiagonal factors. It retains the same models and references and changes only the serving implementation. The following ranges cover all 24 layered cases and include construction; they are not selected best cases.[^1]

| Contracts | Full solver, microseconds | Selected rows, microseconds | Full / selected rows |
| --- | --- | --- | --- |
| 1 | 11.7-41.4 | 12.7-46.4 | 0.80-1.01 |
| 13 | 31.8-97.7 | 21.3-55.8 | 1.35-1.89 |
| 65 | 128.8-328.7 | 67.1-99.4 | 1.80-3.55 |
| 257 | 453.5-1151.5 | 224.9-271.6 | 1.94-4.71 |

The selected-row method loses 23 of 24 single-contract comparisons. At 65 contracts both methods satisfy the predeclared one-millisecond, 256-KiB numeric-state and 16-MiB serving-RSS limits. Selected rows retain 13,992 or 46,632 bytes and peak at 1.75-1.86 MiB RSS; full solves retain less numeric state and peak at 1.86-2.02 MiB RSS. The approximately 56-KB executable uses only the standard library. Compilation and the Python research harness have substantially larger memory costs, reported separately in the archived results.

These measurements establish a dependency-light component on this machine. Process launch, market-data ingestion, calibration, transport and execution are outside the kernel latency. A lower-power-device claim requires that device; an energy claim requires energy measurement. The result is nevertheless practically more informative than an isolated small coefficient count, because the serving process itself has been measured.

### Time-separated individual-request validation

Nine frozen half-hourly windows from 04:00 through 08:00 CEST retain all 24 layered cases and both unchanged native methods. Each case/method/window contributes 1,000 cold-model requests and 1,000 prepared queries. Cold timing includes construction, query, output allocation/destruction and model destruction inside a warm executable; it does not include process launch. All windows complete on time and both methods pass all 216 accuracy, state, RSS and cold-p99 gates.[^28]

| Method | Cold median range, us | Cold p99 range, us | Largest cold request, us |
| --- | ---: | ---: | ---: |
| Full solve | 121-357 | 137-409 | 695 |
| Selected rows | 64-105 | 71-138 | 216 |

These are ranges of case/window empirical quantiles, not pooled quantiles or confidence intervals. Selected rows have lower cold medians in every paired comparison, with ratios 1.72-3.77. No cold or prepared request exceeds one millisecond; the largest prepared full-solve request is 917 microseconds. All observations, including slower windows, remain included. Selected-row serving RSS is 1.66-1.75 MiB, while full solves use 1.64-1.88 MiB. Retained numeric arrays are still larger for selected rows.

The complete per-window record is 522,022 bytes; including the final aggregate raises it to 538,283 bytes, missing the 0.5-MiB compact-record target by 13,995 bytes. The hard campaign budget remains satisfied. This storage-target miss is disclosed separately from the passing measurement gates. The observed headroom supports a narrow research-kernel capability, not market calibration, smooth-gamma hedging, future worst-case latency or a production service guarantee.

## Maturity propagation and confluent stability

For a subsequent interval, the transformed source includes the previous curve: $[-D_x^2+1/4+s/a_{new}]U=w_{old}/a_{new}+\delta_0/s$. Exact cross products propagate two maturity intervals without a dense source interpolation grid. All 28 two-interval cases pass the frozen price, strike-slope and density tolerances. Sixteen-point Gauss-Legendre integration also passes and is slightly faster or effectively tied on every layered case. The analytic calculation establishes a representation capability, not an integration-speed advantage.[^19]

The third interval cannot simply reuse the second curve's nodal values as if that curve were homogeneous on each cell. Its source is inhomogeneous. A formal union of old and new poles has only linear growth in mode count, but that compact algebra conceals conditioning. The distinct-pole expansion passes all 24 ordinary layered checks, abstains at an exact collision and fails slope/density accuracy at a fixed $10^{-10}$ log-volatility perturbation. The largest density cancellation ratio in that stress case is about $2.25\times10^{12}$.[^2]

Keeping the particular response as $[H(\mu)-H(\lambda)]/(\lambda^2-\mu^2)$ avoids expanding it into two large opposing coefficients. At a repeated pole its finite limit is a parameter derivative of the basis, introducing the expected polynomial-exponential form. An independently checked implementation passes 128 high-precision scalar tests and all 34 full-source cases, including the previous collisions. Its ordinary-precision implementation uses cancellation-aware elementary functions and the inhomogeneous equation for second derivatives.

This is a concrete role for operator-derived exponential/Hermite calculus. It is not yet evidence of an efficient arbitrary-depth representation: implicit ancestry, source-evaluation cost and repeated propagation error still require accounting. Analytic source interpolation is also explicit prior art in Itkin-Lipton. A further maturity-propagation result must preserve the distinction between exact source identities, numerical source quadrature and financially admissible model evolution.[^5]

A separately frozen third-interval test makes that distinction explicit. Four constant controls and eight fresh layered cases all qualify against independent references. The layered reference evaluates analytic second divided differences in 60-decimal arithmetic, including repeated-pole limits. Candidate Gauss rules of 32 and 64 points per cell pass every requested output and sampled shape/calendar check. Sixteen points fails density accuracy in two layered cases, with worst density error $5.73\times10^{-4}$ against a $10^{-4}$ target.[^20]

The 32-point method retains only 44.8-78.0 KB, but complete construction and nine strike readouts take 62.6-121.2 ms. Implicit source ancestry therefore creates a material work cost despite compact storage. These outputs are strike derivatives, not the spot Greeks of the native chain. No speedup is claimed against the fine CN reference, which was not designed as a competitive baseline. The result closes a correctness gap while demonstrating that the single-maturity sub-millisecond service does not yet extend to a general term structure.

### Selective confluence changes the measured cost

The next frozen comparison expands only pairs whose relative squared-pole separation exceeds 0.05. All other pairs retain the already verified confluent response. The threshold is fixed before implementation and never swept. The compiled source owns its arrays rather than retaining the preceding two-interval object. Both methods keep the same contour, source quadrature, outputs and tolerances.[^25]

All 34 exposed source cases, including the collision stresses, pass accuracy, strict representation and sampled shape checks. All twelve three-interval cases pass independently qualified references and the same checks. Seven rotated-order cold measurements per method/case give 4.09-12.28 ms for selective compilation versus 63.30-117.35 ms for the original stable implementation. Every case exceeds the predeclared stronger twofold target; ratios range from 5.30 to 17.18. Retained arrays occupy 39-70 KB versus 45-78 KB. Maximum normalized price, strike-slope and density errors are respectively $3.50\times10^{-14}$, $7.18\times10^{-14}$ and $8.17\times10^{-12}$.

The complete saved artifacts occupy 132,999 bytes, exceeding the frozen 100-KiB budget. The driver therefore exits after saving all panels, before final RSS and elapsed-time capture. A separate completeness audit records that failure without deleting, compacting or rerunning measurements. The numerical speed result is valid evidence from the saved paired measurements; the study is not reported as passing every gate. This remains an internal implementation comparison on an exposed cohort, not an external SOTA result or a native multi-maturity service.

Matrix-function theory supplies the broader connection. Repeated eigenvalues correspond to Hermite derivatives; clustered divided-difference algorithms treat close and separated nodes differently. A small triangular cell-cascade prototype verifies that interpretation in ten algebra tests, including repeated-pole chains. It is not an eight-maturity pricing test or a speed result. General nonnormal conditioning and dense matrix-function cost remain obstacles; selective scalar compilation avoids invoking that dense prototype in the pricing loop.[^26]

## Practical research direction

### Bounded source state

An assessment of the measured ancestry bottleneck identifies a structure-aware
compression route. For a convex call curve on a bounded forward-strike domain,
endpoint values and slopes recover each cell's probability mass and first
moment by integration by parts. Replacing that mass by its centroid gives a
lower convex-payoff approximation; distributing it to the cell endpoints with
the same mean gives an upper one. The corresponding call curves are tangent
and secant envelopes. This is established quantization and option-bounding
theory, not a new compression principle.[^21][^22]

For cell $[a,b]$, mass $m$ and centroid $c$, the exact-arithmetic maximum
price-envelope gap is $m(c-a)(b-c)/(b-a)$. Boundary atoms must be retained.
Five algebra tests verify representative identities, including a rational
example and mass/mean accounting. No pricing propagation benchmark has been
run with this helper. Its strict input contract also does not repair noisy
or slightly nonconvex numerical curves.[^23]

A positive real resolvent can propagate such price bounds without expanding
the full preceding source history. Under a common bounded-domain contract,
maximum-principle arguments control accumulated sup-norm price error. That
argument does not automatically apply to the complex weights of a numerical
inverse contour, and an atomic source representation does not supply accurate
final Greeks. The opportunity is therefore conditional: remove the measured
source-work bottleneck while preserving a useful error budget. Ordinary linear
source interpolation and positive tridiagonal solves are essential comparators,
and excessive envelope width or cell count would be a valid negative outcome.

A simple feasibility bound already cautions against assuming tiny state. For
a uniform distribution on $[0,2]$, the tangent/secant gap on a cell of width
$h$ is $h^2/8$. Any partition into $N$ cells therefore has maximum gap at
least $1/(2N^2)$: a $10^{-6}$ budget requires at least 708 cells. Allocating
that total budget equally across 32 steps using only nonexpansive propagation
requires 4,000 cells in this example. This is a warning about this particular
bound and representation, not an impossibility result for better algorithms.

### Calibration-to-risk service

The credible application vision is a compact, inspectable price-and-risk service that can be reconstructed and updated frequently on modest hardware. The service should expose its model and approximation contract, distinguish one-off from batch workloads, return coherent derivatives and abstain or fall back outside validated conditions. This is more defensible than presenting a small neural network, a cardinal basis or an exact derivative formula as an end in itself.

Three missing links determine whether this becomes useful rather than merely elegant. First, the model must support the requested risk quantity: the interface audit rules out an unqualified smooth-gamma claim, while arbitrary-depth maturity propagation remains unverified. Second, calibration must be included: fitting prices within declared quote uncertainty, evaluating unseen strikes/maturities, and testing risk stability under quote perturbations. Third, an actual deployment comparison must include an appropriate conventional library or solver and, for a lower-hardware claim, the target device. Synthetic price accuracy alone cannot substitute for any of these.

Calibration should not be treated as a routine final layer. A small residual may coexist with weakly identified local-volatility parameters and unstable hedging sensitivities. Market quotes are noisy and do not uniquely reveal a continuous coefficient field. The inverse problem and the role of regularization have longstanding mathematical analysis; a faster forward solver does not remove that information limitation.[^24] A meaningful next application study would therefore report fit, out-of-sample surface behavior, parameter/Greek sensitivity and full update cost, including failures, rather than optimize only a historical pricing residual.

The scientific opportunity is to formalize and validate when an operator-defined representation is the appropriate computational state: small enough to deploy, stable under changes, and rich enough to support required calculus. The present evidence supports that program in a bounded European setting. It does not support a claim that a general option-pricing breakthrough has already been achieved. The strongest current assets are the measured native kernel, accuracy-qualified reference machinery, exact finite-change accounting and a diagnosed stability mechanism with a verified confluent alternative.

## Sources

[^1]: Pricing operator investigation, Study 25, FINDINGS_25.md and RESULTS_25.json, 2026. [Archived native serving evidence](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/FINDINGS_25.md). Synthetic local measurements; source, build command and all cases archived alongside the findings.
[^2]: Pricing operator investigation, Studies 26-27, FINDINGS_26.md and FINDINGS_27.md, 2026. [Confluent source evidence](https://github.com/danielschmitter/operator_spline_neural_representation/blob/833453bb0/research/pricing_operator_20260915/FINDINGS_27.md). Distinct-pole failure and alternative-representation checks are separately preserved.
[^3]: Pricing operator investigation, PROTOCOL_01.md through PROTOCOL_19.md, corresponding RESULTS and FINDINGS files, 2026. [Campaign evidence index](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/README.md). Original numerical studies, not external commercial benchmarks.
[^4]: Alexander Lipton and Artur Sepp, Filling the Gaps, 2011. [Author-hosted paper](https://www.city.ac.uk/__data/assets/pdf_file/0015/110085/Filling-the-gaps-Lipton-Sepp.pdf). Layered local-volatility, Laplace-domain Green functions and calibration; main construction reviewed, not every figure or asymptotic detail.
[^5]: Andrey Itkin and Alexander Lipton, Filling the gaps smoothly, Journal of Computational Science, 2018, DOI 10.1016/j.jocs.2017.02.003. [Author preprint](https://arxiv.org/pdf/1608.05145). Source interpolation, explicit spatial solutions and analytic source integrals; reading depth is recorded in READING_LEDGER.md.
[^6]: Lloyd N. Trefethen, J. A. C. Weideman and Thomas Schmelzer, Talbot Quadratures and Rational Approximations, BIT 46, 653-670, 2006, DOI 10.1007/s10543-006-0077-9. [Author-hosted paper](https://people.math.ethz.ch/~hiptmair/Seminars/CONVQUAD/Articles/TWS06.pdf).
[^7]: Anais Badoual, Daniel Schmitter and Michael Unser, An Inner-Product Calculus for Periodic Functions and Curves, IEEE Signal Processing Letters 23(6), 878-882, 2016, DOI [10.1109/LSP.2016.2555139](https://doi.org/10.1109/LSP.2016.2555139). Supplied local source badoual1601.pdf, full text and appendices reviewed.
[^8]: George B. Rybicki and David G. Hummer, Fast Solution for the Diagonal Elements of the Inverse of a Tridiagonal Matrix, Appendix B, Astronomy and Astrophysics 245, 1991. [Author-hosted note](https://numerical.recipes/whp/fast/diagonal.pdf). Classical selected-inverse recurrences; both pages reviewed.
[^9]: Peter Carr and Sergey Nadtochiy, Local Variance Gamma and Explicit Calibration to Option Prices, Mathematical Finance, 2017, DOI 10.1111/mafi.12086. [Published author-hosted paper](https://engineering.nyu.edu/sites/default/files/2019-03/Carr-local-variance-gamma-explicit-calibration-option-prices.pdf). Sections 3.1-3.2 and 4.2 reviewed in full; not the entire calibration algorithm or appendices.
[^10]: Fabien Le Floc'h, An arbitrage-free interpolation of class C2 for option prices, 2020. [Author preprint](https://arxiv.org/html/2004.08650v1). Main text reviewed; model equations were checked against their defining differential equation rather than relying on isolated printed formulas.
[^11]: BENCHOP collaboration, Benchmarking in Option Pricing, 2015. [Original benchmark project](https://www.it.uu.se/research/scientific_computing/project/compfin/benchop/original.html) and [benchmark paper](https://www.it.uu.se/research/scientific_computing/project/compfin/benchop/original/benchop.pdf). Published American reference prices used as external checks.
[^12]: Tinne Haentjens and Karel J. in 't Hout, ADI schemes for pricing American options under the Heston model, 2013 preprint. [Author manuscript](https://arxiv.org/html/1309.0110v1). American obstacle splitting and Heston comparison context; main text and proof reviewed.
[^13]: Mark Broadie and Yusaku Yamamoto, A Double-Exponential Fast Gauss Transform Algorithm for Pricing Discrete Path-Dependent Options, Operations Research 53(5), 764-779, 2005. [Author-hosted paper](https://www.columbia.edu/~mnb2/broadie/Assets/de-fgt-opr-res_2005.pdf). Published barrier values used as a separate price-only check.
[^14]: Pricing operator investigation, Studies 20 and 22, FINDINGS_20.md and FINDINGS_22.md, 2026. [Accuracy-matched focused-polynomial comparison](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/FINDINGS_22.md).
[^15]: Justin Lars Kirkby, Frame and Fourier Methods for Exotic Option Pricing and Hedging, doctoral thesis, Georgia Institute of Technology, December 2016. [Institutional dissertation record](https://repository.gatech.edu/items/53c13dc4-e163-437e-a78d-f41645a33940). Selected spline/Gram, multi-strike, barrier and error sections reviewed; the full dissertation was not read.
[^16]: Kathrin Glau, Mirco Mahlstedt and Christian Pötz, A new approach for American option pricing: the Dynamic Chebyshev method, 2018 preprint. [Author manuscript](https://arxiv.org/html/1806.05579v1). Main text and Appendix A reviewed; not benchmarked as a full competing implementation here.
[^17]: Pricing operator investigation, Study 21, FINDINGS_21.md and RESULTS_21.json, 2026. [Finite deterministic-time book risk](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/FINDINGS_21.md).
[^18]: Pricing operator investigation, Study 24, OPTIONS_BOOK_READOUT_CALCULUS.md and FINDINGS_24.md, 2026. [Fixed-spot chain evidence and model contract](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/FINDINGS_24.md).
[^19]: Pricing operator investigation, Study 23, CROSS_OPERATOR_CALCULUS.md and FINDINGS_23.md, 2026. [Two-interval cross-operator comparison](https://github.com/danielschmitter/operator_spline_neural_representation/blob/0813266d2/research/pricing_operator_20260915/FINDINGS_23.md).
[^20]: Pricing operator investigation, Study 28, FINDINGS_28.md and RESULTS_28.json, 2026. [Three-interval accuracy and source-cost evidence](https://github.com/danielschmitter/operator_spline_neural_representation/blob/c2fcb1cdd/research/pricing_operator_20260915/FINDINGS_28.md). All twelve cases and all three predeclared source rules are retained.
[^21]: Scott B. Laprise, Michael C. Fu, Steven I. Marcus, Andrew E. B. Lim and Huiju Zhang, Pricing American-Style Derivatives with European Call Options, Management Science 52(1), 95-110, 2006, DOI [10.1287/mnsc.1050.0447](https://doi.org/10.1287/mnsc.1050.0447). Secant/tangent bounds and propagation; this is not the same forward-resolvent application.
[^22]: Benjamin Jourdain and Gilles Pages, Quantization and martingale couplings, ALEA 19, 1-22, 2022, DOI 10.30757/ALEA.v19-01. [Primary paper](https://alea.impa.br/articles/v19/19-01.pdf). Selected convex-order and quantization sections reviewed, not the entire paper.
[^23]: Pricing operator investigation, source-envelope assessment and algebra tests, 2026. [Conditional source-state derivation and algebra checks](https://github.com/danielschmitter/operator_spline_neural_representation/blob/6e801a024/research/pricing_operator_20260915/STRUCTURE_PRESERVING_SOURCE_ASSESSMENT.md). Assessment, not a measured propagation advantage or floating-point certificate.
[^24]: Stephane Crepey, Calibration of the Local Volatility in a Generalized Black-Scholes Model Using Tikhonov Regularization, SIAM Journal on Mathematical Analysis, DOI [10.1137/S0036141001400202](https://doi.org/10.1137/S0036141001400202). [Author-hosted manuscript](https://math.maths.univ-evry.fr/crepey/papers/40020SIMA.pdf). Introduction, problem formulation and selected stability/convergence sections reviewed; not all PDE proofs or appendices.
[^25]: Pricing operator investigation, Study 30, FINDINGS_30.md and complete saved measurements, 2026. [Selective confluent source compilation](https://github.com/danielschmitter/operator_spline_neural_representation/blob/8afeb00f9/research/pricing_operator_20260915/FINDINGS_30.md). All accuracy panels pass; the artifact-budget failure and unavailable final resource capture are explicitly retained.
[^26]: Nicholas J. Higham and Awad H. Al-Mohy, Computing Matrix Functions, Acta Numerica, 2010, DOI 10.1017/S0962492910000036, [author-hosted paper](https://eprints.maths.manchester.ac.uk/1451/01/covered/MIMS_ep2010_18.pdf); V. G. Kurbatov and I. V. Kurbatova, Computation of a function of a matrix with close eigenvalues by means of the Newton interpolating polynomial, 2014, [author preprint](https://arxiv.org/html/1402.4003v1). Selected matrix-function and clustered divided-difference sections reviewed; neither general algorithm is benchmarked here.
[^27]: Pricing operator investigation, interface-gamma audit, 2026. [Derivation, complete 28-case diagnostic and interpretation correction](https://github.com/danielschmitter/operator_spline_neural_representation/blob/ae72e5812/research/pricing_operator_20260915/INTERFACE_GAMMA_AUDIT.md). Existing kernels and independent references are unchanged; the audit identifies their shared right-sided convention.
[^28]: Pricing operator investigation, Study 29, FINDINGS_29.md and RESULTS_29.json, 2026. [Complete nine-window native validation](https://github.com/danielschmitter/operator_spline_neural_representation/blob/ea5118e63/research/pricing_operator_20260915/FINDINGS_29.md). All cases/windows, empirical quantile definitions, resource accounting and the compact-record target miss are preserved.