SEP 6 — Frequency-dependent electronic third-order response

SEP	6
Title	Frequency-dependent electronic third-order response
Author	John Travers
Drafting agent	GPT-6 Astra
Status	Final
Review status	Reviewed
Reviewers	Fable 5.1 (AI); Opus 5.5 (AI); John Travers (human)
Type	Feature
Created	2026-09-27
Resolution	Accepted by John Travers, 2026-10-07, after the human and AI reviews, in the SEP 6 final-review discussion; Final, approved by John Travers, 2026-10-08
Requires	SEP 3
Supersedes	—
Superseded-by	—
Implementation	Opus 5.5 (AI) for John Travers; Phases 0–4 merged from feature/sep-0006-dispersive-nonlinearity-0 to -4 (and -1c, -2c) in solprop, Phases 1 and 3 in solprop-kokkos
Manual pages	A frequency-dependent electronic response; materials: the dispersive electronic policies

Part I — Proposal

1. Abstract

Add two opt-in, frequency-dependent electronic third-order responses: a symmetric factorization built from the degenerate Kerr dispersion curve, and an evaluator of the full curated gas dispersion polynomial. Both construct one nonlinear polarization containing all retained mixing processes; neither identifies SPM, XPM, FWM or THG events during propagation. Retain the carrier-evaluated model as the initial default. The factorized model offers low cost with a measurable approximation to nondegenerate mixing; the polynomial offers exact evaluation of the stated material approximation at a larger but still FFT-based cost. Qualify numerical correctness, material validity and propagation sensitivity separately. The first target is transparent gases, especially broadband SPM and widely separated phase-matched FWM, for every gas optmat publishes a hyperpolarizability model of. Two demonstration cases anchor the comparison: deep-ultraviolet FWM in argon and vacuum-ultraviolet resonant dispersive-wave emission in helium. Silica and VUV operation expose additional data and model gaps. Success requires documented errors and costs, appropriate invariants, and parity across NumPy, Numba and Kokkos within the admitted scope; the Kokkos implementation is required, not optional.

2. Motivation and scope

Accepted; implementation not started. This proposal builds on SEP 3’s migration to curated material data, now final, and deliberately goes beyond its carrier-evaluated nonlinearity. The proposal was first audited against solprop 27f2296, optmat b7ee5a70214aaf95c095b4e336449c1d08a61307 and solprop-kokkos 87ac816190ffb0c3978de7e236b6e30d7c068ab8, and re-audited for its final review against solprop b70f2fb, optmat v0.3.0 (3f1eaa4c2e7c18f9eda31e2a6f40677763d9890e, the tag solprop pins) and solprop-kokkos fa5d57c1597e716cb8428bace7e57f53d1e14b29. The fifteen stored third-order fits are byte-identical between the two optmat revisions; v0.3.0 adds model identities and statuses to them. The calculations below are proposal experiments, not a qualified propagator or an implementation-phase report.

The motivating simulations use pumps from 343 to 2000 nm and can generate radiation from the VUV to the IR. Broadband SPM and phase-matched FWM matter more often than explicit THG. A model should represent the coupling of those bands without dividing a continuous spectrum into named processes or assigning unrelated coefficients to forward and reverse conversion.

A full-field \(E^3\) term is general in the frequency combinations it includes. It still assumes the same susceptibility for every combination. Its broad spectral support does not make its material response dispersive. Conversely, introducing material dispersion need not abandon a single broadband field: the susceptibility’s mathematical structure can make its convolution efficient.

Included here are the electronic response, its data interface, frequency-domain evaluation, field/envelope conventions, mixtures, spatial/modal integration, backend cost, provenance and qualification.

Gas scope. The dispersive evaluators are supported for every gas for which optmat publishes a hyperpolarizability model, with no gas list of solprop’s own: the scope follows optmat. At v0.3.0 that is fifteen gases — He, Ne, Ar, Kr, Xe, H2, D2, N2, O2, CO2, CH4, SF6, CO, N2O and O3 — plus mixtures of them. solprop builds twelve of them as species today; the existing air interfaces are compositions of those, not an independent nonlinear fit. SEP 3 (D13) left CO2, CO and O3 as built-in gases to separately qualified follow-up work; when solprop admits them their electronic models resolve through the same path with no change to this proposal, and §4.7–§4.8 already tabulate them. A gas whose optmat record lacks the four dispersion parameters is refused by name, never given a borrowed curve. Where each gas’s fit is supported, and where its resonances lie, are material facts that optmat publishes; how a run uses the fit beyond that support is solprop’s modelling rule (§4.10), which never refuses a run for the frequencies its field occupies: beyond a gas’s data the response is bounded and labelled. The labels differ by gas: CO2, CH4, SF6 and N2O have no curated two-photon limit, so everything beyond their data is best effort; O3 absorbs throughout its range; and air reaches O2’s two-photon resonance at wavelengths shorter than about 305 nm.

Demonstration cases. Two cases, chosen because they sit on either side of the material domain, run through qualification, the tutorial and the manual (§7, §8):

• DUV FWM in argon: a 400 nm pump and an 800 nm seed in an argon-filled capillary generate 266.7 nm, \(\omega_s=2\omega_{400}-\omega_{800}\), the phase-matched deep-ultraviolet source of Durfee et al. [9]. The generating quartet lies inside argon’s fitted frequency sums; the signal’s own Kerr and cross-phase quartets exceed them but stay within argon’s leg bound and far from its resonances, so the rule of §4.10 admits the whole case.

• VUV RDW in helium: the M2 case of the testing plan — a 7.5 fs, 275 µJ, 800 nm pulse in a 125 µm-radius capillary with helium from 0.8 bar to vacuum, emitting a resonant dispersive wave at 100–150 nm [10]. The field extends below helium’s two-photon resonance at a 120.3 nm fundamental, and such runs routinely reach below 80 nm. No off-resonant model is valid there. The dispersive evaluators still run it, with the legs bounded and that band labelled best effort (§4.10), so the comparison is a sensitivity study, not a qualified prediction (§4.7).

Raman and ionisation remain distinct physical responses, with their data migration in SEP 3 and Raman algorithms in SEP 1. This proposal does not supply a new Raman dispersion model, nonperturbative bound-electron model, ionisation law, or generic resonant complex susceptibility. It must compose correctly with them. The helium TDSE response of SEP 12, now final, owns the complete electronic response of its gas, including its cubic policy, and refuses an accompanying KerrResponse; a dispersive evaluator is likewise refused beside it for the same gas. Offering a dispersive cubic as one of the TDSE coupling’s own cubic policies is later work, not part of this proposal. SEP 2 is relevant to nonlinear absorption; independently adding its loss term is not automatically a causal completion of an arbitrary real dispersive Kerr law.

The recommendation is two supported propagation options, not a cheap model with a polynomial used only in tests. Both remain explicit choices while their useful domains and observable-specific accuracy are established.

3. Context

The current nonlinear response chapter derives the four constant-coefficient cubics: real field and envelope, each with and without the harmonic term. Relevant implementation boundaries are:

Component	Present behavior and implication
physdata/selection.py, MaterialSelection.electronic	The electronic policy of a run; ELECTRONIC_POLICIES admits "recommended" and "carrier-kerr", both the Kerr coefficient at the carrier
physdata/nonlinear_data.py, OptmatKerr and CarrierKerr	Adapts optmat’s dispersive hyperpolarizability model and evaluates its Kerr process at the carrier: one number per species and run
physdata/selection.py, MaterialManifest	A run’s material record: model identities, the electronic entries and the policies in force, which enter the resume key
nonlinear/responses.py, KerrResponse	A scalar chi3 per unit density, with thg and a species label; evaluate is the reference cubic
translate/responses.py, KerrCoefficients and _compile_kerr_components	Scalar coefficients and density paths; compatible species can be summed before the cubic
backends/numpy_ref/passes.py	Oversampled time transforms, field synthesis, responses, projection and spectral prefactors
backends/numba_cpu/backend.py, _respond_chunk and _project	Fused time-domain response work and chunked projections; extra transforms can disturb this fusion
backends/numpy_ref/backend.py, _nonlinear_general	Frequency-dependent general modal maps and regional response assembly
solprop-kokkos, src/solprop_kokkos/backend.py and its compiled stage implementation	Host translation followed by resident backend work; new spectral response arrays and workspaces must cross the versioned seam explicitly

These are paths in solprop’s src/solprop/ unless the package is named explicitly. The kernel contract, architecture and backend policy govern the implementation. At the final review the kernel schema is 23 and the output schema 15; an implementation chooses the next available versions rather than reserving numbers in this Draft.

Two later changes bear on cost and qualification. Multistep() is now the default integrator (SEP 16), and its error norm scores every bin of the run’s active band. A dispersive response changes the adaptive trajectory as well as the cost of each nonlinear call, so whole-run comparisons are made at equal achieved error with SEP 16’s work–precision measurements, not per step. The same active band — the grid’s wavelength limits with their collars — is the natural carrier of the band \(B\) of §4.5.

The existing UPPE output factor contains optical frequency. In the appropriate envelope limit this gives self-steepening. It does not supply the omitted input-frequency dependence of the material susceptibility. Similarly, a frequency-dependent effective area or modal overlap is a geometric effect: it must compose with material dispersion, not be counted as its replacement. See GNLSE and general modal propagation.

4. Theory

4.1 One susceptibility, all mixing processes

Use the optics Fourier convention of conventions, \(\widetilde E(\omega)=\int E(t)e^{+i\omega t}\,dt\). For a real, scalar field, \(\widetilde E(-\omega)=\widetilde E(\omega)^*\). Write \(\omega_3=\omega_o-\omega_1-\omega_2\). The third-order polarization is

\[ \widetilde P^{(3)}(\omega_o)=\frac{\epsilon_0}{(2\pi)^2} \iint\chi^{(3)}(-\omega_o;\omega_1,\omega_2,\omega_3) \widetilde E(\omega_1)\widetilde E(\omega_2) \widetilde E(\omega_3)\,d\omega_1\,d\omega_2. \]

For vector fields, replace the scalar susceptibility by \(\chi_{ijkl}^{(3)}\) and contract the three field components. This scalar derivation does not determine those independent tensor functions.

The names describe slices of the same integral:

Contribution	Example inputs and output
SPM	\((\omega,\omega,-\omega)\to\omega\)
XPM / pump-probe	\((\omega_r,\omega_p,-\omega_p)\to\omega_r\)
Degenerate-pump FWM	\((\omega_p,\omega_p,-\omega_i)\to\omega_s=2\omega_p-\omega_i\)
THG	\((\omega,\omega,\omega)\to3\omega\)

All permutations and inverse exchanges belong to the same response. An FFT product automatically sums them with their multiplicities. Different mixing triplets can have the same output frequency: an output-only coefficient cannot distinguish them.

A constant \(\chi_c\) leaves the convolution as \(P^{(3)}(t)=\epsilon_0\chi_c E(t)^3\). A generic unrestricted discretization requires two input sums for each output, approximately \(O(N_\omega^3)\) work for the whole polarization and potentially \(O(N_\omega^3)\) stored kernel values. This is not the proposed implementation. A finite separable expansion of rank \(R\) needs filtered fields, products and transforms, with work scaling as \(O(RN_t\log N_t)\) and workspaces that can be reused between terms.

4.2 Material normalization and the carrier approximation

For a gas, let \(N\) be number density in m\(^{-3}\) and let \(\gamma_T\) be the Taylor-convention second hyperpolarizability in C m\(^4\) V\(^{-3}\). Then

\[ P^{(3)}=\frac{N\gamma_T}{6}E^3, \qquad \chi^{(3)}=\frac{N\gamma_T}{6\epsilon_0}, \qquad \gamma_3=\frac{\gamma_T}{6\epsilon_0}. \]

The last quantity is solprop’s per-particle coefficient in m\(^5\)/V\(^2\); \(\chi^{(3)}\) is in m\(^2\)/V\(^2\). The factor of six follows from the molecular dipole Taylor expansion [1, 2]. For a transparent bulk scalar material with \(E=\mathrm{Re}(A e^{-i\omega t})\) and intensity \(I=n_0\epsilon_0 c|A|^2/2\) (conventions), \(n_2=3\chi_K^{(3)}/(4n_0^2\epsilon_0 c)\); for a dilute gas (\(n_0=1\)) this is optmat’s \(n_2=N\gamma_K/(8\epsilon_0^2c)\) [1]. Normalize a factorization using the electronic susceptibility curve, not the fourth root of \(n_2\) alone when \(n_0\) varies. SEP 3 governs the dilute-gas and local-field approximations.

Define \(\chi_K(\omega)=\chi^{(3)}(-\omega;\omega,\omega,-\omega)\). The carrier approximation uses \(\chi_c=\chi_K(\omega_0)\) everywhere. It is the zeroth-order approximation over the participating frequency region, even when the field itself is represented without a narrowband expansion.

Two separate errors matter: freezing the diagonal Kerr curve, and replacing off-diagonal mixing by that diagonal value. A narrow pump can have a small off-diagonal error for one distant FWM quartet while the Kerr coefficient of the generated band differs much more. No single bandwidth percentage is a universal error estimate.

4.3 Why an output-only curve is inadequate

The inexpensive expression

\[ \widetilde P_{\rm out}(\omega) =\epsilon_0\chi_K(\omega)\mathcal F[E^3](../\omega) \]

matches monochromatic SPM, but assigns the signal’s degenerate Kerr coefficient to every FWM pathway producing that signal. It need not improve upon the pump’s constant coefficient; §4.7 gives a concrete counterexample.

It also generally violates the exchange symmetry of the lossless scalar quartet: the coefficient changes when a different member is treated as the output. An apparent invariant obtained by reweighting energy with \(\chi_K\) would not be solprop’s physical energy. This is distinct from deriving a different field normalization and consistently transforming every operator. Do not implement an output-only material multiplier as the dispersive option.

4.4 Symmetric factorization from nonlinear data

In a transparent domain with a real, positive electronic Kerr curve, choose a positive reference \(\chi_*\) and a real even multiplier

\[ g(\omega)=\left[\frac{\chi_K(|\omega|)}{\chi_*}\right]^{1/4}. \]

Use the same function on all four legs:

\[ \chi_{\rm sym}^{(3)}(-\omega_o;\omega_1,\omega_2,\omega_3) =\chi_*g(\omega_o)g(\omega_1)g(\omega_2)g(\omega_3), \qquad P_{\rm sym}^{(3)}=\epsilon_0\chi_*G[(GE)^3]. \]

Here \(G=\mathcal F^{-1}g\mathcal F\). The input field is filtered, cubed, and the polarization is filtered once more. The result is still one broadband polarization. SPM, FWM and THG acquire their coefficients through the same four-frequency rule. The positive reference cancels out of the final kernel; using a static or carrier reference changes conditioning, not the model.

For a gas use the per-particle curve and apply \(N\) once outside the operator. Do not hide changing density in four field multipliers. A constant positive curve gives the original cubic exactly. A zero, negative or complex curve requires another factorization and is outside this first model’s domain; taking an arbitrary complex fourth root is not a proposed extension.

Matching every degenerate Kerr value does not determine the general susceptibility. This is a separability assumption, with the full permutation symmetry appropriate to the specified lossless scalar approximation. It is not a claim of frequency-independent tensor ratios or of universal Kleinman symmetry near resonances.

Generalized Miller formulas produce a related product of linear susceptibilities under anharmonic-oscillator assumptions [3]. Multiple oscillators and their nonlinear couplings need their own treatment; a measured linear index does not uniquely fix nonlinear dispersion. Gas measurements show that simple linear-dispersion estimates can substantially underestimate nonlinear dispersion [4, 5]. The proposed \(g\) is therefore fitted to or derived from curated nonlinear data. Miller scaling is an alternative model to qualify, not a substitute for that curation.

4.5 Full gas polynomial as a filtered cubic sum

Use signed vacuum wavenumber \(\nu_j=\omega_j/(2\pi c)\) in m\(^{-1}\), with \(j=o,1,2,3\), and define

\[ S_2=\sum_j\nu_j^2,\qquad S_4=\sum_j\nu_j^4, \qquad \gamma_T=\gamma_0[1+A S_2+B S_2^2+B' S_4]. \]

\(A\) has units m\(^2\) and \(B,B'\) have units m\(^4\). The curated tabulations record multipliers of \(10^{-10}\) cm\(^2\) and \(10^{-20}\) cm\(^4\), which become \(10^{-14}\) m\(^2\) and \(10^{-28}\) m\(^4\), respectively. \(\gamma_0\) must also be converted from its recorded units to SI before translation. optmat names the same two sums \(W_2\) and \(W_4\) and evaluates them in cm\(^{-1}\) [1]; \(S_2\) and \(S_4\) here are those sums in SI, \(S_2=10^4\,W_2\) and \(S_4=10^8\,W_4\) numerically.

The expansion follows the low-frequency susceptibility analysis of Bishop and De Kee [6]; their additive coefficients must not be confused with the coefficients normalized by \(\gamma_0\) here. optmat supplies the fit and its uncertainties [1]. The scalar gas fits are not a complete tensor reconstruction: the primary derivation treats diagonal components, and the molecular mean requires the curation’s additional assumptions. Calculations of tensor components show why their distinct dispersion cannot be inferred from one scalar function [7].

Define the spectral derivative operator

\[ D=-\frac{1}{(2\pi c)^2}\partial_t^2, \qquad \mathcal F[Df](../\omega)=d(\omega)\widetilde f(\omega), \qquad d(\omega)=\left(\frac{\omega}{2\pi c}\right)^2. \]

The output-frequency factors differentiate a product; input-frequency factors differentiate its individual fields. Expanding the symmetric sums and counting equal input permutations gives

\[\begin{split} \begin{aligned} \frac{6P_{\rm poly}^{(3)}}{N\gamma_0} ={}&E^3+A\{D(E^3)+3E^2DE\}\\ &+B\{D^2(E^3)+3E^2D^2E+6D(E^2DE)+6E(DE)^2\}\\ &+B'\{D^2(E^3)+3E^2D^2E\}. \end{aligned} \end{split}\]

For example, the \(6E(DE)^2\) comes from twice the three cross-products between squared input frequencies in \(S_2^2\). Grouping output multipliers gives an efficient equivalent:

\[\begin{split} \begin{aligned} \frac{6\widetilde P_{\rm poly}^{(3)}}{N\gamma_0} ={}&[1+Ad+(B+B')d^2]\mathcal F[E^3]\\ &+[3A+6Bd]\mathcal F[E^2DE]\\ &+\mathcal F[3(B+B')E^2D^2E+6BE(DE)^2]. \end{aligned} \end{split}\]

This uses three inverse transforms to obtain \(E,DE,D^2E\), and three forward transforms of product combinations. There is no three-dimensional frequency table. An \(A\)-only truncation uses two of each. A general low-rank representation could later extend the same architecture, but is not required for this polynomial.

This is exact evaluation of the polynomial on the resolved, dealiased grid. The polynomial is itself a truncated off-resonant material model. Its growth at large frequency is not evidence of a physical divergence or a valid VUV response. Do not use finite differences for \(D\) or extrapolate the polynomial onto arbitrary padded FFT frequencies.

The cut-off must be symmetric. The oversampled fine grid reaches three times the coarse band and the cubic’s products land there, so the weights outside the admitted domain need a stated value. Zeroing the output multipliers alone destroys the four-leg symmetry of the kernel, and with it the lossless identity of §4.9. The consistent rule is one band projection \(\Pi_B\) applied identically to the field entering every product and to the polarization leaving it: the restricted kernel \(\Pi_B(\omega_o)\Pi_B(\omega_1)\Pi_B(\omega_2)\Pi_B(\omega_3)\chi^{(3)}\) is still fully symmetric. The same rule applies to the symmetric factorization, whose \(g\) is likewise evaluated only inside \(B\). Which band \(B\) is admitted is the domain question of §5 and Q2; the projection is the numerical consequence, and §7 tests it.

Two bands must not be confused. \(B\) is a numerical band: the run’s active band, outside which the field is negligible and the projection costs nothing physical. The material support is where the fit is qualified. Inside \(B\) but outside the support, §4.10’s rule bounds the legs or refuses. Shrinking \(B\) to the support instead would silently remove the nonlinear coupling of occupied bins — the generated signal in both demonstration cases of §2 — and is not an admissible substitute for that rule.

4.6 Why the symmetric approximation can work well

Normalize by \(\gamma_0\). On the Kerr diagonal,

\[ \frac{\gamma_K(\nu)}{\gamma_0} =1+4A\nu^2+(16B+4B')\nu^4. \]

Expanding its fourth root, \(g(\nu)=1+A\nu^2+(4B+B'-3A^2/2)\nu^4+O(\nu^6)\). Multiplication of four such factors gives

\[ \frac{\gamma_{\rm sym}-\gamma_{\rm poly}}{\gamma_0} =\left(B-\frac{A^2}{2}\right)(4S_4-S_2^2)+O(\nu^6). \]

Thus the factorization agrees through second order for all quartets and exactly on the Kerr diagonal. The leading error measures the spread of the four squared frequencies, since \(4S_4-S_2^2=\sum_{j<k}(\nu_j^2-\nu_k^2)^2\). The exact fourth-root construction also contains higher-order terms; this expansion does not bound its UV error.

Another diagnostic is the quartet’s effective frequency \(\nu_{\rm eff}=\sqrt{S_2/4}\). Evaluating \(\gamma_K(\nu_{\rm eff})\) differs from the polynomial only by \(\gamma_0 B'(S_2^2/4-S_4)\). This can be very accurate, but \(\nu_{\rm eff}\) belongs to the whole quartet. There is no unique effective frequency to assign to an output bin receiving many triplets. It is a useful interpretation of the polynomial, not a justification for an output-only multiplication.

4.7 Numerical scale of the approximations

The following calculations use the optmat fits [1], at the audited revision, of all fifteen gases in scope (§2). Ratios cancel density and the Taylor normalization. They measure disagreement between models, not experimental accuracy. No literature coefficient table is copied into solprop by this proposal.

First, the variation of the degenerate curve itself is substantial. The third column onward compares nondegenerate quartets as described below.

Gas	\(\gamma_K(400\,\mathrm{nm})/\gamma_K(2000\,\mathrm{nm})\)	Symmetric median (%)	Symmetric 95th pct. (%)	Symmetric max. (%)	Output-only median (%)	Output-only 95th pct. (%)	Output-only max. (%)
He	1.1195	0.006	0.063	0.124	0.914	4.622	6.907
Ne	1.1498	0.007	0.079	0.153	1.128	5.745	8.621
Ar	1.3708	0.053	0.494	0.913	2.432	13.700	21.206
Kr	1.5414	0.115	0.966	1.714	3.197	19.573	30.897
Xe	1.8673	0.260	1.921	3.212	4.467	30.224	48.560
H2	1.4198	0.047	0.401	0.718	2.765	15.230	23.493
D2	1.4114	0.046	0.399	0.717	2.718	14.957	23.058
N2	1.3414	0.047	0.450	0.837	2.275	12.689	19.584
O2	1.5476	0.183	1.611	2.908	2.942	20.357	32.781
CH4	1.6282	0.133	1.057	1.831	3.622	22.320	35.321
SF6	1.2516	0.105	1.112	2.141	1.371	10.120	16.394
N2O	1.4441	0.048	0.378	0.656	2.175	10.798	18.890
CO2	1.4428	0.082	0.728	1.325	2.758	16.243	25.408
CO	1.4233	0.019	0.133	0.208	2.895	15.120	23.054
O3	1.0749	0.036	0.193	0.360	0.829	3.172	4.493

Reproduction and interpretation. NumPy default_rng(4) draws a (300000, 3) array of independent wavelengths uniformly on 400–2000 nm. Convert to SI vacuum wavenumbers and form \(\nu_o=\nu_1+\nu_2-\nu_3\). Keep positive outputs in the same wavelength interval and \(S_2\) no larger than that gas’s largest fitted value, with no 10% warning margin. This leaves 208109 quartets for each gas except N2O (175846) and O3. O3’s fit is declared only from 700 nm, so its row draws and keeps wavelengths on 700–2000 nm instead (204331 quartets), and its diagonal ratio is \(\gamma_K(700\,\mathrm{nm})/\gamma_K(2000\,\mathrm{nm})\). For each retained quartet compute the polynomial, the geometric mean of the four degenerate Kerr values, and the Kerr value at the output. Errors are \(100|\gamma_{\rm candidate}/\gamma_{\rm poly}-1|\); percentiles use NumPy’s default linear interpolation. Python 3.14.5 and NumPy 2.5.2 were used; the final review reproduced every earlier row exactly from the stored v0.3.0 fits and added the CO2, CO and O3 rows by the same procedure. The diagonal ratio column is a separate curve evaluation: in particular, N2O’s 400 nm value exceeds its fitted-\(S_2\) support and rests on borrowed dispersion assumptions [1]. The quartet columns enforce the stated cutoff.

This is neither a uniform-frequency sample nor a pulse-weighted distribution, and the sampled maximum is not a worst-case bound. Fitted-\(S_2\) admission alone does not certify a previously unmeasured process. A map of fit uncertainty and primary-data support is needed alongside these comparisons.

For selected degenerate-pump FWM quartets, define \(\omega_s=2\omega_p-\omega_i\). The error columns below are signed relative to the polynomial. The ratio column shows how much the actual quartet coefficient differs from the pump’s degenerate Kerr value.

Pump / idler / signal (nm)	Gas	\(\gamma_{\rm FWM}/\gamma_K(\omega_p)\)	Output-only error (%)	Symmetric error (%)
800 / 2000 / 500	He	1.0052	+4.130	+0.028
800 / 2000 / 500	Ar	1.0139	+12.005	+0.237
800 / 2000 / 500	Xe	1.0266	+25.777	+1.023
800 / 2000 / 500	N2	1.0130	+11.134	+0.213
1030 / 2000 / 693.603	Ar	1.0053	+5.200	+0.066
1030 / 2000 / 693.603	Xe	1.0094	+10.045	+0.325

All frequencies in these rows lie in the declared fundamental-wavelength window, and their \(S_2\) values are below fitted maxima. For the argon 800 nm case, the frozen pump coefficient is only about 1.37% low for this quartet, whereas substituting the signal’s Kerr coefficient makes it 12% high. This directly answers why a frequency-dependent output coefficient can be worse than the constant approximation.

Two tempting extensions illustrate the validity trap:

Pump / idler / signal (nm)	Formal calculation	Why it is not qualified
515 / 2000 / 295.552	Ar: output-only +46.595%, symmetric +0.951%	The signal’s degenerate curve, needed by both constructions, is below its declared 400 nm range; a small symmetric error against the polynomial is not a physical validation
343 / 1030 / 205.760	Ar: \(S_2/S_{2,\max}=1.453\); Xe: 1.650	The pump itself is below the declared range and these quartets also exceed fitted frequency sums

The first row does not prove the polynomial is invalid solely because its output is below 400 nm: the published process range is a fundamental range, not a universal cutoff on every output. It proves that a scalar wavelength window and one \(S_2\) threshold are insufficient descriptions of arbitrary quartet validity. A factorization sampling a degenerate curve at each leg has different data needs from a process fit evaluated at its fundamental.

The demonstration cases. The same comparison for the two cases of §2, with the carrier coefficient \(\gamma_K(\omega_c)\) added. Errors are signed relative to the polynomial. \(S_2/S_{2,\max}\) measures the quartet against the fitted sums; a ratio above one is beyond the data. The last two columns anticipate §4.10: the polynomial under its leg bound, as a ratio to the unbounded polynomial, and whether the quartet’s frequencies lie inside the gas’s band edge.

Quartet (nm)	Gas	\(\gamma_{\rm poly}/\gamma_K(800\,\mathrm{nm})\)	Carrier error (%)	Symmetric error (%)	Output-only error (%)	\(S_2/S_{2,\max}\)	Leg-bounded / polynomial	Inside band edge
400 + 400 − 800 → 266.7	Ar	1.345	−4.25 at 400; −25.65 at 800	+0.556	+50.00	0.98	1	yes (189.4 nm)
SPM of 266.7	Ar	2.018	−36.17 at 400; −50.44 at 800	0	0	1.97	1	yes
XPM of 266.7 by 400	Ar	1.612	−20.13 at 400; −37.98 at 800	−0.025	+25.13	1.42	1	yes
THG of 800 → 266.7	He	1.060	−5.66 at 800	+0.311	+20.60	0.32	1	yes (120.3 nm)
250 + 250 − 1500 → 136.4	He	1.485	−32.68 at 800	−0.012	+88.95	1.49	0.949	yes
200 + 200 − 600 → 120.0	He	1.785	−43.96 at 800	−1.229	+106.15	2.11	0.878	no, 120.0 < 120.3 nm
XPM of 120 by 800	He	1.963	−49.06 at 800	−2.284	+87.43	2.46	0.771	no
SPM of 120	He	3.679	−72.82 at 800	0	0	4.81	0.628	no; its pair reaches 1.002 \(E_2\)
SPM of 80	He	11.27	−91.13 at 800	0	0	10.81	0.205	no; past \(E_2\)

In argon the generating quartet is inside the fitted frequency sums. The pump’s own Kerr coefficient is only 4% low for it, whereas a carrier at the 800 nm seed is 26% low; the output-only construction is 50% high, and the symmetric model is within 0.6% of the polynomial. The signal’s own SPM and cross-phase quartets exceed the fitted sums, but every leg stays within argon’s leg bound (215.9 nm), so the bounded polynomial is the polynomial itself, a lower bound on \(\gamma\) there (§4.10). Every frequency is far inside argon’s band edge: two 266.7 nm photons carry 9.3 eV, below its first excited states, whose resonance lines are at 104.8 and 106.7 nm [11]. The whole case is admitted by default.

In helium the M2 case’s harmonic sector is inside the fitted sums, and the fit is exact for legs down to the leg bound (151.9 nm, the third harmonic of Bishop and Pipin’s 455.6 nm calculation [12]). But the RDW band, and a run’s field more generally, reaches past helium’s band edge: the degenerate two-photon \(1s^2\to1s2s\,{}^1S\) resonance at a 120.3 nm fundamental. Helium’s other low-order resonances follow from NIST’s levels [11, 12]: three-photon \(1s2p\,{}^1P\) and \(1s3p\,{}^1P\) resonances at 175.3 and 161.1 nm, and three-photon ionisation below 151.3 nm, which agree with SEP 12 §4.4. A field with frequencies beyond the edge has an SPM quartet past the resonance, which no resonance-free expansion represents. The dispersive evaluators run the case anyway, with each leg held at the leg bound (§4.10) and the band beyond the edge recorded as best effort. The comparison measures how sensitive the RDW’s wavelength and energy are to the treatment of its VUV legs: held at the 800 nm value by the carrier model, or raised along the fit to the leg bound and held there by the dispersive ones. It is reported as that sensitivity, never as a qualified prediction. A resonant helium response, such as SEP 12’s TDSE coupling, is the physical reference there.

Pump dynamics and VUV emission in one response. An RDW is emitted through four-wave mixing: three components of the broadened soliton produce a VUV photon. Neither dispersive evaluator gives a band one coefficient, so modelling the emission does not cost accuracy at the pump. The soliton’s own quartets — SPM at 800 nm, and its broadened wings down to about 250 nm at maximum compression — lie inside helium’s fitted data and are evaluated exactly. The SPM quartet at 800 nm keeps \(\gamma_K(800\,\mathrm{nm})\), exactly the carrier value. A generating quartet such as 250 + 250 − 1500 → 136.4 nm takes a coefficient set by all four of its legs: 1.41 times the 800 nm value under the leg bound (1.49 unbounded), where the carrier model is about 30% low. In the symmetric model each leg contributes \(\gamma_K^{1/4}\), so the single VUV leg enters only as a fourth root. The largest coefficients belong to quartets with two or more VUV legs — cross-phase modulation of the RDW by the pump (1.51), its own SPM (2.31) — and they scale with the weak RDW field, so they hardly act back on the pump. The RDW’s wavelength is set by phase matching to the soliton’s nonlinear wavenumber, which comes from the exactly evaluated SPM, so it is expected to move little between the models. The conversion efficiency, and the RDW’s evolution after emission, are what the comparison tests. This is the output-only construction’s failure (§4.3) avoided: that construction would give every quartet producing a VUV photon the VUV leg’s degenerate value.

These rows reuse the §4.7 procedure on the stated quartets. The quartets with a VUV leg are illustrative members of a broad soliton spectrum, not a weighting of it.

4.8 Material validity, causality and missing data

The fits of fourteen of the fifteen gases in scope declare fundamental wavelengths from 400 nm to 20 µm [1]; O3’s declares 700 nm. Their fitted frequency sums and independent evidence differ by gas; optmat’s 10% warning margin is a software warning policy, not an accuracy bound. 343 nm pumping is already outside that declared interval. “VUV–IR” is the desired application range, not this Draft’s qualified material range.

The harmonic sector has a different domain in each evaluator. When the third harmonic is retained, the polynomial evaluates the THG quartet \((\nu,\nu,\nu)\to3\nu\) at \(S_2=12\nu^2\), which stays within the fitted frequency sums down to the pump wavelengths below [1]. The symmetric factorization instead samples the degenerate curve on the output leg at \(3\nu\), so it needs \(\gamma_K\) at one third of the pump wavelength, which is inside the declared 400 nm range only for pumps of 1200 nm or longer, and inside the fitted Kerr sums, \(4(3\nu)^2\le S_{2,\max}\), only for the pumps of the third column.

Gas	Shortest pump whose THG quartet is within fitted \(S_2\) (nm)	Shortest pump whose harmonic leg is within fitted Kerr \(S_2\) (nm)	Shortest pump whose harmonic leg is within the declared Kerr range (nm)
He	456	789	1200
Ne, Ar, H2, N2, SF6	648	1122	1200
D2, O2, CH4	674	1167	1200
Kr, Xe, CO2	690	1196	1200
CO	695	1203	1200
N2O	1150	1992	1200
O3	1203 (resonant; optmat gives no THG)	2084	2100

The second and third columns are Kerr-diagonal criteria, which §4.10 shows are too strict: the data reach harmonic legs well beyond the Kerr diagonal of the fitted sums. Under §4.10’s leg bound both evaluators share the first column, since \(3\nu\le\nu_c\) exactly when \(12\nu^2\le S_{2,\max}\). For pumps at or above it the harmonic sector is evaluated without bounding; the symmetric model’s \(g\) then samples the Kerr curve up to \(3S_{2,\max}\), a polynomial extrapolation of the diagonal whose departure from the polynomial is measured in Phase 0. For shorter pumps the harmonic legs are bounded. O3’s third harmonic falls in its Hartley and Huggins absorption bands for fundamentals up to about 1050 nm [1], so its harmonic sector is resonant whatever the evaluator. The harmonic-sector decision of §4.9 and Q4 still precedes acceptance.

The relevant gaps are:

Gap	Needed before the corresponding claim
Arbitrary-quartet validity	Largely closed by optmat’s leg support and resonance limits (§4.10); its public record (OEP 5) must be released, and polyatomic two-photon limits curated
UV/VUV electronic response	Curated measurements or calculations of nondegenerate response, truncation error and, where needed, its imaginary part
Nonlinear absorption and causality	Compatible dissipative/resonant terms; a real polynomial and an independently chosen loss fit are not automatically one causal response
Tensor dispersion	Independently supported isotropic tensor components or a named scalar-times-tensor approximation
Molecular electronic separation	Preserve the residual vibrational contamination and borrowed-prior qualifications of the curated fits; do not double count them as new exact electronic data
Silica	A coherent electronic/Raman split and evidence for off-diagonal susceptibility, beyond a degenerate \(n_2(\omega)\) curve

A real even \(g\) gives a symmetric, self-adjoint band operator. It is not a globally causal absorbing material law. The polynomial is likewise a low-frequency expansion, not a resonant susceptibility. Near resonances, a causal pole/auxiliary-oscillator model or a more general symmetry-constrained low-rank response may be appropriate. Those require additional data and a separate derivation; neither scalar extension here promises them.

Optmat’s gas fits carry different absolute-scale, statistical and systematic uncertainties [1]. For example, several molecular fits retain inseparable vibrational contributions, and N2O’s dispersion is constrained partly by priors. Being exact relative to those fits does not remove those uncertainties. The silica Kerr curve is useful input for a candidate symmetric approximation, but does not uniquely supply its nondegenerate electronic response. Do not manufacture silica polynomial coefficients from that curve alone.

4.9 Lossless consistency and the meaning of THG selection

For a periodic or decaying real field and a real, even, self-adjoint \(G\), which commutes with \(\partial_t\), the proposed symmetric response obeys

\[ \int (\partial_t E)P_{\rm sym}\,dt =\epsilon_0\chi_*\int \partial_t(GE)(GE)^3\,dt=0. \]

By integration by parts this also gives zero net \(\int E\partial_tP\,dt\). It is a necessary lossless-work identity. The polynomial obeys the same identity because all four frequency legs enter symmetrically and their signed sum is zero. A test must use dealiased products and compatible projection. Temporal windows and absorbers can introduce physical or numerical losses; disable them for this test and account for them separately in a run.

An independent discrete convolution check used 128 samples, Fourier bins \(\pm3,\pm7,\pm11\), Hermitian random amplitudes and dimensionless \(d=(k/10)^2\), \(A=0.05\), \(B=0.02\), \(B'=-0.003\). The polynomial FFT expression agreed with the explicit 216 input-triplet sum at relative \(L^2\) error \(4.3\times10^{-15}\). The normalized work residual was below \(7\times10^{-17}\) for the constant, symmetric and polynomial forms, versus about \(1.5\times10^{-3}\) for the output-only form. These synthetic coefficients check algebra; they are not material recommendations or test tolerances.

Broadband process selection requires another careful distinction. Let \(u=E_a\) be the optical analytic signal, so \(E=(u+u^*)/2\), and let \(\Pi_+\) retain strictly positive optical frequencies. Ignoring a DC component, the analytic signal of the full cubic is

\[ 2\Pi_+[E^3]=\frac14\Pi_+[u^3+3u^2u^*+3u(u^*)^2]. \]

The last term can have positive-frequency support for a sufficiently broad field. It must not be discarded merely because a conventional narrowband envelope derivation would put it outside the band. Analytic-signal formulations make this issue explicit [8]. The same projection must be derived after the electronic input filters, or term by term for the polynomial.

The existing field-grid thg=False rule is \(3|E_a|^2E/4\), whereas the envelope rule keeps \(3|A|^2A/4\). Their documented narrowband parity does not establish arbitrary-band parity or lossless exchange. A direct diagnostic with dimensionless \(E(t)=\cos t+0.2\sin3t\) gives \(\langle E' E^3\rangle=0\) but \(\langle E'(3|E_a|^2E/4)\rangle=-0.075\). Evaluating the current KerrResponse on 128 equally spaced samples reproduces those values to roundoff after removing its constant multiplier. The partial cubic retains part of inverse harmonic exchange without its forward partner. This is a qualification gap in the existing switch when those bands are occupied, not a reason to change unrelated operators in this documentation PR.

For the new dispersive response, recommend defining harmonic exclusion by retaining the two-positive/two-negative quartets, consistently for every output. In analytic variables the scalar number-conserving cubic is then \(\Pi_+[3|u|^2u/4]\); apply the corresponding symmetric kernel to each quartet. The complete response retains all allowed quartets, including harmonic and inverse harmonic terms. There is still no runtime SPM/FWM classification. This choice and its relationship to the existing thg interface must be settled before acceptance (Q4). Until derived and qualified, a backend refuses the unsupported new response/grid/THG combination.

For full lossless electronic propagation test physical energy, but do not assert conservation of total photon number across THG. For an explicitly number-conserving four-wave approximation test its photon invariant as well. Raman and plasma require their own energy-transfer accounting. The discrete UPPE normalization and modal projections must preserve the relevant continuum identity; the local work identity alone does not prove an entire solver correct.

4.10 Beyond the fitted data: a leg bound, never a refusal

A run’s field routinely extends far beyond any gas’s data: helium RDW grids reach below 80 nm. The carrier model runs there today with no domain at all, a constant value extrapolated from the carrier to every quartet. The rule for the dispersive evaluators is therefore: never be less physically sound than the carrier, and never refuse a run for the frequencies its field occupies. optmat’s analysis of where its fits are supported [12] gives three facts that make such a rule possible.

• The legs of the data. The four signed legs of a quartet sum to zero, so by the Cauchy–Schwarz inequality \(\nu_j^2\le\tfrac34 S_2\), with equality for the third-harmonic output. No quartet within the fitted sums has a leg beyond \(\nu_c=\sqrt{3S_{2,\max}/4}\): 151.9 nm for He; 215.9 nm for Ne, Ar, H2, N2 and SF6; 224.7 nm for D2, O2 and CH4; 230.1 nm for Kr, Xe and CO2; 231.5 nm for CO; 383.4 nm for N2O; 401.0 nm for O3. The largest fitted single legs are harmonic legs at 229–244 nm (151.9 nm for He), 1.6–1.7 times further out in wavenumber than the Kerr diagonal of the fitted sums.

• The resonances. In the sum over states a single leg reaches the lowest dipole-coupled level \(E_1\), and a pair sum \(|\nu_a+\nu_b|\) the lowest level two dipole steps away, \(E_2\). Every frequency \(\nu\) of a field has an SPM quartet whose pair sum is \(2\nu\), so a field stays clear of both only while it occupies wavelengths longer than the band edge, \(\min(E_1, E_2/2)\): He 120.3 nm; Ne 133.5 nm; Ar 189.4 nm; Kr 216.7 nm; Xe 256.0 nm; H2 201.7 nm; D2 201.1 nm; N2 290.1 nm; O2 304.6 nm; CO 308.9 nm. These are for a field of one linear polarisation. A vector field also reaches the even \(J=1\) levels of the atoms and CO’s \(\Sigma^-\) state, which move the edges of Ne, Ar, Kr, Xe and CO by 1–3 nm. No two-photon state of CO2, CH4, SF6 or N2O is curated, and O3 absorbs throughout its range, its Chappuis and Wulf bands reaching 1047 nm.

• Monotonicity. Every fit has \(A>0\), \(B\ge0\) and \(B+B'>0\), so it rises with each \(\nu_j^2\) at least as fast as \(\gamma_0A\), inside its data and beyond. No theorem makes physical \(\gamma\) do the same [2], but every curated calculation rises toward the first resonance, and faster than the polynomial: beyond the data the polynomial is low by 0.7–3.6% at about twice \(S_{2,\max}\) for Ne, Ar and N2, about 5% for H2 and D2, and 28% for helium’s THG at four times it. The rise fails above a two-photon resonance: xenon’s \(n_2\) changes sign at 248.4 nm.

The rule. The rule is solprop’s modelling decision, not a material property: optmat supplies the facts it rests on. Bound every leg of both evaluators at the gas’s \(\nu_c\): \(\nu_j\to\min(|\nu_j|,\nu_c)\). For the polynomial this replaces the multiplier \(d(\omega)\) of §4.5 by \(\min(d(\omega),\nu_c^2)\); for the symmetric model it evaluates \(g\) at \(\min(|\nu|,\nu_c)\). Both remain real, even and identical on all four legs, so the lossless identity of §4.9 and the FFT cost are unchanged. The bound divides every run’s band into three regions, which are recorded with the run and can be marked on its outputs:

Region	Where	What the evaluators give	Standing
Supported	Every quartet within \(S_{2,\max}\), and every quartet whose legs are all within \(\nu_c\)	The exact model: the bound never touches a quartet within the fitted sums	Exact within the fit; beyond \(S_{2,\max}\) a polynomial extrapolation that is a lower bound below resonance
Bounded	Legs beyond \(\nu_c\), field inside the band edge	Each such leg held at \(\nu_c\)	A lower bound on \(\gamma\). Against carriers at 400, 800 and 1030 nm it is never worse within the data, and is shown no worse beyond them for every quartet except a few dozen Xe and CO quartets that mix legs either side of a 400 nm carrier, and N2O against a 400 nm carrier, itself beyond N2O’s single fitted value
Best effort	Field beyond the band edge, or beyond the data of a gas with no curated two-photon limit	The same bounded evaluation	No off-resonant model is valid here. The response is held at the gas’s value at the edge of its data: the constant extrapolation the carrier model makes everywhere, anchored at the last supported leg instead of at the carrier, and never the polynomial’s \(\nu^4\) growth. Recorded and labelled, never refused

The helium SPM quartet at 80 nm shows the difference: the raw polynomial gives 11.3 times the carrier’s coefficient, the bounded evaluation 2.31 times (§4.7). Refusing the best-effort region would push such runs back onto the carrier model. That model is worse wherever the bound is shown better, and no better founded beyond the edge. A resonant response — SEP 12’s TDSE coupling for helium, or a future resonant model — is the physical route there.

Three other bounds were evaluated and rejected [12]:

• The Kerr diagonal of the fitted sums. It changes up to 27 fitted values of a gas, by up to −26%, and is worse than a 400 nm carrier for 11–16% of the quartets within the fitted sums: −9.3% for the argon DUV quartet of §4.7, against the carrier’s −4.3%.

• The largest fitted leg. It is exact on the data but moves quartets within the fitted sums by up to −5.4% (Xe).

• The band edge as a leg bound. It cuts the data of Xe, N2, O2 and CO. A pair resonance is reached by two legs together, so it is not a per-leg limit at all.

The raw polynomial beyond the data is not offered: its \(\nu^4\) growth through the resonances is not a physical picture.

Ownership. optmat states the material facts as well as it can, in the support record of each hyperpolarizability model proposed by its OEP 5 [13]: the largest fitted \(S_2\) and the legs of the fitted data, the levels \(E_1\) and \(E_2\) for one and for any polarisation, the ionisation energy, and their sources and uncertainties. solprop never hard-codes them. From them solprop derives \(\nu_c\), the band edges and the regions, and it alone sets the rule: bounding beyond the data, labelling, and never refusing are modelling choices. The figures above are optmat’s at the cited revision, quoted for this proposal’s argument. The record carries its own identity, which enters the run’s manifest and resume key. Phase 1 needs an optmat release that provides it.

5. Proposed interface

Extend SEP 3’s material-backed electronic selection. MaterialSelection.electronic exists and ELECTRONIC_POLICIES holds "recommended" and "carrier-kerr"; add "symmetric-kerr" and "gas-polynomial". "recommended" keeps meaning "carrier-kerr" until Q6 is decided. The two new values are proposed names, not APIs present in this Draft:

selection = MaterialSelection(electronic="gas-polynomial")  # or "symmetric-kerr"
argon = species("Ar", selection=selection)
fill = Mixture.fill({argon: 1.0}, 3.0, temperature=294.0)
run = prop_capillary(radius, length, "Ar", 3.0, lambda0=..., materials=selection)

Both dispersive options resolve optmat’s hyperpolarizability model of the gas, with its identity, units, density convention and domain. gas-polynomial is refused for a material without that model; it does not silently turn into a diagonal fit. A positive real supplied electronic susceptibility curve can support the symmetric model under its declared assumptions. The existing explicit scalar KerrResponse remains a constant-coefficient response; it does not acquire dispersion merely because a built-in gas shares its label.

Reference wavelength selection continues to matter for the carrier model: MaterialSelection.carrier_wavelength remains the wavelength of the carrier Kerr coefficient, and of the Raman pump dispersion and the ionisation rate under every electronic policy. For the dispersive choices, changing the envelope carrier while holding the same physical spectral samples must not change the material response. A reference used to normalize \(g\) is numerical metadata, not a new physical choice. Polynomial coefficients and filters are evaluated at absolute optical frequencies, including the sign treatment required by the representation.

The resolved response should carry enough information to identify the data model, evaluation strategy, tensor approximation, material state convention, support record and the regions of §4.10. The draft does not require a general public susceptibility-tensor array API. A structured material response is preferable to adding several unrelated coefficient arguments to KerrResponse.

Validate the requested active spectral support including collars, not only the pump or the full padded transform array. Padded zeros do not justify querying the material model outside its domain. Static/DC limits, the Nyquist bin, Hermitian completion and any permitted interpolation are explicit. Validation must consider the interactions retained by the selected operator: a valid pump alone cannot certify every output. The leg bound of §4.10 is the only modification of the model, and it is recorded. No silent clipping, blanket warning suppression, or automatic model switching is proposed.

Record the extent of each region with the output, and distinguish permission to evaluate a formula from physical qualification. The region check is made once, on the active band at translation, never per quartet in the hot loop.

§4.10 replaces a selectable extrapolation policy with one rule. Beyond a gas’s data the legs are bounded, and the run is never refused for the frequencies its field occupies. What a run records is the region each part of its active band falls in — supported, bounded or best effort — with the support record’s identity; these enter the manifest and the resume key, and outputs and plots can mark the regions. A run whose field enters the bounded or best-effort region logs that once, naming the gas and the band, as SEP 3’s range notices do; it does not warn per step. A vector run takes the any-polarisation band edge.

6. Implementation strategies

Ownership and numerical representation

Optmat owns material coefficients, source selection, covariances, validity and identity, including where each model is supported and where the material resonates. solprop owns the derived polarization operator, its numerical execution and the modelling rule for using a model beyond its support (§4.10). SEP 3’s adapters resolve the public records; no database or fit implementation is copied into solprop. Optmat callbacks and Python objects never enter a backend stage.

The physics layer defines reference operators. Translation constructs spectral weights on the actual fine grid, response descriptors and density paths. The seam carries only supported, validated numerical representations. A new spectral response bundle, rather than an overloaded scalar KerrCoefficients, makes the changed scheduling explicit. The first implementation phase that changes these fields bumps SCHEMA_VERSION, its literal contract tests and the changelog. An out-of-tree backend checks capability and schema by name.

The compiled implementation uses preallocated buffers, planned transforms and deterministic sums. It keeps optmat identities on the host and response arrays resident where the backend executes. Every additional field/product workspace appears in the memory ledger. No new third-party numerical dependency is proposed: the existing FFT and array facilities suffice.

Response composition and mixtures

The evaluators reach a run through the existing route: the translation layer asks each species’ electronic model for weights on the fine grid, exactly as it now asks OptmatKerr for one carrier value. A run whose electronic response is SEP 12’s TDSE coupling refuses a dispersive policy for the same gas (§2).

For a pure gas with only electronic Kerr, the symmetric multiplier can be folded into the existing input spectrum and output polarization. It need not add any FFTs. If Raman or plasma also needs the unfiltered physical field, both fields must exist or be recomputed, and the electronic polarization may need a separate output transform/projection before summation. Applying \(g\) to the total polarization would spuriously alter Raman and plasma.

For a mixture, in general

\[ P_{\rm sym}=\sum_s\frac{N_s\gamma_{*,s}}{6} G_s[(G_sE)^3]. \]

Different \(G_s\) prevent the current reduction to one scalar \(\sum_sN_s\gamma_{3,s}\), and independent filters cost approximately one response path per species. The recommended alternative is one filter built from the mixture’s own diagonal curve, \(\gamma_{K,\rm mix}(\nu)=\sum_sN_s\gamma_{K,s}(\nu)\), factorized exactly as in §4.4. Its separability assumption is the pure-gas one, it keeps a single response path, and its error against the per-species polynomial sum is measured by the procedure of §4.7. The filter depends on the composition, not only on the total density: where composition varies along \(z\) it must be rebuilt per region or per stage, and the recorded model identity names the composition it was built for. Any other approximate common filter needs its own demonstrated error and recorded identity.

The polynomial has a useful mixture advantage. When species share the same spatial field and scalar operator, form the four stage coefficients \(\sum_sN_s\gamma_{0,s}(1,A_s,B_s,B'_s)/6\) and use the same \(E,DE,D^2E\). Thus its FFT count need not grow with species count. Keep regional/spatially varying coefficients at their proper nodes; the ability to sum four coefficients does not make different material regions uniform.

Geometry and transform counts

Setting	Symmetric strategy	Polynomial strategy
Scalar uniform material, electronic response only	One inverse and one forward transform, as for the constant cubic; two spectral multiplications	Three inverse and three forward transforms with grouped products
Uniform legacy modal material	A common scalar filter commutes with frequency-independent synthesis/projection; transform modal columns and reuse chunking	Derivative fields can likewise be transformed on modal columns; extra synthesized fields and product projections remain
General modal maps / regional materials	Respect frequency-dependent synthesis and region-specific filters; a common scalar multiplier can commute only where genuinely common	Share derivative work where possible, but retain regional tensors/coefficient paths and the existing frequency-dependent maps
Radial, Cartesian, planar or line spatial routes	Filter temporal spectra at each material field point; existing spatial work remains	Additional temporal transforms and derivative/product buffers at each field point
Raman/plasma and multiple distinct gas filters	Preserve bare fields and separate source assembly; cost can exceed the isolated two-transform estimate	Reuse the bare field, but retain all electronic product channels and unfiltered other sources

Complex envelope transforms, positive-frequency projections and harmonic selection change the constants in this table. Six real transforms is the simple scalar full-field polynomial count, not a promise for every route. Likewise, a common filter does not imply that all regional maps commute.

With \(C\) transformed columns and \(N_t\) fine-grid samples, both methods remain \(O(CN_t\log N_t)\) for fixed response rank. The polynomial needs at least the additional derivative fields or an equivalent recomputation schedule; temporary products can be streamed. Chunk sizes should follow the actual memory ledger, not a fixed multiplier on current allocations. For many modes, extra dense synthesis/projection work can matter more than the FFT count. On GPUs, memory bandwidth, batch size and transform plans must be measured; CPU ratios cannot be transferred to device runs.

Exploratory timings and what they establish

An initial isolated CPU experiment measured the following medians per cubic polarization evaluation. These are not solprop propagation benchmarks.

Real field shape	Constant (µs)	Symmetric (µs; relative)	Polynomial (µs; relative)
(1, 4096)	20.0	25.0; 1.26×	78.6; 3.94×
(8, 16384)	1008.3	1078.9; 1.07×	3311.1; 3.28×

The experiment used SciPy real FFTs with one worker, NumPy products and allocated temporaries, single-threaded external math libraries, warm-up calls, and the median of seven groups of twenty evaluations. It excludes material loading, propagation, modal projection, Raman, plasma and adaptive stepping. Those figures establish plausible kernel-scale cost, not production speed.

A repeat during drafting with all fixed spectral weights precomputed gave 23.1/24.0/79.8 µs for the first shape and 1023.2/1109.1/3440.6 µs for the second (constant/symmetric/polynomial). The corresponding relative costs are 1.04/3.45 and 1.08/3.36. The variation cautions against promising an exact percentage from a small microbenchmark.

For reproducibility, the repeat used Python 3.14.5, NumPy 2.5.2 and SciPy 1.18.1; default_rng(42) generated standard-normal real arrays in the shape order above, transformed once before timing. It used dimensionless \(d_k=(k/(N_t/2))^2\), \(A=0.05\), \(B=0.02\), \(B'=-0.003\), \(g_k=[1+4Ad_k+(16B+4B')d_k^2]^{1/4}\), and the grouped formula in §4.5. Each function was warmed five times. Spectral weights were constructed outside the timed call; field/product arrays were allocated inside it. Each of seven samples used twenty calls timed with perf_counter_ns. The synthetic weights measure the arithmetic paths and make no material claim. No propagation was timed, and these results should not enter the production benchmark ledger as though an implemented backend had been measured.

Whole-stage cost depends on the fraction spent in these operations, extra projections and response coexistence. Whole-run cost also depends on accepted and rejected steps, since changing the physics can change the adaptive trajectory. Measure those separately on short windows. Do not extrapolate a microbenchmark or ten-step timing to a full fibre run.

7. Verification and validation

Separate three layers of evidence: numerical implementation against the specified kernel; approximation error against an independently evaluated material model; and material-model agreement with physical measurements or calculations. An extremely accurate FFT implementation can still use an unsupported susceptibility.

Test family	Comparison and acceptance basis
Units and normalization	Taylor factor of six, SI coefficient conversion (\(10^{-10}\) cm\(^2\), \(10^{-20}\) cm\(^4\) and the \(W_2/S_2\) scale), electronic \(n_2/\chi^{(3)}\) conversion against both forms in §4.2, and density linearity; algebraic tolerances from floating-point reductions
Constant and monochromatic limits	Recover existing constant cubics in their shared scope; recover the Kerr diagonal and the distinct polynomial THG quartet, including degeneracy factors
Explicit convolution	Small signed-frequency direct sums for arbitrary multitone fields, not just one pump; include FWM, THG and inverse exchange; reduction-length error bounds
Symmetry and reality	Input permutation, lossless four-leg exchange within the model assumptions, Hermitian symmetry, negative-frequency completion and DC/Nyquist handling
Work and propagation invariants	The local identity in §4.9 and physical energy for lossless full-field propagation, with the band projection of §4.5 in place, and a negative check that an output-only cut-off breaks it; photon number only for the corresponding number-conserving sector
Mixture filter	The mixture-curve symmetric filter of §6 against the per-species polynomial sum, by the §4.7 procedure, for at least one binary mixture
Grid and carrier	Dealiasing convergence, larger time windows, absolute-frequency derivative weights, carrier-reference independence, and field/envelope parity beyond a narrowband pulse
Mixtures and regions	Explicit per-species sum versus aggregated polynomial coefficients; identical versus distinct filters; regional material response and density-profile variation
Response coexistence	Electronic changes do not filter or renormalize Raman/plasma sources; verify combined energy-transfer accounting
Backend parity	One nonlinear evaluation, one step and about eight to ten steps across NumPy, Numba and Kokkos CPU, on reduced forms of both demonstration cases among others; the Kokkos device build under pytest -m gpu; explicit capability refusal only while a backend’s phase is still open
Gas coverage	Every gas optmat fits resolves both evaluators, or is refused by name when its record or domain does not admit the request; the CO2, CO and O3 records are exercised at the model level until solprop builds them as species
Leg bound and regions	The bound leaves every fitted value and every quartet within \(S_{2,\max}\) exact, and holds legs beyond \(\nu_c\); the lossless identity holds with it; the region of each part of the active band is recorded, a field reaching the best-effort region runs and logs once, and a vector run takes the any-polarisation edge
Identity and errors	Changed material fit, support record, evaluator, band or tensor approximation invalidates resume/cache compatibility; malformed inputs fail before kernels

Use independent convolution code for the oracle: reproducing the same grouped FFT expression in a second function does not independently check the algebra. Include the §4.9 fundamental-plus-third-harmonic example and quartets that cross the analytic-signal projection boundary. A narrowband test alone cannot qualify the proposed THG semantics.

For model comparisons, publish frequency/quartet maps with 0.1%, 1% and 5% contours as descriptive levels, not predetermined acceptance thresholds. Show supported and extrapolated regions separately. Include maximum error, representative spectral weighting and the fit’s uncertainty; preserve correlations when comparing values from the same fitted curve. A covariance alone is not a bound on a truncated model or its borrowed priors.

Propagation qualification should include a short broadband SPM case, seeded widely separated phase-matched FWM, and a weaker THG/reverse-exchange check. The two demonstration cases of §2 serve: the argon DUV FWM case as the seeded, phase-matched FWM witness, and a reduced helium RDW case — the reduced M2 size of the testing plan, about ten steps — as the broadband, harmonic-retaining field-grid witness, whose band reaches the bounded and best-effort regions of §4.10. Use phase matching obtained from the admitted linear model and geometry. Compare complex spectral phase, nonlinear phase/B-integral, signal/idler conversion, pump depletion and energy in specified weak UV bands. Also vary nonlinear phase matching: an error in coupling or nonlinear phase can strongly alter gain and conversion near a sensitive operating point. A coefficient error of 1% does not imply a spectral-energy error of 1%.

Keep material parameters, input pulse and linear operator fixed when comparing the electronic evaluators. First establish step/grid convergence within each model; then attribute the remaining difference to the model. Do not retune pressure or pulse energy to restore an old spectrum. SEP 3’s data changes need their own comparison axis. Short-window tests establish local numerical and physical behavior; scientifically representative longer runs belong in the worked-example gallery under the repository’s existing policy.

Benchmark isolated response, full nonlinear stage and a short propagation window separately. Cover one mode, enough modes to expose projection cost, one spatial route, a mixture, and Raman/plasma coexistence. Report grid, oversampling, modes/nodes, threads, buffers/peak memory, accepted/rejected steps and timings; compare evaluators at equal achieved error with SEP 16’s work–precision measurements under the default Multistep(). Record implemented hot-path results in the benchmark ledger, with no full-run extrapolation.

8. Documentation

Implementation extends the nonlinear response chapter with the unified susceptibility, both operators, source conventions and the exact scope of the THG choice. Add a material-validity section and examples showing how to choose an evaluator and inspect the model identity. Update the GNLSE and general modal pages to distinguish material dispersion, modal dispersion and the UPPE frequency factor. Update the architecture and kernel contract when the spectral response bundle actually exists.

A quick tutorial compares carrier, symmetric and polynomial responses on the two demonstration cases of §2, exposing both coefficient and propagation differences, with the extrapolated region marked:

• DUV FWM in argon: the 400 nm + 800 nm → 266.7 nm capillary, single mode, at the phase-matching pressure of the admitted linear model; it compares the signal energy and spectrum and shows the coefficient table of §4.7.

• VUV RDW in helium: the case of the existing RDW tutorial (docs/tutorials/capillaries/plot_02_rdw_emission.py), which is M2’s in a shorter, two-mode form; it compares the RDW band’s energy and centre wavelength as a sensitivity to the extrapolated VUV legs (§4.7).

Each run stays as short as its point allows, as tutorials run on every docs build. The full-size M2 comparison belongs with the existing worked example (docs/examples/processes/plot_00_vuv_dispersive_wave.py), re-run by hand with the three evaluators; an argon DUV worked example beside the existing helium FWM one is optional. The nonlinear manual page carries both cases’ coefficient tables and the domain maps of §7. Physical lengths follow the gallery rules; tests and benchmark lanes remain short. Public docstrings state units, domain, approximations and error behavior. Record changed operator semantics on the Luna differences page if an accepted implementation introduces them.

9. Compatibility and impact

The carrier-evaluated electronic model remains the initial default inherited from SEP 3. Both dispersive options are opt-in and are never selected by backend availability, species count or the generated bandwidth. No material data return to solprop. An unsupported backend names the missing response and refuses it.

New propagation results are expected to differ: nonlinear phases, spectral transfer, phase matching, compression and adaptive step histories can all change. A model change is not a numerical regression to conceal by relaxing tolerances. Existing validated carrier cases keep their model selection; new dispersive cases receive their own qualified witnesses. Reproduce older published results with their recorded environments and model identities.

Store the electronic evaluator and version, material fit identity/digest, normalization, tensor and harmonic-sector assumptions, domain/extrapolation policy, and any user-supplied model identity. Include these in scan and resume keys. A file whose existing metadata cannot establish those identities cannot silently resume with a different response. SEP 3 delivered the carrier for these: a run’s MaterialManifest records each model’s optmat identity and the electronic policy, and they enter the resume key. The new policies add the evaluator, the band and the extrapolation record to the manifest’s electronic entry, with the support record’s identity and the regions of §4.10; that changes the persisted layout, so an output-schema bump (from 15 at the final review) is expected, and the kernel seam change requires its own kernel-schema bump (from 23). Keep older files readable under the established output policy.

The thg=False issue in §4.9 is explicit compatibility work. Acceptance must choose how a consistently restricted new response is distinguished from the existing constant partial cubic. This Draft does not silently redefine the old response or claim broadband parity that its current tests do not establish.

10. Alternatives considered

Alternative	Assessment
Keep one carrier coefficient	Useful baseline and initial default; cannot represent arbitrary broadband material dispersion
Multiply the final cubic spectrum by \(\chi_K(\omega)\)	Cheap but generally incorrect for nondegenerate quartets and lossless exchange; not recommended
Separate coefficients for SPM, XPM, FWM and THG	Artificial for a continuous spectrum; easy to break inverse-process consistency; the joint kernel already handles the processes
Quartet effective-frequency Kerr curve	Often a good interpretation of the gas polynomial; cannot be implemented by assigning one effective frequency to each output
Miller scaling from linear refractive index alone	Useful under additional oscillator assumptions; not reliable enough to replace curated nonlinear dispersion
Symmetric factorization only	Attractive cost, but a sampled few-percent error may matter; the agreed recommendation also supports polynomial propagation
Polynomial only	Removes the factorization error within the model, but adds transforms/projections where the cheaper option may be sufficient
General dense frequency kernel	Too expensive for the intended grids; retain only as a small-grid independent oracle
General low-rank or causal oscillator model	Plausible future treatment of richer tensor/resonant response; requires data, symmetry and causality constraints beyond this proposal

11. Phases

Phase 0 — Mathematical and material qualification

Confirm SI adapters and public material records with optmat. Reproduce the quartet comparisons, direct-convolution reference and invariant diagnostics. Derive broadband field/envelope and harmonic-sector forms for both evaluators. Set a supported transparent-gas domain and document per-species evidence, including molecular/tensor approximations. Agree observable-specific accuracy and runtime targets without confusing fit uncertainty with numerical tolerance. Resolve Q1–Q4 sufficiently for a first supported implementation, and confirm the leg bound and region labels of §4.10 on the two demonstration cases. Name remaining UV/VUV and silica work explicitly; do not defer a fundamental model decision to a backend engineer.

Phase 1 — NumPy reference and public response

Implement both electronic choices, their translation and the versioned seam, starting with scalar gases on field and envelope grids, for every gas optmat fits (§2). Preserve the carrier response and explicit user data. Add metadata/resume handling, manual pages and the short example together with direct-sum, invariant, convergence and short propagation tests. Qualify mixtures and route support explicitly; refuse any combination not yet supported. Implement a tensor extension only under the accepted Phase 0 model, not by inferring it from a scalar fit. NumPy is the oracle.

Phase 2 — Numba CPU

Add the same operators with preallocated workspaces, transform planning, species aggregation and deterministic kernels. Qualify all Phase 1 routes, mixed response cases and memory accounting against the reference. Record isolated, stage and short-window performance, including cases where projections dominate. No fastmath or silent fallback. Update the benchmark ledger.

Phase 3 — Kokkos CPU, then device qualification

Extend solprop-kokkos to the new seam and operators, keeping arrays resident through each stage. This phase is required, not optional: no exception for solprop-kokkos is written into this SEP, and it does not go Final while Kokkos refuses any combination that numpy_ref admits. Match NumPy and Numba across admitted geometries, mixtures, response combinations and metadata on the CPU build. The device build compiles the same operators from the one source and is qualified with pytest -m gpu under the existing separate GPU policy; CPU results do not establish device performance or parity. Record memory and short-window cost for both.

Phase 4 — Scientific qualification and closing scope

Publish the agreed accuracy/cost comparisons and application guidance for both options, including the full-size demonstration cases of §8. Decide whether either becomes a default in a separately explicit compatibility decision. Silica and resonant UV/VUV claims require their own curated inputs and evidence; unresolved work becomes named roadmap items or issues. All required backends must support the accepted scope before Final, with no exception for solprop-kokkos (Phase 3). Append each phase’s verification, source audit, limitations and handoff to Part II when delivered.

12. Open questions

• Q1 — Acceptance by observable. Which coefficient, nonlinear phase and conversion errors are acceptable for the priority SPM and widely separated FWM cases? Maintainers decide during Phase 0. No fixed 1% promise follows from the sample table; weak UV output and sensitive phase matching need separate criteria.

• Q2 — Material domains beyond the diagonal. Answered in substance: optmat’s support analysis and its draft OEP 5 [12, 13] supply the material facts, and §4.10 sets solprop’s rule on them. Remaining before Phase 1:

  • optmat’s release of the support record, carrying material facts only;

  • for optmat: whether O2’s spin-forbidden \(a\) and \(b\) states, which pair differences reach inside the range, count as resonances, and the record’s units;

  • for solprop: whether the bounded region is reported as a lower bound or as an estimate, and whether the few Xe and CO quartets not shown better than a 400 nm carrier need a narrower bound. The region check is solprop’s, made on its active band from optmat’s levels.

  Resonant response in the best-effort region, including helium’s VUV, is a named curation/model workstream.

• Q3 — Tensor and silica scope. Which scalar-times-isotropic-tensor approximation, if any, is acceptable for the first vector release? What independent nondegenerate silica data justify its factorization? Resolve admitted scope before implementation; leave unsupported claims explicit.

• Q4 — Harmonic-sector compatibility. Adopt the complete quartet response and the consistently number-conserving restriction of §4.9 for the new choices? How should these map to thg and general-modal kerr_form without silently changing legacy partial cubics? This is a pre-acceptance physics and interface decision, supported by the diagnostic already identified and by the harmonic-leg domain table of §4.8: under the leg bound, both evaluators keep the full cubic exact only for pumps at or above its first column (648–695 nm for most gases, 456 nm for helium).

• Q5 — Performance targets. What extra time and memory are acceptable for the priority single-mode, multimode and mixed-response workloads? Measure complete stages before choosing buffer/recomputation trade-offs.

• Q6 — Eventual default. Should a qualified dispersive model replace carrier evaluation, and for which materials/domains? Keep carrier as the initial default; decide only after scientific and backend qualification.

13. References

1. optmat, Third-order nonlinearity, Third-order accuracy and Fused silica. Public material-model and curation reports; numerical comparisons use the model revision identified in §2, not an independent experimental dataset.

2. D. P. Shelton and J. E. Rice, “Measurements and calculations of the hyperpolarizabilities of atoms and small molecules in the gas phase,” Chemical Reviews 94, 3–29 (1994), doi:10.1021/cr00025a001.

3. W. Ettoumi, Y. Petit, J. Kasparian and J.-P. Wolf, “Generalized Miller formulae,” Optics Express 18, 6613 (2010), doi:10.1364/OE.18.006613.

4. V. Mizrahi and D. P. Shelton, “Dispersion of nonlinear susceptibilities of Ar, N2, and O2 measured and compared,” Physical Review Letters 55, 696–699 (1985), doi:10.1103/PhysRevLett.55.696.

5. J. J. Pigeon, S. Ya. Tochitsky, E. C. Welch and C. Joshi, “Experimental study of the third-order nonlinearity of atomic and molecular gases using 10-µm laser pulses,” Physical Review A 97, 043829 (2018), doi:10.1103/PhysRevA.97.043829.

6. D. M. Bishop and D. W. De Kee, “The frequency dependence of nonlinear optical processes,” Journal of Chemical Physics 104, 9876–9887 (1996), doi:10.1063/1.471752.

7. C. Hättig and P. Jørgensen, “Coupled cluster calculations of the frequency-dependent second hyperpolarizabilities of Ne, Ar, N2, and CH4,” Journal of Chemical Physics 109, 2762–2778 (1998), doi:10.1063/1.476833.

8. M. Conforti, A. Marini, T. X. Tran, D. Faccio and F. Biancalana, “Interaction between optical fields and their conjugates in nonlinear media,” Optics Express 21, 31239–31252 (2013), doi:10.1364/OE.21.031239.

9. C. G. Durfee III, S. Backus, H. C. Kapteyn and M. M. Murnane, “Intense 8-fs pulse generation in the deep ultraviolet,” Optics Letters 24, 697–699 (1999), doi:10.1364/OL.24.000697.

10. J. C. Travers, T. F. Grigorova, C. Brahms and F. Belli, “High-energy pulse self-compression and ultraviolet generation through soliton dynamics in hollow capillary fibres,” Nature Photonics 13, 547–554 (2019), doi:10.1038/s41566-019-0416-4.

11. A. Kramida, Yu. Ralchenko, J. Reader and NIST ASD Team, NIST Atomic Spectra Database, National Institute of Standards and Technology, doi:10.18434/T4W30F.

12. optmat, Where the fits are supported and Beyond the fitted data: bounds on the legs, at optmat a9e5dbf: the fitted legs, resonance limits, monotonicity evidence and per-leg bound comparisons that §4.10 adopts.

13. optmat, OEP 5 — a public record of where the hyperpolarizability fits are supported, Draft, at optmat 0d5a0ab.

Source audit. The nonlinear dispersion expansion and its diagonal-component scope were checked in [6], the distinct tensor dispersions in [7], and the oscillator assumptions and experimental limitations of linear-index scaling in [3–5]. The analytic-signal issue was checked against [8] and independently against the current solprop cubic. The Taylor convention and fit/domain interpretation use [1, 2] and SEP 3’s source audit. The model ratios, quartet survey and algebraic diagnostics were calculated independently for this Draft. The demonstration cases rest on [9, 10] for their configurations, on solprop’s own M2 definition and SEP 12’s helium resonance positions, and on [11] for argon’s resonance lines and helium’s levels; their bibliographic details were checked, their measured results are not used. The §4.10 support figures are optmat’s [12]: the leg bound and the per-row bounded values were re-derived independently from the stored fits and agree, while the resonance energies and the carrier comparison counts were taken from [12] and not recomputed. No new primary coefficient table was transcribed. This work does not re-fit the underlying measurements or validate VUV extrapolation, nonlinear absorption, a complete tensor model, or an implemented dispersive propagator.

Part II — Implementation record

Phase 0 — mathematical and material qualification

• Scope: Phase 0 of §11, 2026-10-07, implemented by Opus 5.5 (AI) for John Travers, on feature/sep-0006-dispersive-nonlinearity-0. No package code changes: Phase 0 fixes the algebra and the material evidence that Phase 1 implements.

• Delivered:

  • tests/dispersive_kerr_oracle.py: the independent oracle, an explicit sum over signed input triplets of a band-limited periodic field with the kernel evaluated leg by leg, scalar and isotropic vector; and the FFT forms derived below, for both evaluators, with the leg bound, in both sectors, on field and envelope frames.

  • tests/nonlinear/test_dispersive_kerr_algebra.py: 37 tests holding the FFT forms to the oracle and to the invariants of §4.9, the band rule of §4.5 and the bound of §4.10, with negative checks of the output-only weighting and of the existing field-grid thg=False cubic.

  • tests/sep6_quartets.py, which writes tests/data/sep6_quartets.json, and tests/test_sep6_quartets.py, which recomputes it. The record holds §4.7’s survey, FWM and demonstration tables, §4.8’s harmonic table, §4.10’s leg bounds and the SI conversion check. Each is evaluated through optmat’s public HyperpolarizabilityModel at the pinned v0.3.0, with the identity digest of every model.

The operators Phase 1 implements

Write the bounded leg weight \(\tilde d(\omega)=\min\{(\omega/2\pi c)^2,\nu_c^2\}\), a real even spectral multiplier, with \(\nu_c=\sqrt{3S_{2,\max}/4}\) per gas (§4.10). For a product of three real fields \(X,Y,Z\) with analytic signals \(x,y,z\) (\(X=\mathrm{Re}\,x\)), each factor is \((x+x^*)/2\). The analytic polarisation \(u_P=2\Pi_+P\), with \(\Pi_+\) the projection onto absolute frequencies above zero, is then

\[ u_P=\tfrac14\Pi_+\big[\,xyz+(xyz^*+xy^*z+x^*yz)+(xy^*z^*+x^*yz^*+x^*y^*z)\,\big]. \]

The three groups are the quartets whose signed legs, the output counted negative, have three, two and one positive members. A zero-frequency leg belongs to none; the active band never contains \(\omega=0\).

• The full cubic keeps all three groups. On a field grid it is the real product \(XYZ\) itself.

• The number-conserving sector keeps the middle group: \(P_{\rm NC}=\mathrm{Re}\{\tfrac14\Pi_+[xyz^*+xy^*z+x^*yz]\}\). For \(X=Y=Z=E\) this is \(\mathrm{Re}\{\tfrac34\Pi_+[|u|^2u]\}\), §4.9’s form.

Each sector restricts the kernel to a set of quartets closed under every permutation of the four signed legs, so a symmetric kernel stays symmetric and §4.9’s work identity holds in both sectors. In the conserving sector the sum \(\sum\chi\,a_o^*a_3^*a_1a_2\) over quartets \(\omega_o+\omega_3=\omega_1+\omega_2\) is real, being its own conjugate under \((o,3)\leftrightarrow(1,2)\), so the photon-number rate vanishes too. In the full sector it does not, because of the harmonic quartets.

Both evaluators are sums of such products:

• The polynomial is §4.5’s grouped form with \(D\) replaced by the multiplier \(\tilde d\): four products, \(E^3\), \(E^2\tilde DE\), \(E^2\tilde D^2E\) and \(E(\tilde DE)^2\), each with its output multiplier.

• The symmetric model is one product of \(GE\), with \(g(\omega)=[\gamma_K(\tilde\nu)/\gamma_0]^{1/4}\). The reference is the static \(\gamma_0\), so \(g(0)=1\) and \(g\ge1\) for every fit (§4.10).

A sector is applied to each product, and the output multipliers commute with \(\Pi_+\) and with taking the real part.

Envelope grids. With \(A=u\,e^{-i\omega_0t}\), the convention of nonlinear.responses.thg_carrier, the same expression holds with every multiplier evaluated at the absolute frequency \(\omega_0+\Delta\omega\). The first group carries \(e^{+2i\omega_0t}\), as the existing harmonic does, and the third carries \(e^{-2i\omega_0t}\). The third group is new. It is the one-into-three exchange, and reaches positive absolute frequencies only for fields wider than an octave, so an envelope fine grid must hold it or remove it by the band projection, which acts on inputs and output alike.

Vector fields. The scalar kernel multiplies the isotropic tensor \(T_{ijkl}=(\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})/3\). Three distinct filtered fields contract as \(\tfrac13[(X\!\cdot\!Y)Z+(X\!\cdot\!Z)Y+(Y\!\cdot\!Z)X]\), which is \((E\cdot E)E\) when they coincide. This is a named approximation: Kleinman’s ratio \(\chi_{xxyy}=\chi_{xxxx}/3\) is held at every frequency. optmat’s \(\gamma\) is the mean of parallel linear polarisations, and the components’ distinct dispersion [7] is not represented.

Decisions

Proposed in this phase and accepted by John Travers on 2026-10-07; they bind Phase 1 onward.

• Q4, the harmonic sector. For the two new policies, thg=True and kerr_form="real" select the full cubic, and thg=False and kerr_form="analytic" select the number-conserving sector, with \(\Pi_+\). The existing constant cubic is left as it is.

  • The existing rules equal the new sector whenever \(|u|^2u\) has no negative-frequency content, that is for any band narrower than an octave (\(\omega_{\max}<2\omega_{\min}\)). This is tested.

  • Beyond an octave the field-grid thg=False cubic lacks \(\Pi_+\) and does net work, also tested.

  • Adding \(\Pi_+\) to it is a small correction to an existing operator, outside this SEP’s scope. It is proposed as a separate change (open question 2).

• Q3, vector scope. The scalar kernel times the isotropic Kleinman tensor, recorded as such in the manifest. Silica keeps its carrier-evaluated cubic: no curated nondegenerate silica data exist to factorise (§4.8).

• Q1, accuracy by observable. Numerical acceptance is separated from model comparison.

  • Implementations meet the oracle to 1e-12 relative on the Phase 0 grids.

  • Backends meet the reference under the existing parity rules.

  • Propagations converge on step and grid ladders.

  • Differences between the evaluators and the carrier model are reported, not gated. For the demonstration cases they cover: argon’s 266.7 nm energy, conversion and spectral phase; helium’s RDW band energy and centre wavelength; the B-integral. Phase 4 turns them into application guidance.

• Q5, performance. Targets are proposed from §6’s kernel measurements, to be measured on whole stages in Phases 1–2:

  • Time: the symmetric nonlinear stage within 1.3 times the constant cubic’s on the same route, the polynomial within 4 times.

  • Memory: the symmetric model adds at most one field-sized buffer, needed only when Raman or plasma also needs the bare field; the polynomial adds two derivative fields and one product buffer.

  • A missed target opens an optimisation slice. It never refuses a run.

• Q2, what remains. The bounded region’s standing becomes a bounded estimate, read as a lower bound below resonance by solprop and not certified by optmat (its OEP 5 review). The region check runs on the whole active band, which contains every generated output the grid can carry.

Deviations from Part I

Part I is frozen; these qualify it, and Phase 1 follows them.

• “A lower bound on γ” (§4.7, §4.10) reads bounded estimate: solprop’s interpretation of the evidence, which optmat does not certify. Just beyond the data the polynomial lies 0.1–0.2% above two calculations (argon’s and N2’s THG); farther out it lies below every calculation. “Shown no worse than the carrier” beyond the data rests on the same interpretation.

• The band edge (§4.10) is a sufficient bound: below it no leg reaches \(E_1\) and no pair reaches \(E_2\). For one linear polarisation the SPM pair reaches \(E_2\) exactly at it. For a vector field the any-polarisation edge is conservative: two photons of equal frequency do not reach an even \(J=1\) level. A field beyond the edge is labelled best effort, which stays correct.

• The polyatomic limits (§4.10) are evidence, not levels, so the edge is undefined for CO2, CH4, SF6, N2O and O3:

  • no CH4 or SF6 absorption is detected at the sampled wavelengths down to 197.7 nm;

  • N2O’s 217.5 nm is where an integration over its B band begins;

  • O3’s 1047 nm is the end of the curated table, not an onset; O3 absorbs to at least there.

  The policy is unchanged: these gases are best effort beyond their data.

Verification

• The algebra: 37 tests.

  • The FFT forms of both evaluators, bounded and unbounded, in both sectors, scalar and vector, agree with the triplet sum to \(1.3\times10^{-14}\).

  • The work identity holds to \(4\times10^{-17}\) in every case. The photon-number identity holds in the conserving sector to \(6\times10^{-17}\), and fails in the full sector, as it must.

  • The envelope frame equals the field frame to \(1.5\times10^{-15}\) at two carriers, which is §5’s carrier independence.

  • Monochromatic fields give \(\tfrac34\gamma_K\) and \(\tfrac14\gamma_{\rm THG}\) at the polynomial’s own sums, and \(\tfrac14 g(3\omega)g(\omega)^3\) for the factorisation’s harmonic.

  • The constant limit is KerrResponse(thg=True) exactly, and thg=False below an octave.

  • Projecting the band on every leg keeps the work identity; projecting the output alone breaks it.

  • The leg bound changes no quartet within the fitted sums.

• The material record.

  • solprop’s SI evaluation — the Taylor factor of six, \(10^{-10}\) cm² → \(10^{-14}\) m², \(10^{-20}\) cm⁴ → \(10^{-28}\) m⁴ — equals optmat’s gamma3 for the Kerr and harmonic processes, and its gamma_au for the pump-probe process, to \(3.3\times10^{-16}\) for all fifteen gases.

  • Every row of §4.7’s survey, FWM and demonstration tables, §4.8’s harmonic table and §4.10’s leg bounds reproduces through optmat’s public API.

• Not run: no propagation, since none exists yet; no timing; no Kokkos or Numba work.

Per-species evidence for the first implementation

The leg bound, the largest fitted leg and the band edge for one linear polarisation, in nm, from optmat at 18ad60a [12, 13]; “—” is undefined.

Gas	Leg bound	Largest fitted leg	Band edge	Notes
He	151.9	151.9	120.3	Fitted to Bishop and Pipin’s calculation alone; measurements validate
Ne	215.9	229.0	133.5
Ar	215.9	229.0	189.4
Kr	230.1	244.1	216.7	\(B'\) held at its prior
Xe	230.1	244.1	256.0	\(n_2\) changes sign at 248.4 nm, past the edge
H2, D2	215.9, 224.7	229.0, 231.5	201.7, 201.1	Vibrational part subtracted
N2	215.9	229.0	290.1	Vibrational part subtracted; edge set by the two-photon \(a\,{}^1\Pi_g\) state
O2	224.7	238.3	304.6	Vibrational part subtracted; \(B'\) at its prior; limits air
CO2, CH4, SF6	230.1, 224.7, 215.9	231.5, 238.3, 229.0	—	Fits include a small vibrational part
CO	231.5	231.5	308.9
N2O	383.4	580.0	—	One measured value; dispersion borrowed from CO2
O3	401.0	694.6	—	Theory only; absorbs throughout its range

The rest of this report

• Sources checked: optmat’s manual at 18ad60a, “Where the fits are supported” and “Quantities derived from the facts”, and OEP 5, read for the facts quoted above. No new literature.

• Limitations:

  • The FFT forms are references on periodic dimensionless grids, not the package’s grids, windows or modal projections; Phase 1 qualifies those.

  • The proposed targets are unmeasured on whole stages.

  • The band edges are quoted from optmat’s manual until its support record is released and pinned.

• Open questions:

  1. Resolved: the proposals for Q1, Q3, Q4 and Q5 were accepted on 2026-10-07.

  2. Add \(\Pi_+\) to the existing field-grid thg=False cubic and to kerr_form="analytic", as a separate change; it alters results only for fields wider than an octave.

  3. The optmat release that carries OEP 5’s support record, and the pin move to it, before Phase 1 reads \(\nu_c\) and the edges.

  4. Whether an envelope fine grid built for harmonics also holds the one-into-three group, or projects it away, measured in Phase 1 against the field grid on a super-octave case.

• Handoff: Phase 1 starts from main with this branch merged.

  • It implements the operators above in numpy_ref behind the two new policies, tested against tests/dispersive_kerr_oracle.py.

  • It reads \(\nu_c\) and the edges from optmat’s support record, never from this report.

  • It records the regions and the record’s identity in the manifest.

  • tests/sep6_quartets.py is rerun whenever the optmat pin moves; its test reports a changed model.

Phase 1a — the dispersive response on numpy_ref

• Scope: slice 1a of Phase 1 (§11), 2026-10-07, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-1 in solprop and in solprop-kokkos. Both dispersive policies through the material layer, the response, the capillary driver routes, translation and kernel schema 24, executed by numpy_ref on its fused node path; every other route and backend refuses the response by name. Slice 1b adds the general modal, slab and Cartesian free-space routes; slice 1c the manual chapter, the tutorial and the demonstration tests.

• Delivered:

  • physdata/selection.py: ELECTRONIC_POLICIES gains "symmetric-kerr" and "gas-polynomial"; DISPERSIVE_POLICIES maps them to their evaluators; "recommended" is still the carrier policy.

  • physdata/nonlinear_data.py: DispersiveKerrModel, optmat’s fit in SI — gamma3_static = gamma0 au / (6 eps0) in m⁵/V², a in m² (×1e-14), b, b' in m⁴ (×1e-28) — with leg_bound = support.leg_max_in_w2 and both band edges from optmat 0.4.0’s support record, the model and support records, and regions(); ElectronicRegions; OptmatKerr.dispersive_model().

  • nonlinear/responses.py: DispersiveKerrResponse (evaluator, the SI model, thg, species; traits: convolution, field and envelope grids, vector-isotropic), DispersiveWeights, and dispersive_cubic, the Phase 0 forms on a fine grid’s spectral layout at absolute frequency, the arithmetic both the reference evaluate and the numpy_ref pass run.

  • nonlinear/mixtures.py: electronic_response, the response a selection’s policy names; responses_for uses it, so the mixture route and scans (ResponseSelection) follow the policy.

  • translate/responses.py: the leaf DispersiveKerrTerms on CompiledResponses.dispersive_kerr, dispersive_terms, the dispersive_kerr stage, dispersive_regions; translate/stage.py: the bundle row dispersive_kerr_densities; translate/problem.py: schema 24 and the leaf’s validation.

  • backends/numpy_ref/passes.py: dispersive_kerr_polarisation; backends/driver.py: refuse_dispersive_kerr, used by numpy_ref (route refusals) and numba_cpu (the leaf); solprop-kokkos: a refusal by name in _refuse_unsupported and the native build record at schema 24.

  • driver/simulation.py, driver/material_output.py: the capillary routes build the response; the GNLSE and general modal routes refuse a dispersive policy; the manifest records the law.

  • Tests: tests/physdata/test_dispersive_kerr_model.py, tests/nonlinear/test_dispersive_kerr_response.py, tests/translate/test_dispersive_kerr_terms.py, tests/backends/test_dispersive_kerr.py, tests/driver/test_dispersive_policy.py, and the Kokkos refusal in solprop-kokkos tests/test_backend.py.

  • docs/dev/kernel-contract.md (schema 24, the leaf, the bundle row), the CHANGELOG.md entries of both repositories, and one-line updates of the solver and materials pages.

Decisions

• The leaf carries sampled multipliers, not coefficients. DispersiveKerrTerms holds each channel’s multipliers on the problem’s fine spectral layout — g, or the polynomial’s bounded leg weight, its two output multipliers and its two constant product weights — with Π₊ and the static coefficient, so no backend evaluates material physics. The pass and the reference share dispersive_cubic; the independent check of that arithmetic is Phase 0’s triplet sum.

• The band projection of §4.5 is the existing one. The state is masked to the active band and the prefactor is zero outside it, so inputs and output are projected alike; the multipliers are evaluated on the whole fine grid, bounded there by the leg bound.

• One electronic response per population. A constant KerrResponse and a dispersive one on the same label are refused; on different labels both stages run. All electronic responses of a problem agree on thg (one harmonic sector) and all dispersive ones on their evaluator (one law). Raman’s field-grid drive form reads the dispersive sector as it reads the constant thg.

• The policy is the run’s, not the caller’s response alone. A constant KerrResponse declared beside a dispersive selection is refused, so a record never names one law and propagates another; from_gnlse (a fitted gamma) and the general modal route (regional Kerr tensors) refuse a dispersive policy by name.

• Regions and the record. At translation each dispersive population’s active band, ω_min/2 to 1.1 ω_max, is classified per frequency: supported (within ν_c and inside the band edge), bounded (beyond ν_c, inside the edge) and best effort (beyond the edge, or beyond ν_c where the edge is undefined), with the any-polarisation edge when the node field has two components. A band beyond the support is logged once at INFO, naming the gas and the band; nothing is refused. The manifest’s electronic entries record evaluator, sector, tensor approximation, model and support digests, leg bound, both band edges and the regions; policies["electronic_response"] names the law with each population’s support digest and regions, and the support record joins the manifest’s records — all in the resume key.

• No output-schema bump. The persisted layout is unchanged: the new content lives in the manifest’s existing policies, electronic and records entries, a carrier run’s record and digest are bit for bit what they were, and older files keep resuming. §9 expected a bump; John Travers may still prefer one (open question 3).

Deviations from Part I

• Mixtures are the exact per-species sum. Each population is its own channel with its own multipliers, leg bound and density: P = Σ_s ε₀γ₀,s ρ_s cubic_s(E). §6’s mixture-curve symmetric filter and aggregated polynomial coefficients are not built; they become Phase 2 optimisations to be measured against this sum, which is the reference they approximate.

• Kokkos refuses the leaf in this phase, by name, as §11 allows while a backend’s phase is open; Phase 3 implements it.

Verification

All on macOS (Apple silicon), Python 3.14.5, optmat 0.4.0; tolerances are Phase 0’s 1e-12 (decision Q1) unless stated.

• The response against the triplet sum (direct_sum), on dimensioned grids with argon’s model and the bound active (bin 12 at 200 nm, ν_c at 215.9 nm): both evaluators, both sectors, scalar and isotropic vector, field frame and envelope frame at two carriers — worst relative error 2.4e-15 over 24 cases.

• Zero dispersion: the full cubic equals KerrResponse to two FFT round trips; the conserving sector equals thg=False below an octave (field and envelope) and departs from it beyond. The whole numpy_ref stage derivative equals the constant cubic’s to 4.3e-16 (full) and 9.1e-16 (conserving) on the mode-averaged capillary, and to 4.2e-16 on the radial route.

• The pass against the reference: the compiled pass equals DispersiveKerrResponse.evaluate bit for bit on the capillary’s node field, field and envelope grids, both sectors, both evaluators; a He/Ar mixture’s pass equals the sum of its species’ evaluations to 4e-16 of the largest sample.

• Work and energy: the response does no net work (both evaluators, both sectors, below 1e-12). Ten fixed DOPRI5 steps over 2 mm of argon at 2 bar, lossless, no absorber, 10 fs and 30 µJ at 800 nm: the conserving sector conserves energy to 1.4e-14; the full cubic to 1.5e-9 at ten steps and 1.0e-12 at twenty, the integrator’s error falling with the step.

• Open question 4 of Phase 0: on EnvelopeGrid.from_limits(..., thg=True) (150 nm to 4 µm) a super-octave field (250 nm, 800 nm and 2.5 µm) evaluated in the full sector agrees with a four times finer evaluation across the active band to 1.2e-14 (carrier 800 nm) and 7.5e-14 (carrier 400 nm): the harmonic grid holds the one-into-three group, as it holds the constant cubic’s harmonic.

• Material: the SI model equals optmat’s gamma3 for the Kerr and harmonic processes to 1e-14 (measured 3.3e-16) for all fifteen gases; the leg bound changes no quartet within the fitted sums; regions: helium to 90 nm reaches all three (bounded 120.3– 151.9 nm), argon’s DUV band to 1.1 × 210 nm only supported and bounded, argon’s any-polarisation edge 192.1 nm for a vector run, N2 and SF6 best effort as stated.

• Refusals by name: numba_cpu and Kokkos refuse responses.dispersive_kerr; numpy_ref refuses it on the general modal, slab and free routes; a supplied species and air under a dispersive policy, a constant Kerr beside it, the GNLSE and the general modal driver routes are refused; changing the policy, the evaluator or the sector changes the resume key, and check_cache refuses naming the changed materials.policies entries.

• Runs (pytest -n auto, Kokkos CPU installed from this branch): the five new modules, 103 tests (with Phase 0’s algebra and the SEP checks, 267) in 15 s; tests/backends 2660 passed (after clearing a stale numba disk cache, below), tests/translate, tests/nonlinear and tests/physdata 3320 passed, tests/driver, tests/output, tests/physdata and tests/test_seps.py 2924 passed; solprop-kokkos’s whole suite against this branch, 234 passed and 4 skipped. Two numba_cpu vector-Raman cases (test_vector_complete_run_matches_reference_and_modal_routes, SF6, analytic field) failed once against a numba disk cache the session had filled under xdist; they pass on the base commit and here after the worktree’s .nbi/.nbc files were removed, and no code they run changed. ruff, basedpyright (no errors in the changed files), numpydoc lint and codespell on the changed files.

• Cost (indicative only): the numpy_ref stage on the 2 mm argon capillary, one thread, same-process medians on a busy macOS workstation: symmetric 2.5–5.8 times and polynomial 6.8–8.6 times the constant cubic’s stage. The reference is written for clarity — full complex transforms, allocations in every product — and decision Q5’s targets (1.3 and 4 times) are for the optimised stages of Phase 2, where they are measured.

• Not run: no full gate (poe check, slow lane, coverage, docs build) — they run at the end of the SEP; no Numba or Kokkos execution of the response; no demonstration case.

The rest of this report

• Sources checked: optmat 0.4.0’s HyperpolarizabilityModel.support (HyperpolarizabilitySupport: leg_max_in_w2, band_edge, model_digest, identity) and AU_HYPERPOLARIZABILITY, read from the installed package; Part I and the Phase 0 report. No new literature.

• Limitations:

  • The reference evaluates every product with full complex transforms; its cost is well above the targets of Q5 (above).

  • The region classification is per frequency of the active band; it does not identify which quartets a field actually drives.

  • On an envelope grid the harmonic groups can still wrap when the field itself reaches the edges of the active band, as the constant cubic’s harmonic does; the measurement above is for fields inside the band.

  • The manual chapter, the tutorial and the demonstration cases are slice 1c’s.

• Open questions:

  1. Phase 0’s open question 4 is answered above for fields inside the active band; the lead may want the edge case (content in the collars) written into the manual.

  2. Q5’s targets are missed by the reference, as expected; Phase 2 measures the optimised stage. Confirm that the reference need not meet them.

  3. Whether to bump the output schema although the persisted layout did not change (§9 expected one); a bump would make every schema-15 file unresumable.

  4. The policy check refuses a constant KerrResponse beside a dispersive selection on every route, including the spatial ones whose responses are the caller’s; confirm that this strictness is wanted.

• Handoff:

  • Slice 1b adds the general modal route (its regional responses and kerr_form), the slab and the Cartesian free-space pipelines: each evaluates responses.dispersive_kerr per node through dispersive_cubic, and its refusal in refuse_dispersive_kerr and in the driver’s _refuse_dispersive_policy for the general route is removed as it lands.

  • Slice 1c writes the nonlinear manual chapter (both operators, the sectors, the regions, the record), the tutorial with the argon DUV and helium RDW cases, and their demonstration tests.

  • Phase 2 (Numba) implements the leaf against numpy_ref, with the mixture-curve filter and aggregated polynomial coefficients measured against the per-species sum.

Phase 1b — the remaining numpy_ref routes

• Scope: slice 1b of Phase 1 (§11), 2026-10-07, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-1 (solprop only; solprop-kokkos unchanged). numpy_ref evaluates responses.dispersive_kerr on every route it runs: the Cartesian free-space and line pipelines, the hybrid planar pipeline and the general modal route, where a region’s gas fill takes the run’s electronic policy. numba_cpu and Kokkos still refuse the leaf by name. Slice 1c (the manual chapter, the tutorial, the demonstration tests) runs in parallel.

• Delivered:

  • backends/driver.py: refuse_dispersive_kerr(problem, backend) refuses the leaf on a backend without it; numpy_ref no longer calls it and refuses no route.

  • Free space, line and planar: no code change beyond the refusal. _nonlinear_free and _nonlinear_slab already run the fused response pass, with the stage bundle and the harmonic carrier, at every node of a chunk, so the dispersive stage of slice 1a is evaluated per field point, between the unchanged spatial transforms.

  • modes/material.py: GasKerr(fill), a region’s gas fill as its electronic response (a Mixture of constant densities); MaterialRegion.gas; a region declares one electronic response, MaterialKerr or GasKerr, and a SilicaTensorResponse still needs a MaterialKerr. Exported from solprop.modes.

  • translate/general.py: general_modal_route(..., selection=). _regional_responses resolves each gas region through electronic_response at the carrier 2 pi c / omega_reference: a carrier policy adds it to the constant regional cubic with chi3 = sum_s N_s gamma3_s(lambda_0); a dispersive policy makes each species a channel of the dispersive leaf, labelled "<region>:<population>", with the region as its node region, kerr_form="real" its full cubic and "analytic" its number-conserving sector, under the same harmonic-room rule. The route carries the leaf (GeneralModalRoute.dispersive) and the responses it was sampled from (electronic); the guide’s identity names the fill (key gas, written only for a gas region) and the modal identity the channels (key dispersive, written only when present). Each channel’s spectral regions are logged once at INFO with the any-polarisation band edge.

  • translate/responses.py: DispersiveKerrTerms.regions (int32, one node region per channel, present exactly on the general modal route), dispersive_terms(..., regions=), and compile_responses(..., regional_dispersive=); translate/problem.py: the leaf’s regions validated against the node rule, int32 admitted for them.

  • backends/numpy_ref/general_modal.py: dispersive_polarisation, called from general_source with the stage bundle’s densities, adds each channel’s coefficient rho cubic(Re E_a) at its region’s nodes, by dispersive_cubic, to the polarisation the regional tensors give; the projection is unchanged. backends/numpy_ref/admission.py admits the leaf beside the general route’s tables when it names regions.

  • driver/simulation.py: the general modal entry point passes its selection to the route and no longer refuses a dispersive policy; the manifest takes the any-polarisation edge for a general route’s channels.

  • Tests: tests/backends/test_dispersive_kerr_routes.py (39); the three route refusals of tests/backends/test_dispersive_kerr.py are replaced by it.

  • docs/guide/waveguides/general-modal.md (the gas region), docs/dev/kernel-contract.md (the leaf’s regions, every route on numpy_ref), one changelog line.

Decisions

• A gas region is declared by its fill, GasKerr. The general modal route had no gas region: its regions’ laws are linear indices and their responses one absolute chi3. The slice’s scope, “a region whose material is a gas optmat fits”, is read as a region declaring its gas fill as its electronic response, beside its own linear law, rather than a new permittivity law: the providers’ material-agreement checks are untouched, and a carrier-policy run of the same guide is the constant cubic every other route evaluates for a gas. Proposed, not yet agreed (open question 1).

• The leaf is amended at schema 24. regions joins DispersiveKerrTerms without a new schema version: schema 24 has not left this branch, and Kokkos refuses the whole leaf by name whatever its fields. The general route reuses the population leaf and its stage-bundle densities rather than adding a regional leaf, so every backend that refuses the leaf refuses it on this route too.

• The general route filters in time, as the reference does. The route’s frequency-domain synthesis would let the multipliers act on the synthesised spectrum before the inverse transform. The reference instead takes Re E_a and calls dispersive_cubic, so its arithmetic is the response’s own; applying the filter during synthesis saves one transform per node and component and is Phase 2’s to measure.

• The vector sector is the isotropic Kleinman cubic on three components (Q3). On the real field it is the real form’s tensor and in the conserving sector the analytic form’s, which is what makes the zero-dispersion limit exact on this route.

• A mixture region is one channel per species, as on the population routes; two regions of one gas are two channels.

Deviations from the slice’s scope

• “A gas core and a constant solid region.” The step-index provider guides only with its core above its cladding, so the tested guide is a 1.5 µm strand of a constant solid (MaterialKerr) in an argon exterior (GasKerr); no shipped provider has a gas core.

• The general route’s reference at a node is evaluate’s arithmetic, not evaluate. DispersiveKerrResponse.evaluate admits the state’s one or two components and a general node has three; the test applies its expression, (rho coefficient) dispersive_cubic(E, weights), with the response’s own weights.

• Energy. None of the three routes’ energy measures is conserved exactly by the constant cubic itself (below), so a lossless run is held to the constant cubic’s change on the same route rather than to zero.

Verification

All on macOS (Apple silicon), Python 3.14.5, optmat 0.4.0, numpy_ref, ten fixed DOPRI5 steps where a run is stepped.

• One evaluation at a node, bit for bit: the pass the free pipeline calls equals evaluate at the brightest node for the field grid’s full and conserving sectors and the envelope grid’s conserving sector, both evaluators; the planar pipeline, both sectors and evaluators; the general route, both forms and evaluators, at the gas region’s brightest node, with the solid core’s nodes untouched.

• Zero dispersion against the constant cubic, whole stage derivative, as a fraction of the largest bin: free 2.9e-16 (full) and 4.9e-16 (conserving), line 2.9e-16 and 2.5e-16, planar 1.8e-16 and 3.1e-16, general modal 1.2e-16 (real) and 2.3e-16 (analytic, against a region declaring chi3 = N gamma3_0). The conserving sector is compared on a launch confined to 300–450 THz: a Gaussian’s tails reach beyond an octave at the 1e-6 level, and there the sector departs from the thg=False rule by 3.7e-6 of the largest bin, as decision Q4 says it must.

• Two laws in one guide: a solid core (chi3 = 1e-29 m²/V², weak so that the gas is a fifth of the source) and a gas exterior give the sum of each alone to 1.4e-15. Under the carrier policy the gas region is the constant cubic with N gamma3(835 nm) to 1e-14; a supplied species without optmat’s model is refused under a dispersive policy naming the region and the population, and runs under the carrier one.

• Energy: each route’s own measure moves for the constant cubic by an amount independent of the step and proportional to the pulse energy — the free route’s paraxial eps0 c / 2 int |E|^2 against the k_z-weighted flux (-1.40066e-11 for a 15 fs, 3 µJ beam; a beam three times wider moves it a hundredth as much), the planar route’s truncated slab basis (6.484e-9), the general route’s identity metric (-6.246e-12 for 20 µJ in the strand). The dispersive runs move it by -1.40060e-11, 6.493e-9 and -6.253e-12 (both sectors): within 1% of the constant cubic’s, which is the test’s bound.

• Refusals by name: numba_cpu and Kokkos refuse responses.dispersive_kerr on the free, planar and general routes (six cases).

• Runs (pytest -n auto, Kokkos CPU installed from the branch): the new module, 39 tests in 17 s (19 s serially, the slowest 4.0 s; the four energy tests are medium); tests/backends route and general modules with slice 1a’s, 410 passed; tests/translate, 1168 passed; the general modal, policy, manifest, moving, GRIN, split-block, solid multimode, waveguide and resume driver modules, 182 passed; tests/modes with the general modal and manifest driver modules, 1039 passed. Two numba_cpu cases of test_free_nonlinear.py failed once in a combined -n auto run and passed on rerun, alone and combined; no code they run changed, and slice 1a saw the same with the numba disk cache under xdist. ruff, basedpyright (no errors), numpydoc lint and codespell on the changed files.

• Cost (indicative only): the numpy_ref stage, one thread, same-process medians on a busy workstation, against the constant cubic on the same route: free 1.5 times (symmetric) and 2.3 times (polynomial), planar 3.0 and 5.9, the general strand 5.7 and 6.5. The reference allocates in every product; Phase 2 measures the targets.

• Not run: no full gate, slow lane or coverage lane (the SEP’s end); no solprop-kokkos suite (unchanged); no moving general guide with a gas region (its route takes the same regional compile).

The rest of this report

• Sources checked: Part I, the Phase 0 and Phase 1a reports, and the code of slice 1a. No new literature or material data.

• Limitations:

  • The memory ledger counts none of the dispersive response’s buffers, on any route.

  • A gas region’s density is constant along z, as a general guide’s geometry is; a moving guide’s gas region is compiled but not tested.

  • The general route’s energy measure, like the free and planar ones, moves with the nonlinearity by itself; the tests bound the dispersive response’s share of it, not the measure.

• Open questions:

  1. Resolved: GasKerr(fill) in solprop.modes is the way a general modal region declares a gas (accepted by John Travers, 2026-10-08).

  2. Resolved: the unreleased schema-24 leaf keeps regions without a new schema version (accepted, 2026-10-08).

  3. Resolved: DispersiveKerrResponse.evaluate admits three components (accepted, 2026-10-08; done at the Phase 1 merge, with the triplet-sum test extended to three components).

• Handoff:

  • Slice 1c may now run its demonstrations on any numpy_ref route; a general modal example needs a GasKerr region.

  • Phase 2 (Numba) implements the leaf on every route, the general route’s regions included, with the in-synthesis filter of the general route and the memory-ledger rows measured there.

Phase 1c — manual, tutorial and demonstration tests

• Scope: slice 1c of Phase 1 (§11), 2026-10-07, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-1c, from slice 1a’s head. The manual chapter, the quick tutorial on both demonstration cases of §2 and their short demonstration tests (§7, §8). No package code changed.

• Delivered:

  • docs/guide/media/dispersive-kerr.md, the manual chapter: one susceptibility for every mixing process; why an output-frequency coefficient is wrong; the two evaluators, their forms, error and cost; the harmonic sectors and how thg and kerr_form select them, with the envelope edge case of Phase 1a’s open question 1; the leg bound, the band edge and the three regions, with every gas’s figures read from optmat 0.4.0’s support records; how pump dynamics and VUV emission share one response (§4.7); the demonstration-case coefficient table and a map of the symmetric model’s error over degenerate-pump FWM in argon and helium with the fitted-sum, leg-bound and band-edge boundaries; mixtures (the exact per-species sum); vectors (the isotropic Kleinman tensor); choosing an evaluator and reading the manifest; limitations.

  • docs/guide/media/nonlinear.md (a section, the traits table and the mixture rule), materials.md (the electronic policies and a section on the dispersive ones), gnlse.md and general-modal.md (material dispersion, modal dispersion and the UPPE frequency factor kept apart), docs/guide/workflows/luna.md (a capability beyond Luna, whose \(\gamma_3\) is one constant per gas).

  • docs/tutorials/capillaries/plot_11_dispersive_kerr.py: the argon DUV FWM case and the helium RDW case under the three policies.

  • tests/driver/test_dispersive_demonstrations.py: 15 tests, medium.

  • The CHANGELOG.md entry of slice 1a extended; this report.

Decisions

• The argon case. A 50 µm-radius capillary, 0.5 m, a 400 nm pump and an 800 nm seed of 10 µJ and 30 fs each, the grid’s reference at the pump, wavelengths from 210 nm so that the active band (to 190.9 nm) stays inside argon’s 189.4 nm band edge. The radius is a choice of this tutorial, not Durfee et al.’s geometry. The pressure is the linear phase matching of the HE₁₁ mode, the zero of \(2\beta_p-\beta_i-\beta_s\) formed from MarcatiliMode.neff_minus_one with brentq: 209.4 mbar (93.1 mbar at 75 µm, 52.3 at 100 µm). The coefficient table of §4.7 is computed in the tutorial from DispersiveKerrModel and reproduces §4.7’s argon and helium rows to the printed digits.

• The helium case is not reduced. plot_02_rdw_emission.py’s case — 1 m, 1.2 bar to vacuum, two modes, 100 nm to 1.5 µm, 200 fs — runs in 8–11 s per policy on numpy_ref, so it is used whole, changed only in backend and policy.

• Every run selects backend="numpy_ref", the carrier runs too, so the comparison is on one backend; the default would choose Kokkos, which refuses the leaf by name.

• The demonstration tests run ten fixed steps over 2 mm with the split-step absorber scheme: the default collar of a guide that short is too stiff for an explicit step of a tenth of it (the run refuses with a stability warning otherwise). The helium test takes M2’s geometry, grid, pulse and two modes at its constant entrance pressure, not the gradient.

• Zero dispersion through the whole driver. optmat’s with_parameters model has no support record of its own, which DispersiveKerrModel.from_optmat refuses, so the test substitutes an equivalent constructed model at the material layer (OptmatKerr.dispersive_model, monkeypatched): a = b = b' = 0, the static value the carrier’s Kerr coefficient, the real model’s leg bound and edges.

• The near-infrared bound is computed, not quoted. §4.7’s survey procedure (default_rng(4), uniform wavelengths, output kept in band) on the run’s own active band, 545 nm to 2.4 µm, gives the symmetric model’s largest sampled error, 0.34 %, below §4.7’s 400–2000 nm argon maximum of 0.913 %; the test compares the evaluators’ nonlinear change of the field (the run minus a linear one) against it.

Deviations from Part I

• §8’s “domain maps of §7” are one figure, the symmetric model’s error over degenerate-pump FWM in two gases with the region boundaries. Maps of the fit’s uncertainty and of other quartet families are not drawn; they belong with Phase 4’s application guidance.

• The full-size M2 comparison in the worked example and the optional argon worked example (§8) are not done: they are Phase 4’s, run by hand.

Verification

All on macOS (Apple silicon), Python 3.14.5, optmat 0.4.0, Kokkos CPU installed from the slice 1a branch (not used by these runs).

• Demonstration tests (pytest -n 4, 13 s; the module’s fixture runs the six demonstration runs once per worker, 6.3 s): 15 passed.

  • Both cases under all three policies complete, finite, on numpy_ref.

  • Regions: argon supported and bounded only (bounded 190.9–215.9 nm); helium all three, supported to its 151.9 nm leg bound and best effort below its 120.3 nm edge.

  • With the dispersion zeroed, both evaluators reproduce the carrier run’s exit field to 4e-16–8e-16 of its largest bin (bound 1e-13); the real models move it by 4e-4 to 1e-3, asserted above 1e-5.

  • Near-infrared window (2 cm of 1 bar argon, 20 µJ, 30 fs; the nonlinear change is 0.20 of the linear field): symmetric and polynomial differ by 6.3e-6 of the nonlinear change, against the bound of 0.34 %.

• Tutorial, 39.6 s in the docs build and 41 s run as a script, on numpy_ref, deterministic (no noise):

  • Argon: the 250–285 nm signal peaks at 3.17 µJ at 0.44 m under the carrier policy and at 2.62 µJ at 0.32 m under both dispersive ones; at the exit 3.08 µJ against 2.15 µJ (symmetric) and 2.17 µJ (polynomial). The dispersive runs lead by 7–8 % in the first centimetres, as the square of the generating quartet’s 4 % predicts, and then saturate earlier: the signal’s cross-phase modulation by the pump is 20 % stronger than the carrier’s.

  • Helium: the 100–140 nm band at the exit carries 3.63 µJ (carrier), 3.98 µJ (symmetric) and 3.97 µJ (polynomial), −8.5 % for the carrier; its centre moves from 133.2 to 134.2 nm.

  • Per-run cost: argon 7.1 / 8.5 / 10.4 s, helium about 7.5 / 9 / 11 s (carrier / symmetric / polynomial), on a shared workstation.

• Step tolerance: the tutorial’s argon and helium runs under the carrier and polynomial policies at the default rtol=1e-6, atol=1e-10 and at 1e-8, 1e-12: the exit signal and RDW-band energies agree to the fourth or fifth digit (argon 3.0779 and 2.1694 µJ at both; helium 3.6348 against 3.6347, and 3.9738 at both), so the policy differences are the models’, not the stepper’s.

• Docs build (poe docs, every tutorial executed): the first build failed on two warnings from slice 1a’s dispersive_cubic docstring, which named the private _isotropic and _product_spectrum with :func: roles that cannot resolve; they are now literals (src/solprop/nonlinear/responses.py, docstring only), and the rebuild succeeds with no warning. The tutorial took 39.6 s in the build, the longest of the gallery (the next is 6.6 s): every other tutorial runs on Kokkos, this one on numpy_ref.

• ruff, basedpyright (the test module), codespell and the public-tree scan on the changed files; tests/test_seps.py, 127 passed.

• Not run: grid convergence of the tutorial’s runs; no Numba or Kokkos execution; no full gate.

The rest of this report

• Sources checked: optmat 0.4.0’s support records, through DispersiveKerrModel.from_optmat, for every leg bound and band edge the manual quotes (all fifteen gases, both polarisations); Part I’s §4.7 tables, recomputed. Durfee et al. [9] is cited for the process only; its geometry was not used. No new literature.

• Limitations:

  • The tutorial’s runs are converged in step tolerance (above) but not checked on a finer grid or wider window.

  • The demonstration tests are ten-step functional checks; they do not test conversion, phase matching or the RDW.

  • The manual’s statements about the general modal route (its kerr_form mapping, its refusal) describe this branch; slice 1b changes them.

• Open questions:

  1. In a one-mode exploration of the helium case (not in the tutorial) the 100–140 nm band at the exit differed by about 60 % between the carrier and the dispersive policies (2.2 against 3.5 µJ), against 9 % with two modes: the sensitivity depends strongly on the mode set. Phase 4’s full-size comparison should report it per mode count.

  2. Resolved: the 40 s tutorial, the gallery’s slowest by a factor of six, was accepted by John Travers on 2026-10-08 until a compiled backend runs the dispersive response; Phase 2 moves it off numpy_ref.

• Handoff:

  • Slice 1b’s routes were reconciled with these pages at the merge of the two slices: dispersive-kerr.md’s harmonic-sector paragraph and limitations, materials.md’s route sentence, the general-modal.md paragraph and the CHANGELOG entry now state the general modal, free-space and planar routes.

  • Phase 4 reruns docs/examples/processes/plot_00_vuv_dispersive_wave.py with the three evaluators at full size, and may add the argon worked example.

Phase 2a — numba_cpu’s fused path

• Scope: slice 2a of Phase 2 (§11), 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-2 (solprop only; solprop-kokkos unchanged at kernel schema 24, which it still refuses by name). numba_cpu executes responses.dispersive_kerr on its fused node path: the single-mode, mode-averaged multimode and radial routes, field and envelope grids, both evaluators, both sectors, one or two components, mixtures as Phase 1’s exact per-species sum, beside Raman, field plasma and envelope plasma. Slice 2b adds the free-space, planar and general modal routes, §6’s mixture optimisations and the tutorial’s move off numpy_ref.

• Delivered:

  • backends/numba_cpu/dispersive.py: DispersiveWorkspace (buffers, plans and the per-stage prepare, scale, respond_chunk, finish, finish_coarse, memory_ledger), DispersiveLayout, DispersiveMatrices, the product-kernel factories make_dispersive_self and make_dispersive_pairs (closure-specialised, never disk-cached), the spectral kernels filter_spectra, weigh_terms, combine_spectra, add_windowed, apply_window, and DISPERSIVE_LEDGER_ROWS.

  • backends/bodies.py: twelve bodies of the sector products — the isotropic contraction, its number-conserving and harmonic groups for two components, and the closed forms of (u, u, u), (u, u, f) and (u, f, f) for one, in each form.

  • backends/numba_cpu/backend.py: the workspace on the fused path (_allocate_dispersive, _modal_to_time, _over_nodes, _on_accepted_step), its ledger rows and metadata (dispersive_alone, dispersive_planning_seconds); kernels.column_spec takes the analytic family for the conserving sector on a field grid; backends/driver.py: refuse_dispersive_kerr now refuses only the routes off the fused path.

  • Tests: tests/backends/test_numba_dispersive_kerr.py (82) and tests/backends/test_numba_dispersive_kernels.py (36, in the coverage lane’s interpreted pass, pyproject.toml); the product factories in test_numba_cache_policy.py; Phase 1’s refusal tests narrowed to Kokkos and to numba_cpu’s remaining routes.

  • benchmarks/sep6/dispersive_cost.py, the Q5 measurement; the ledger entry in docs/dev/benchmarks.md; docs/dev/backends.md §3.11, its route table and limits; docs/dev/kernel-contract.md and docs/guide/media/dispersive-kerr.md on what runs where; a changelog line.

Decisions

• The multipliers act on the modal columns, not at the nodes. The legacy transform’s synthesis and projection are real matrices independent of frequency, and the temporal collar is the same at every node, so P G[(G E)^3] = G[P (G E)^3]. The input multipliers act on the padded modal spectrum the stage combination already formed — the field spectrum before the stage’s inverse transform, so the response forms no forward transform of the node field — and one batched inverse transform gives every filtered field; the products are formed at the nodes and projected by the same GEMMs as the polarisation; one batched forward transform, the output multipliers and the channels’ densities summed in the spectrum, one inverse transform and the collar finish them on the columns. On the 33-node capillary rule that is a few transforms of a few columns where the reference takes them per node.

• The collar stays after the output multiplier, as the reference applies it to the summed polarisation (w G[(G E)^3]). It costs the output’s forward and inverse transforms of the columns, which G[w (G E)^3] would fold into the stage’s own; that fold changes the operator at the window’s edges and is not this backend’s to make (open question 1).

• A lone response owns the node and projected buffers. With no constant Kerr term, Raman, plasma or envelope plasma beside it, the response column and the polarisation’s projection do not run and the response writes the projected buffer. A symmetric response of one species also takes the stage’s forward plan and buffers for its product and, with no in-stage absorber reading the bare modal field, filters the stage’s spectrum in place: no field-sized buffer of its own. Beside other responses the bare field stays theirs, so Raman and plasma are never filtered.

• The embedded rule is finished lazily: its products are projected at every stage and transformed when a step is accepted, from the last stage’s buffers, once per evaluation, before the alarm transforms the embedded projection — the envelope currents’ scheme.

• The polynomial subtracts the bare analytic signal’s zero-frequency constant. It reads the bare field the analytic family synthesised, which keeps the zero-frequency bin once; Π₊ excludes it. The state carries some: the split-step absorber’s product in time puts content at every bin after the band mask. Without the correction one evaluation on a state with such content disagreed at 1e-6 (a ten-step plasma run at 3.5e-7); with it, at a few ε. The symmetric evaluator’s filtered fields carry Π₊ in their multipliers.

• The node budget is unchanged. The response adds one node buffer (symmetric beside other responses) or two (polynomial) of the chunk’s size, which node_chunk, sized at translation, does not count (open question 3).

Deviations from Part I

• Q5’s targets are missed in places (below): the symmetric stage on one node and on a mixture, the polynomial on a mixture, and the memory targets beside Raman or plasma and for the polynomial’s products. Q5 makes a miss an optimisation task, never a refusal; the causes and the measured remedies are in the open questions.

• The tutorial still runs on numpy_ref; moving it is slice 2b’s, with the routes it needs.

Verification

macOS (Apple silicon, 18 cores), Python 3.14.5, optmat 0.4.0, pyFFTW 0.15.1 on FFTW 3.3.11, Kokkos CPU installed from the kokkos main checkout (schema 24, not exercised but for its refusal).

• Parity with numpy_ref (test_numba_dispersive_kerr.py, relative L2; bounds from the transforms and node reductions both backends perform, asserted positive):

  • one evaluation’s projected polarisation on the 33-node, three-mode argon capillary, both evaluators × field and envelope × both sectors: 2.3–4.0 ε on the fine rule, 2.1–3.9 ε on the embedded one, against about 1600 ε (symmetric) and 2400 ε (polynomial); the whole stage source agrees as the constant cubic’s does on the same problem (150–1400 ε from the prefactor’s band edges, identical for the three policies);

  • one Runge–Kutta step (seven evaluations) on the same capillary: 0.9–1.7 ε against 7400–13000 ε; ten replayed steps of the default integrator: 3.0–8.2 ε;

  • the quadrature alarm after ten steps, which reads the lazily finished embedded rule: alarms of 5e-15 and 7e-13 agreeing to 3.4e-17;

  • the mode-averaged route (one node), a two-component elliptical launch, the radial route on field and envelope grids (the latter on the real radial GEMM), a He/Ar mixture, nitrogen’s Raman beside it (both sectors), field and envelope plasma beside it, five node chunks of eight, a state with zero-frequency content, and the Runge–Kutta march: one evaluation and ten steps each within their bounds; beside the plasma column the bound adds the carrier policy’s own disagreement on the same problem (916 ε on the fine rule);

  • bitwise across two runs and across one and four threads with the parallel regime forced, with and without Raman.

• Kernels (test_numba_dispersive_kernels.py, compiled and interpreted): the closed forms against the general contraction and the reference’s _isotropic groups to 8 ε, the real products bit for bit; the product kernels against the reference’s sector groups, every form, one and two components, with and without the offset, to 16 ε of the largest sample; the spectral kernels against NumPy; one small evaluation end to end against numpy_ref.

• Memory ledger (tested): every dispersive_* row named at zero on every problem and released on close; a lone symmetric response holds no field-sized buffer beside the split-step absorber and one (its filtered modal field) beside the in-stage one; the polynomial two derivative fields, two node buffers, three projected products per rule and their spectra; beside Raman, one node buffer, one projected product per rule, their spectrum and its output.

• Q5, cost (benchmarks/sep6/dispersive_cost.py at fec4e09, the ledger entry of docs/dev/benchmarks.md): the RDW grid (8192 fine samples), a 125 µm helium capillary at 0.8 bar; same-process medians of seven interleaved rounds on a workstation at a load average of 16 to 30, so ratios only. Stage ratios to the constant cubic, one thread; default pools:

  case	symmetric, thg / no thg	polynomial, thg / no thg
  one mode-averaged node	1.51 / 1.64; 1.58 / 1.55	2.94 / 3.55; 2.96 / 3.64
  four modes, 33 nodes	1.17 / 1.11; 1.28 / 1.20	2.56 / 3.45; 2.62 / 2.69
  He/Ar mixture, four modes	2.17 / 2.08; 2.34 / 2.05	4.04 / 5.67; 3.72 / 3.92

  Met: the symmetric evaluator on the multimode capillary, the polynomial on one species and on the mixture in the default pools. Missed: the symmetric evaluator on one node and on the mixture, the polynomial on the mixture at one thread. The ten-step windows, which carry the integrator’s own per-step work, move by 1.04–1.62 (symmetric) and 1.26–3.34 (polynomial). The isolated response — the constant cubic’s column against the dispersive response’s filter, transforms, syntheses, products, projections and output — is 4–10 µs against 40–71 µs (symmetric) and 109–240 µs (polynomial) on one node, 207–315 µs against 423–675 µs and 1010–2450 µs on four modes. An earlier measurement at a load average of 38, before the stage’s views were made once, gave the same ratios within 0.1 but for the mixture’s polynomial (4.25 and 5.81; 3.72 and 4.11).

• Runs (pytest -n auto): the new modules, 118 tests in about a minute; with the numba_cpu modules, the cache policy, Phase 1’s dispersive modules, test_driver.py, test_bodies.py, test_memory.py, test_vector_raman.py, test_envelope_plasma.py and test_radial_nonlinear.py, 1071 passed at the branch’s head. An earlier run of the set saw one case of test_numba_cpu_plasma_scan.py (the blocked plasma column, which this slice does not touch) fail once and pass on rerun, alone and under xdist, as Phase 1 saw against a numba disk cache filled under xdist. The kernel module with test_bodies.py under NUMBA_DISABLE_JIT=1, 80 passed; dispersive.py 100 % covered by the new modules (interpreted kernels, compiled rest). The docs build with -W -n passes with the galleries unexecuted; executed, the planar-hybrid tutorial (docs/tutorials/free_space/plot_04_planar_hybrid.py) fails translation on the spatial collar’s stability limit (a fixed 1 cm step against 6.96 mm), which nothing in this slice touches (open question 5). ruff, basedpyright (no errors) and the public-tree scan on the changed files.

• Not run: the full gate, the slow lane and the coverage lane (the SEP’s end); no Linux or quiet-machine timing; no Kokkos change.

The rest of this report

• Sources checked: Part I §6 and §11, the Phase 0 and Phase 1 reports, the Phase 1 code (dispersive_cubic, the numpy_ref pass, the leaf). No new literature or material data.

• Limitations:

  • The symmetric stage on one node costs about 1.5 to 1.6 times the constant cubic’s, most of it the output’s transform pair (open question 1).

  • A mixture costs a response path per species (Phase 1’s exact sum).

  • Beside Raman or plasma the response holds its own node field, projected products, their spectrum and its output: four real modal fields on the full cubic and ten on the analytic family, against Q5’s one. The polynomial’s three projected products per rule and their spectra exceed its one product buffer (eleven and twenty-two units beside its two derivative fields).

  • The node chunk is sized without the response’s node buffers.

  • Timings are same-process ratios on a heavily loaded workstation.

• Open questions:

  1. Resolved (John Travers, 2026-10-08): the collar is applied before the output multipliers, G[w (G E)^3] for w G[(G E)^3] and likewise for the polynomial’s output weights, in the reference and the backends; slice 2b makes the change.

  2. Resolved: whether an optimisation slice is needed is decided on slice 2b’s measurements, after the mixture optimisations land.

  3. Resolved: node_chunk counts the response’s node buffers; slice 2b makes the change.

  4. Resolved: the split-step absorber’s out-of-band content is raised as a separate issue, outside SEP 6.

  5. The planar-hybrid tutorial no longer translates (above); it needs shorter steps, RungeKutta() or a longer boundary, outside this SEP (raised separately).

• Handoff:

  • Slice 2b adds the free-space and line routes (their nodes are the columns, so the input multipliers can act on each column’s spectrum before its temporal inverse transform, the per-node scheme of §6), the planar route (rows to time a block at a time, likewise), the general modal route (frequency-dependent synthesis: the in-synthesis filter of Phase 1b’s decision, measured there), §6’s mixture optimisations measured against the exact sum, and the tutorial off numpy_ref. It removes refuse_dispersive_kerr as the routes land.

  • Phase 3 (Kokkos) can take this slice’s structure: the multipliers on the modal columns, the lazy embedded rule and the zero-frequency offset.

Phase 2b — the remaining routes, the decided changes and the mixtures

• Scope: slice 2b of Phase 2 (§11), 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-2 (solprop only; solprop-kokkos unchanged at kernel schema 24, which it still refuses by name). The two changes decided on 2026-10-08 — the collar before the output multipliers, and the node budget counting the response — in the reference and numba_cpu; numba_cpu on the Cartesian free-space, line, planar and general modal routes, so that it runs the response wherever numpy_ref does; §6’s mixture optimisations, the exact polynomial grouping adopted and the symmetric mixture-curve filter measured; the tutorial off numpy_ref; Q5 measured again. Phase 2 is complete with this slice.

• Delivered:

  • nonlinear/responses.py: dispersive_cubic(..., window=) and DispersiveKerrResponse.evaluate(..., window=) apply the temporal collar to each product in time, before Π₊ and the output multipliers (G[w (G E)^3]); None is a unit collar, bit for bit the operator without one. backends/numpy_ref/passes.py: dispersive_kerr_polarisation(..., window=), and nonlinear_polarisation adds the dispersive term after windowing every other response’s, never windowing it twice. The general modal route has no collar.

  • backends/numba_cpu/dispersive.py: the response’s windowed products are transformed and their weighted spectra land in the stage’s own polarisation spectrum (add_spectra, weigh_spectrum), so it has no inverse transform; a lone symmetric response of one species projects into the stage’s projected buffer and takes the stage’s forward transform; DispersiveTerms and dispersive_terms_layout group the polynomial’s species of equal leg bound and give each region (on the general route) its own cube; StageWeights weighs the channels and groups once per stage. The dispersive_total_spectrum and dispersive_total_time ledger rows are gone with the buffers.

  • backends/numba_cpu/dispersive_spatial.py: SpatialDispersive for the Cartesian, line and planar routes, per chunk of columns or block of rows. backends/numba_cpu/dispersive_general.py: GeneralDispersive and its three-component kernels (make_general_dispersive), the multipliers in the route’s synthesis. backends/bodies.py: dispersive_isotropic3 and dispersive_conserving3. backends/numba_cpu/general_modal.py: the general workspace runs the response and skips an identically zero regional contraction. backends/driver.py: refuse_dispersive_kerr removed.

  • translate/problem.py: _dispersive_chunk, the node chunk’s admission of the response’s buffers.

  • Tests: tests/backends/test_numba_dispersive_routes.py (new, 103); test_numba_dispersive_kerr.py (the spectral comparison, the krypton–xenon group, 85), test_numba_dispersive_kernels.py (the new kernels and bodies, the layout, 43), tests/nonlinear/test_dispersive_kerr_response.py (the collar’s two orders), tests/translate/test_dispersive_kerr_terms.py (the node admission), test_dispersive_kerr_routes.py (the window in the node checks, the unit collar, Kokkos’s refusal alone), test_numba_cache_policy.py.

  • benchmarks/sep6/dispersive_cost.py (the new routes and mixtures), benchmarks/sep6/mixture_curve.py (new); the ledger entry in docs/dev/benchmarks.md; docs/dev/backends.md §3.11, the route table and the limits; docs/dev/kernel-contract.md; docs/guide/media/dispersive-kerr.md (the collar, the cost, mixtures, backends), materials.md, general-modal.md, solver.md; docs/tutorials/capillaries/plot_11_dispersive_kerr.py on numba_cpu; the changelog.

Decisions

• Where the collar sits in the conserving sector. The decision places the collar on the cubic, before the output multipliers. In the conserving sector the cubic in time is the complex inner product of analytic signals, Re{¼ Π₊[S]}, and Π₊ is applied in the spectrum with the output multipliers, as the Phase 0 forms apply it; so the collar multiplies S, before Π₊: Re{¼ Π₊[w S]} under G. This keeps the output strictly on positive frequencies and lets a backend finish the response in the stage’s own transforms. With the dispersion removed, it is the constant thg=False rule exactly under a unit collar; under the free route’s real collar it departs from it by 1.2e-9 of the largest bin, the collar’s leakage below zero frequency that Π₊ removes and the constant rule keeps (open question 1).

• The general route keeps its absent collar: it has none on either backend.

• The polynomial’s exact grouping is numba_cpu’s default, as the slice’s scope decided: a group is the channels whose leg-weight rows are equal bit for bit (and, on the general route, whose region is the same). numpy_ref keeps the per-species sum.

• The node budget counts, on the fused path, no node buffer for a lone symmetric response of one species, one beside other responses and two for the polynomial; a Cartesian or planar column adds two complex-equivalent buffers per product (in time and in frequency) to the free route’s twenty. The general route sizes its chunk from its maps and is not changed (open question 4).

• The mixture-curve filter is measured, not offered, as the slice’s scope requires: no policy, option or leaf selects it; the benchmark builds its one-channel leaf by hand.

Deviations from Part I

• §6 recommends the mixture-curve filter for symmetric mixtures; it is measured and left for a decision, not adopted (the slice’s scope).

• §6’s polynomial aggregation is adopted only for species of equal leg bound, the exact case. Air’s three species have three leg bounds and gain nothing (open question 2).

Verification

macOS (Apple silicon, 18 cores), Python 3.14.5, optmat 0.4.0, pyFFTW 0.15.1 on FFTW 3.3.11, Kokkos CPU installed from the kokkos main checkout (schema 24, exercised only for its refusal); a load average between 6 and 43.

• The collar (test_dispersive_kerr_response.py): a unit collar gives the operator without one bit for bit, both evaluators, sectors and grids. An 8 fs pulse in the window’s flat middle gives the two orders to 5e-15–3.3e-12 of the largest sample; one at nine tenths of the half-window separates them by 3.2e-3–5.9e-3, of which the part inside the flat middle is 0.4–1.1 % of the part where the collar varies in the full sector and on envelope grids, and a third on a field grid’s conserving sector, whose Π₊ is a Hilbert transform. The free and planar routes’ node checks equal evaluate(..., window=) bit for bit; the free route’s zero-dispersion check runs at a unit collar (above).

• Parity with numpy_ref (relative L2; bounds from the transforms and reductions both backends perform, asserted positive):

  • the fused path (test_numba_dispersive_kerr.py, now comparing the stage’s polarisation spectra, where the response joins): unchanged in kind from Phase 2a, every case within its bound, and the krypton–xenon polynomial, one group, one evaluation and ten steps in three sectors;

  • free space and the line, both evaluators, field full and conserving, envelope conserving and full: one evaluation 2.2–4.9 ε against bounds of 640–3100 ε, ten steps 0.65–1.5 ε; beside Raman 3.1–5.0 ε and 0.8–1.3 ε; a helium–argon medium; a partial chunk; one Runge–Kutta step;

  • the planar route, both sectors: 3.7–5.2 ε against 1800–3500 ε, ten steps 0.78–1.3 ε; beside Raman 3.3–5.4 ε; beside the in-stage collar, which reads the bare rows; a partial block; one Runge–Kutta step;

  • the general route, gas alone, beside a solid core and a helium–argon region, real and analytic forms: 0.47–1.8 ε against 560–1100 ε, ten steps 0.53–1.0 ε; one Runge–Kutta step;

  • bitwise across runs and across one and four threads with the parallel regime forced on the fused, free-space, planar and general routes.

• Kernels (test_numba_dispersive_kernels.py, compiled and interpreted): the three-component bodies against the reference’s _isotropic and conserving group to 8 ε; the general product kernels against the reference at their region’s nodes and exactly zero at another’s; the layout’s groups, cubes and terms; the spectral kernels against NumPy.

• Node budget: a chunk admits two nodes at twice the per-node bytes and refuses one byte short, for the four fused-path cases and a free-space column, as Raman’s admission is tested.

• Mixtures: the krypton–xenon polynomial is one group (two filtered fields, three products) and agrees with the reference’s per-species sum as the one-species cases do.

• Runs (pytest -n auto): the dispersive, route, general, free, planar, radial, cache-policy, driver, memory and bodies modules, 676 passed and 8 skipped; every tests/backends/test_numba_*.py, 1007 passed; tests/nonlinear, 1012 passed; tests/translate, 1174 passed and 3 skipped; the driver’s dispersive modules, 25 passed. The kernel modules under NUMBA_DISABLE_JIT=1, 87 passed. ruff, basedpyright (no errors) and the public-tree scan on the changed files. The test counts above preceded the in-place filter of the last commit, after which the route module (103) was run again.

• The tutorial on numba_cpu prints exactly Phase 1c’s figures: argon’s signal 3.17 µJ at 0.44 m (carrier) and 2.62 µJ at 0.32 m (both dispersive), 3.08, 2.15 and 2.17 µJ at the exit; helium’s 100–140 nm band 3.63, 3.98 and 3.97 µJ, centres 133.19 and 134.18 nm. Its time in the docs build: 15.9 s, against 39.6 s on numpy_ref (Phase 1c), at a load average near 10; run as a script under a load average of 43 it took 116 s of wall time for 23 s of CPU time.

Q5 measured again

Same-process stage ratios to the constant cubic (benchmarks/sep6/dispersive_cost.py, seven interleaved rounds, medians; the full table, the windows and the response’s own times are the ledger entry of docs/dev/benchmarks.md). One thread; default pools in brackets where they differ by more than 0.1.

case	symmetric, thg / no thg	polynomial, thg / no thg
one mode-averaged node	1.12 / 1.31 (1.35)	2.49 / 3.51
four modes, 33 nodes	0.97 / 0.92	2.58 / 3.52 (2.09 / 2.38)
He/Ar, four modes	2.02 / 1.93 (1.78 / 1.68)	4.16 / 5.98 (3.26 / 3.71)
Kr/Xe, one leg bound	2.07 / 1.89	2.42 / 3.21 (2.27 / 2.68)
air, three leg bounds	2.98 / 2.67	5.38 / 7.55 (4.32 / 4.77)
Cartesian, 16 × 16	0.96 / 1.14	2.19 / 3.17 (1.95 / 2.46)
line, 64	1.01 / 1.19	1.88 / 2.65 (1.69 / 2.13)
planar	1.15 / 1.04 (1.21 / 1.23)	2.56 / 3.72 (2.57 / 3.08)
general modal, real / analytic	1.15 / 0.51	2.32 / 1.44

• Met: the symmetric evaluator for one species on every route (one node’s analytic sector at 1.31–1.35, at the target, where Phase 2a measured 1.51–1.64); the polynomial for one species everywhere, and for krypton and xenon, now one group.

• Missed: the symmetric evaluator on every mixture, 1.68–3.00: the exact response is one product per species, which no backend can merge; the polynomial on mixtures of distinct leg bounds, helium and argon at one thread and air in every pool.

• The ten-step windows move by 0.98–1.90 (symmetric) and 1.17–3.87 (polynomial).

• Where the fold of the collar into the stage’s own transforms was estimated (Phase 2a: 1.15 times on one node, full cubic), it measured 1.09–1.12.

The mixture-curve filter, measured

benchmarks/sep6/mixture_curve.py: helium and argon, 50/50, and air as nitrogen, oxygen and argon (78.08/20.95/0.97). The filter is g_mix = [γ_K,mix/γ_K,mix(0)]^(1/4) with γ_K,mix(ν) = Σ_s N_s γ_K,s(ν̃_s), each species’ leg bounded inside its own term; for a pure gas it equals that gas’s own filter to 2.2e-16.

• The coefficient (§4.7’s procedure: default_rng(4), 300 000 draws on 400–2000 nm, positive outputs in band, S₂ within the smallest fitted maximum of the species; 208 109 quartets for each mixture), in per cent, median / 95th percentile / maximum:

  mixture	curve vs polynomial sum	exact symmetric sum vs polynomial sum	curve vs exact symmetric sum
  He/Ar	0.052 / 0.488 / 0.903	0.052 / 0.480 / 0.887	0.0005 / 0.0077 / 0.0157
  air	0.078 / 0.728 / 1.350	0.077 / 0.708 / 1.305	0.0005 / 0.0206 / 0.0446

  The mixing error alone is fifty times below the factorisation’s own: the curve is as good an approximation of the polynomial as the exact per-species symmetric sum is.

• A short propagation (numba_cpu, a 125 µm capillary at 1 bar, 3 cm, 30 fs and 150 µJ at 800 nm, one mode, the default integrator), the departure of each exit field from the polynomial’s as a fraction of the polynomial’s nonlinear change (the field minus a linear run’s), and the 250–290 nm band’s share of the exit energy:

  mixture	carrier	exact symmetric	curve	band: carrier / symmetric / curve / polynomial
  He/Ar	5.80 %	0.27 %	0.29 %	1.58e-3 / 1.780e-3 / 1.780e-3 / 1.755e-3
  air	5.01 %	0.29 %	0.36 %	1.67e-4 / 1.475e-4 / 1.502e-4 / 1.367e-4

• The cost (one thread, stage ratio to the constant cubic on that capillary): the curve 1.26 (He/Ar) and 1.24 (air), the exact symmetric sum 2.24 and 3.29, the polynomial 3.82 and 5.23.

So the curve costs what one gas costs and departs from the exact symmetric sum by about a tenth of the symmetric factorisation’s own error in these cases; John Travers may decide whether to offer it (open question 3).

The rest of this report

• Sources checked: Part I §4.4, §4.7, §6 and §11; the Phase 0, 1 and 2a reports; the Phase 1 and 2a code. No new literature or material data; every leg bound and coefficient is read from optmat 0.4.0 through DispersiveKerrModel.

• Limitations:

  • A symmetric mixture costs one product per species, and a polynomial mixture one group per distinct leg bound; air’s species have three.

  • The general route’s node chunk is sized by its maps without the response’s buffers: per chunk it holds the filtered fields, the products and their spectra, two complex-equivalent buffers per field and product of three components per node.

  • The general modal guide tested has a narrow 815–860 nm band, over which the multipliers vary little; the kernels are held to the reference on random fields, and the fused and spatial routes on broad bands.

  • The tutorial’s wall time depends on the machine’s load; it was measured on a busy workstation.

  • Timings are same-process ratios on a loaded workstation; no Linux or quiet-machine timing.

• Open questions:

  1. Resolved (John Travers, 2026-10-08): the conserving sector’s collar acts on the analytic product before Π₊, as implemented.

  2. Resolved: no grouping by approximate equality; species group only when their leg bounds are equal, so the grouping stays an exact rewriting.

  3. Resolved: the mixture-curve filter becomes the symmetric evaluator’s operator for mixtures, on every backend, recorded in the manifest; the exact per-species sum stays the test reference that measures it. Slice 2c implements it.

  4. Resolved: the general route’s chunk counts the response’s buffers too (slice 2c).

  5. Resolved: the polynomial’s Q5 miss on mixtures of distinct leg bounds is a roadmap item, not a slice of this phase.

• After the merge of main (the split-step absorber now masks the band after its product, so an accepted state carries no out-of-band content): the zero-frequency offset the stage subtracts is kept as a guard. It costs one reduction per row, is zero on a clean state, and keeps the conserving sector’s exclusion of the zero-frequency bin independent of how a state was produced.

• Handoff:

  • Phase 3 (Kokkos CPU, then device) implements the leaf against numpy_ref and numba_cpu, taking this phase’s structure: the multipliers off the nodes (on the modal columns, on each chunk’s columns or rows, in the general route’s synthesis), the collar on the products and the response’s spectrum added to the stage’s, the lone symmetric response on the stage’s own transforms, the polynomial’s grouping, the lazy embedded rule and the zero-frequency offset. The kernel schema is unchanged by this slice.

  • Phase 3 copies the structure after slice 2c (the mixture-curve filter).

Phase 2c — the symmetric mixture filter and the general route’s budget

• Scope: slice 2c of Phase 2 (§11), 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-2c (solprop only; solprop-kokkos unchanged at kernel schema 24, which it still refuses by name). The two decisions of Phase 2b’s open questions 3 and 4: §6’s mixture-curve filter as the symmetric evaluator’s operator for a mixture, on numpy_ref and numba_cpu and every route, recorded in the manifest and the resume key, with the exact per-species sum kept as the test reference; and the general modal route’s node chunk counting the response’s buffers.

• Delivered:

  • nonlinear/responses.py: mixture_filter(filters, factors), the operator’s definition: F = Σ_s f_s and g_mix = [Σ_s f_s g_s⁴ / F]^(1/4) with f_s = ε₀ γ₃,ₛ ρ_s; one species returned unchanged; a unit filter at F = 0.

  • translate/responses.py: DispersiveKerrTerms.groups() (the symmetric evaluator’s channels acting at the same nodes; the polynomial’s of equal leg weight and region, the grouping numba_cpu already took) and evaluations(densities), the cubics the reference evaluates at a stage with their factors. The leaf’s fields are unchanged.

  • backends/numpy_ref/passes.py (dispersive_kerr_polarisation) and general_modal.py (dispersive_polarisation) evaluate evaluations: one cubic per symmetric group, one per polynomial channel.

  • backends/numba_cpu/dispersive.py: DispersiveTerms lays out one field and one product per symmetric group (mixtures, quartics, input_scale, output_scale); StageWeights holds the stage’s own input and output rows, rebuilds a mixture’s with the kernel mixture_rows and skips an update whose densities equal the last; the workspace weighs the channels in prepare, before it filters (the backend’s _modal_to_time(bundle)), and a symmetric mixture takes the lone path. dispersive_spatial.py and dispersive_general.py read the stage’s rows.

  • translate/problem.py: _dispersive_node_buffers counts groups. backends/numpy_ref/general_modal.py: general_dispersive_buffers(leaf) and general_node_chunk(..., dispersive=), used by _general_maps, the chunk both backends take; backends/memory.py estimates the same chunk and, on numba_cpu, its dispersive bytes.

  • driver/material_output.py: MIXTURE_FILTER_RULE; material_manifest(..., leaf=) writes policies["electronic_response"]["mixture_filters"] for each symmetric group of several species – the rule, the general modal region (None elsewhere), the composition as each population’s label and the SHA-256 of its density path’s record – and each such population’s electronic entry names its filter (filter); driver/simulation.py passes the translated leaf.

  • Tests: tests/nonlinear/test_dispersive_kerr_mixture.py (new, 11, with exact_symmetric_sum, the test reference); test_numba_dispersive_kernels.py (the layout, the stage’s rebuild and the kernel, bit for bit against mixture_filter, 4); test_numba_dispersive_kerr.py (a mixture of varying composition, ten steps; a lone mixture’s ledger, 3); test_dispersive_kerr.py (the numpy_ref pass, 1); test_dispersive_kerr_routes.py (a general mixture region bit for bit, the general chunk, the region’s manifest, 7); test_dispersive_kerr_terms.py (the fused admission of a mixture, 2); tests/driver/test_dispersive_policy.py (the record and its change with the composition, 2).

  • benchmarks/sep6/dispersive_cost.py (the filter’s rebuild timed); benchmarks/sep6/mixture_curve.py (the policy against Phase 2b’s hand-built curve); the ledger entry of docs/dev/benchmarks.md; docs/guide/media/dispersive-kerr.md (the mixtures section, the cost, the record), nonlinear.md, general-modal.md; docs/dev/backends.md §3.11 and §11; docs/dev/kernel-contract.md; the changelog.

The operator, and its normalisation

For channels s acting at the same nodes, with f_s = ε₀γ₃,ₛρ_s at the stage and K̂_s = g_s⁴ = γ_K,s(ν̃_s)/γ₃,ₛ each species’ Kerr diagonal under its own leg bound,

P = F G_mix[w (G_mix E)³], F = Σ_s f_s, g_mix = [Σ_s f_s K̂_s / F]^(1/4).

This is §6’s filter factorised as §4.4 factorises a pure gas: on the Kerr diagonal F g_mix⁴ = Σ_s f_s K̂_s = ε₀ γ_K,mix, the mixture’s degenerate coefficient ε₀ Σ_s N_s γ_K,s(ν̃_s). The reference is the mixture’s static coefficient F, so g_mix(0) = 1; any other positive γ_* moves (F/γ_*)^(1/4) into the filter and γ_* out of the coefficient, and the cubic, of fourth degree in the filter, is unchanged (§4.4: the reference cancels). Both are tested: the diagonal identity to 8 ε and the reference’s cancellation to 64 ε of the largest sample.

Decisions

• The leaf is unchanged; the schema stays 24. Each channel keeps its g_s, and K̂_s is recovered as (g_s²)², a few roundings from the curve the translation sampled. The leaf’s meaning for a symmetric group of several channels changed – one cubic under the mixture filter, no longer the per-species sum – and docs/dev/kernel-contract.md states it; no out-of-tree backend runs the leaf (Kokkos refuses it whole by name), as when Phase 2b changed the collar’s order at the same version. John Travers may still want a version for it (open question 1).

• Groups by node region. Every channel of a fused, free-space or planar problem is one mixture; on the general modal route each region’s gas fill is its own. The polynomial is unchanged: the exact sum on numpy_ref, the exact grouping on numba_cpu.

• Rebuilt per stage, when the densities change. The filter depends on the composition. StageWeights.update returns at once when the stage’s densities equal the last update’s, so a fill of constant composition builds it once (every capillary fill at one pressure, every general modal gas region), and a composition varying along z rebuilds it at every stage: O(n_omega × n_species) and two square roots per bin, 4–8 µs on the benchmark grids (below). A shared fill under a pressure gradient has one composition in exact arithmetic, but its species’ density tables are fitted separately and their ratio moves in the last bits, so a filter precomputed once would not be identical to the rebuilt one; it is not precomputed. No cheaper exact scheme was found: the fourth root of a convex combination does not factor, and the rebuild is about 2 % of the symmetric stage.

• The arithmetic is the reference’s, in its order. Species summed in channel order from a zero, the quotient by F, the fourth root as two correctly rounded square roots: mixture_rows equals mixture_filter bit for bit (tested), so the backends build the same filter.

• A vacuum gives a unit filter. Densities are non-negative (the problem’s domain check), so g_mix⁴ is a convex combination of the K̂_s; where every density is zero the response is zero whatever the filter, which is taken as one rather than 0/0.

• The fused path weighs the channels before it filters. The input multipliers of a mixture are the stage’s, so prepare (before the stage’s inverse transform) updates the weights; begin still calls the update, now a no-op.

• A mixture takes the lone path. One product, so alone it is the lone symmetric response: no field-sized buffer of its own beside the split-step absorber (tested).

• The record. mixture_filters is written only when a symmetric group has several species, so a pure gas’s and a polynomial run’s records and resume keys are what they were. The composition’s identity is each population’s label and the SHA-256 of its density path’s record as the run’s configuration records a mixture (Mixture.serialise): sampled values, breakpoints and declarations, independent of the translation’s Chebyshev fit, which could differ in the last bits between machines. The policy enters the resume key; the configuration already carried the densities. The §4.10 regions stay per species.

• The general chunk counts two buffers per node and component for each filtered field and each product of the compiled general workspace: the symmetric evaluator one of each per gas region, the polynomial two fields and two products per group and one cube per region (general_dispersive_buffers), over the longer of the fine time and spectral axes, as general_node_chunk counts its own. numpy_ref, whose reference pass allocates per call, takes the same chunk.

Deviations from Part I and the slice’s scope

• The exact per-species symmetric sum is no longer a backend’s operator. It is the test reference (exact_symmetric_sum) and the survey’s column; Phase 2b’s propagation of it cannot be repeated through a backend, so mixture_curve.py now compares the policy with Phase 2b’s hand-built curve instead.

• The memory estimator (backends/memory.py) takes the general chunk with the response’s buffers and counts its bytes on numba_cpu, so that its chunk is the backend’s; on the fused and spatial routes it still counts none of the response’s buffers (open question 3).

Verification

macOS (Apple silicon, 18 cores), Python 3.14.5, NumPy 2.5.2, optmat 0.4.0, pyFFTW 0.15.1 on FFTW 3.3.11, Kokkos CPU installed from the kokkos main checkout (schema 24, exercised only for its refusal).

• The form against Phase 0’s triplet sum of the kernel F Π_j g_mix(ν_j), g_mix evaluated leg by leg from helium’s and argon’s models (argon’s leg bound active): both sectors, one and two components, below 1e-12 (decision Q1).

• One species is its own operator, bit for bit: mixture_filter returns the row itself; a one-channel leaf’s evaluations is its weights and coefficient ρ; every pre-existing single-species parity and reference test passes unchanged. Helium split into two populations keeps its own filter to 4 ε.

• Against the exact per-species sum (test_dispersive_kerr_mixture.py): a three-colour field (500, 800 and 1600 nm) confined to 400–2000 nm, the conserving sector, on the capillary’s own grid. At an output bin on 400–2000 nm every contributing quartet has its four legs in §4.7’s survey domain; each contribution is a positive coefficient times field amplitudes, so the departure at that bin is at most the largest relative departure of a coefficient times the exact sum of the spectral magnitudes, which for this zero-phase field is the exact sum itself. The bound is Phase 2b’s measured mixing error, the curve against the exact symmetric sum over 208 109 quartets: 1.57e-4 (He/Ar), 4.46e-4 (air). Measured: 9.2e-5 and 2.8e-4 of the bin’s value at worst, margins 1.7 and 1.6 (a random-phase field of the same spectrum: 3.2e-5 and 8.9e-5 against the magnitudes’ sum); relative L2 over the whole spectrum 4.2e-5 and 1.6e-4.

• Backends. numba_cpu against numpy_ref: the He/Ar mixture’s one evaluation and ten steps (both evaluators, field full and envelope conserving) within Phase 2a’s bounds; a mixture of varying composition, 2 to 0.5 bar of helium against 0.5 to 2 bar of argon over the 2 mm guide, ten steps under both evaluators within the same bounds; the free-space and general routes’ helium–argon cases (both evaluators, one evaluation and ten steps) within theirs; the general route’s mixture region bit for bit against dispersive_cubic under mixture_filter. Bitwise across runs and threads as before.

• Budget: the fused admission of a symmetric He/Ar mixture is that of one species (no node buffer alone), the polynomial’s two; the general chunk admits two nodes at twice a node’s bytes and one a byte short, counting 4 (symmetric) and 10 (polynomial) buffers on the strand, and _general_maps takes it.

• The record: a He/Ar capillary names its filter, the composition and two distinct density identities, and its entries the filter; a pure gas and a polynomial mixture record none; changing the composition changes the policy; a general region’s filter names region 1.

• Runs (pytest -n auto, per directory): tests/backends 3519 passed and 9 skipped (8 min at a load average near 30); tests/nonlinear 1023 passed; tests/translate 1176 passed and 3 skipped; tests/driver 1452 passed. After the last code commit (the general layout’s keys, the manifest’s membership) tests/driver with test_numba_dispersive_routes.py again, 1555 passed, and the changed modules’ doctests, 42 passed; tests/test_sol.py and tests/test_structure_figures.py 273 passed; tests/test_seps.py 127 passed and 1 skipped. The kernel module with test_bodies.py under NUMBA_DISABLE_JIT=1, 91 passed, and with the compiled dispersive modules (417 passed) the changed lines are covered: dispersive_spatial.py 100 %, dispersive.py and dispersive_general.py 99 % with only pre-existing branches missed. ruff, basedpyright (no errors) and the public-tree scan (no finding in a changed line) on the changed files. poe docs (every tutorial executed) succeeds with no warning.

• Not run: the full gate, the slow lane and the coverage lane (the SEP’s end); no Linux or quiet-machine timing; no Kokkos change.

Q5 measured again

Same-process stage ratios to the constant cubic on the four-mode capillaries (benchmarks/sep6/dispersive_cost.py at 8a6583d, seven interleaved rounds, medians, a load average of 16 to 20; the full tables are the ledger entry of docs/dev/benchmarks.md). One thread; default pools in brackets.

case	symmetric, thg / no thg	rebuild of the filter	polynomial, thg / no thg
helium alone (reference)	1.00 / 1.03 (1.06 / 1.09)	–	2.55 / 3.42 (2.34 / 2.54)
He/Ar	1.00 / 0.93 (1.04 / 1.09)	7.7 µs	4.10 / 5.91 (3.42 / 3.84)
Kr/Xe	1.03 / 0.92 (1.11 / 1.00)	4.3–5.0 µs	2.36 / 3.18 (2.36 / 2.49)
air	1.02 / 0.90 (1.06 / 1.03)	5.2 µs	5.20 / 7.27 (4.35 / 5.04)

• Met: the symmetric evaluator on every mixture, 0.90–1.11, where Phase 2b measured 1.68–3.00 for the exact per-species sum; a mixture costs what helium alone costs on the same capillary. Its rebuild, paid only where the composition changes, is about 2 % of the stage. Ten-step windows: 0.94–1.07.

• Unchanged: the polynomial, which this slice does not touch, within the machine’s noise of Phase 2b; its miss on mixtures of distinct leg bounds is the roadmap’s.

• mixture_curve.py: the policy’s filter equals Phase 2b’s hand-built curve to one ulp (2.2e-16); 3 cm adaptive runs under the two give exit fields 4.9e-9 (He/Ar) and 1.2e-15 (air) apart, the first the step controller’s response to that ulp, and the same departures from the polynomial as Phase 2b’s curve (0.287 % and 0.361 %).

The rest of this report

• Sources checked: Part I §4.4, §4.7, §6 and §11; the Phase 0 to 2b reports and their decisions; the Phase 2b code and benchmarks. No new literature or material data: every coefficient and leg bound is read from optmat 0.4.0 through DispersiveKerrModel.

• Limitations:

  • The bound on the operator’s departure from the exact sum is Phase 2b’s sampled maximum over 400–2000 nm quartets, not a supremum, and holds only for quartets in that domain; quartets with an ultraviolet leg (the RDW band) depart more, and are not bounded by a test here.

  • The rebuild runs at every stage of a composition varying along z, including a shared fill under a pressure gradient whose composition is constant in exact arithmetic.

  • The memory estimator counts the response’s buffers on the general route only.

  • Timings are same-process ratios on a loaded workstation; no Linux or quiet-machine timing.

• Open questions:

  1. Resolved (John Travers, 2026-10-08): schema 24 stays; no version for the changed meaning of a symmetric group, as for the collar’s order.

  2. Resolved: the exact per-species symmetric sum stays a test reference only; no leaf or option reaches it.

  3. Resolved: the memory estimator counts the response’s buffers on the fused and spatial routes too; done at the start of Phase 3 (solprop side).

  4. Resolved: slice 2c merged, completing Phase 2.

• Handoff:

  • Phase 3 (Kokkos CPU, then device) implements the leaf against numpy_ref and numba_cpu with this phase’s structure, now including the symmetric mixture filter: one product per group of channels at the same nodes, its filter rebuilt by mixture_filter’s arithmetic (channel order, two square roots, a unit filter at a vacuum) whenever the stage’s densities change, the general chunk counting the response’s buffers. The kernel schema is unchanged by this slice.

Phase 3a — Kokkos CPU’s fused path

• Scope: slice 3a of Phase 3 (§11), 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-3 in solprop and in solprop-kokkos (from solprop fabc01b and solprop-kokkos 895af49). Kokkos CPU executes responses.dispersive_kerr on its fused node path: the mode-averaged, multimode and radial routes, field and envelope grids, both evaluators, both sectors, one or two components, mixtures (the polynomial’s equal-bound groups; the symmetric mixture filter), beside Kerr, Raman, plasma and envelope plasma, against numpy_ref and numba_cpu. The Cartesian free-space, line, planar and general modal routes stay refused by name for slice 3b; the device build is slice 3c’s. With it, Phase 2c’s decision 3: the memory estimate counts the response’s buffers on the fused and spatial routes. The kernel schema stays 24.

• Delivered, solprop-kokkos:

  • include/solprop_kokkos/dispersive.hpp, src/dispersive.cpp: DispersiveResponse (the layout, the per-stage weights and the mixture filter’s rebuild, the filter, the synthesis and projection GEMMs, the product functors, the forward transform, the landing and weighing kernels, the lazy embedded finish, the ledger), the sample bodies of bodies.py’s dispersive forms, and dispersive_mixture_rows, mixture_filter’s arithmetic.

  • include/solprop_kokkos/core.hpp: the layout’s fields on Problem, TrialScalars::dispersive_densities, GeneralState::projected_to_host (a diagnostic: the last evaluation’s projected spectra on both rules) and dispersive_path. src/general.cpp: the response inside the stage (evaluate_stage_source), its validation, plans, densities, the embedded finish at the score, the ledger and close. src/bindings.cpp, _core.pyi: the layout, the trial’s densities, the two accessors.

  • src/solprop_kokkos/backend.py: _dispersive_payload (the layout of numba_cpu’s dispersive_terms_layout, from the leaf’s groups()), the analytic family for the conserving sector on a field grid, each stage’s densities in the trial’s material rows, _refuse_dispersive_route (the routes off the fused path, by name), the dispersive_path metadata. __init__.py: DISPERSIVE_KERR_FIELDS (new), dispersive_kerr in RESPONSE_FIELDS, dispersive_kerr_densities in STAGE_SCALARS_FIELDS, all checked by check_build.

  • Tests: tests/native/test_dispersive.cpp (8), tests/test_dispersive.py (13), the contract and refusal tests of test_backend.py and test_api_contract.py, the native tests’ trial scalars; CMakeLists.txt, scripts/check.sh; CHANGELOG.md, README.md, AGENTS.md (the public tuple).

• Delivered, solprop:

  • tests/backends/test_kokkos_dispersive_kerr.py (87) and kokkos_dispersive_thread_probe.py; test_dispersive_kerr.py’s Kokkos test now admits the leaf on the fused path.

  • backends/memory.py: dispersive_memory, in the base group of estimate_memory; translate/problem.py: dispersive_node_buffers, now public; tests/backends/test_memory.py (15 new).

  • benchmarks/sep6/dispersive_cost.py --backend kokkos; the ledger entry of docs/dev/benchmarks.md; docs/dev/backends.md (§4, §9, §11), docs/dev/kernel-contract.md, docs/guide/media/dispersive-kerr.md, the tutorial’s note on backends; the changelog.

Decisions

• numba_cpu’s structure, natively (the Phase 2 handoff). Before the stage’s inverse transform the response weighs its channels at the stage and filters the padded modal spectrum, and one batched inverse transform gives every filtered field; per node chunk the real synthesis GEMM puts them at the nodes, a functor forms the products in the leaf’s sector, and the projection GEMMs take them to both rules’ columns; one batched forward transform and a landing kernel add their weighted spectra to the stage’s own polarisation spectrum. Alone, the response replaces the response column, the projection and the stage’s forward transforms; a lone symmetric response reads the stage’s node buffer, filters the stage’s spectrum in place when no in-stage absorber reads the bare field, and outside a field grid’s analytic family is the stage’s own polarisation, weighed by g in the stage’s spectra on both rules. The polynomial subtracts the bare analytic signal’s zero-frequency constant (the guard Phase 2b kept).

• The collar acts at the nodes, in the product functors. numba_cpu multiplies the projected products; the collar is the same at every node and the projection is linear, so P (w X) = w (P X) and the operator is the decided G[w (G E)^3] either way. At the nodes it costs no pass of its own, and the borrowed product, the stage’s polarisation, carries it already; the two orders differ by rounding.

• The embedded rule’s products are finished when the step is scored. The engine transforms its own embedded projection at every stage, so a borrowed product’s embedded spectrum is weighed there; the response’s own embedded products are projected every stage and transformed and landed once, when the accepted step’s quadrature is scored, from the last stage’s products and weights.

• The layout is the adapter’s, the arithmetic native. The adapter packs the leaf as dispersive_terms_layout does (every multiplier carries the sector’s Π₊ and factor; a mixture’s term names its own factor), and the engine executes the tables. The stage’s weights – ε₀ γ₀ ρ per channel, each symmetric mixture’s coefficient and filter rows, the polynomial group’s pair constants, the terms summed per product in term order from zero – are rebuilt only when a stage’s densities differ from the last rebuild’s, compared on the host; a mixture’s rows by mixture_filter’s arithmetic (channel order, two square roots, a unit filter at a vacuum), held bit for bit to its closed form by a native test. The factors cross through a ring of four host staging buffers, as the stage scalars do, so a rebuild allocates nothing and needs no fence.

• The dense transforms take the build’s GEMM: the real matrices times the samples’ interleaved real views through selected_gemm (the vendor BLAS, the native-order kernel or the custom one, as the build’s provider says); with more than one node chunk each chunk’s columns of the scaled projections are packed once a stage.

• The contract tuples. DISPERSIVE_KERR_FIELDS is a new public tuple, as MODAL_TABLES_FIELDS is for its leaf: the leaf’s members the adapter reads (carrier, coefficients, evaluator, filters, n_channels, outputs, positive, products, regions, sector); its grouping is solprop’s own groups(), which the adapter calls. The schema stays 24: no solprop field changed.

• Phase 2c’s decision 3, the memory estimate. dispersive_memory(problem, backend) returns rows of the base group. On the fused path both compiled backends hold the response’s filtered fields, node slots, products, embedded products and spectra by one rule, and the rows are those ledgers’ to the byte. On the Cartesian, line and planar routes (numba_cpu) the row is the complex-equivalent buffers per node and component the node budget admits (dispersive_node_buffers) times the chunk, as the general route’s chunk counts its own; Kokkos refuses the response there. numpy_ref, whose pass allocates per call, and the general route, whose chunk already counts the response, add nothing. The multipliers, scratch and packed projections are named among what is not counted.

Deviations from Part I and the slice’s scope

• The collar’s place in the arithmetic (above): the same operator, a different rounding from numba_cpu’s.

• Envelope plasma is compared on the whole stage source. The engine assembles the envelope currents into the projected spectrum, which the reference keeps beside it, so an envelope-plasma problem’s projected spectra differ by the currents themselves (a relative 1.8); the stage source holds them on both.

• Thread invariance is bitwise where every transform is short. The response’s batched transforms follow the engine’s rule: a transform of fewer than 8192 real-side samples is planned on one FFTW thread. On the mode-averaged route all are, and ten steps are bitwise at one and four threads; the multimode capillary’s batched products exceed it and may move by rounding with the thread count, as the engine’s own long transforms may.

Verification

macOS (Apple silicon, 18 cores), Python 3.14.5, NumPy 2.5.2, optmat 0.4.0, pyFFTW 0.15.1 on FFTW 3.3.11; solprop-kokkos at 41ca10e (the branch’s native code), the local profile (Homebrew Clang 23.1.2, Accelerate, native architecture), Kokkos 5.2.1, Kokkos-FFT 2.0.0, OpenMP, OMP_PROC_BIND=false; a load average of 16 to 44.

• Parity (test_kokkos_dispersive_kerr.py; relative L2; numba_cpu’s bounds, from the transforms and node reductions both backends perform, asserted positive):

  • one evaluation’s projected polarisation on the 33-node three-mode argon capillary, both evaluators × field and envelope × both sectors: 2.3–4.3 ε on both rules against numpy_ref and 0.5–2.3 ε against numba_cpu, under bounds of 1064–1870 ε;

  • one Runge–Kutta step (seven evaluations): 1.2–1.5 ε against 7400–13100 ε; ten steps of the default integrator (the resident Adams engine): 5.8–6.7 ε against numpy_ref and 5.6–6.7 ε against numba_cpu, under 155 000–273 000 ε; ten Runge–Kutta steps 5.7–5.9 ε;

  • the quadrature alarm after ten steps, which reads the embedded products finished at the score: 7.2e-13–7.5e-13 (full cubic) and 4.9e-15 (conserving), within 4.7e-17 of the reference’s;

  • the mode-averaged route 2.4–4.2 ε (ten steps below 0.1 ε), a two-component launch 2.8–3.8 ε, the radial route 2.7–5.0 ε; a He/Ar mixture 2.9–4.1 ε (ten steps 5.9–6.8 ε), krypton and xenon as one polynomial group 2.9–4.1 ε (6.6–7.3 ε), a varying composition (the filter rebuilt per stage) over ten steps, nitrogen’s Raman beside it 2.5–3.4 ε (6.4–7.2 ε), five node chunks of eight, a state with zero-frequency content;

  • beside plasma, the stage source: the carrier policy’s own disagreement with the reference is 268 ε (the field’s plasma column) and 178 ε (envelope plasma), and the dispersive runs sit within 0.2 ε of it; ten steps 6.8–19.5 ε, the carrier’s 6.6–6.7 ε, under 155 000–273 000 ε;

  • bitwise: two runs of one build (Raman beside the polynomial), and ten steps of both evaluators at one thread, again, and at four (the mode-averaged route).

• Native (tests/native/test_dispersive.cpp): the contraction against the tensor sum and the closed forms against the general groups to 16–32 ε; the mixture rows against mixture_filter’s closed form bit for bit, and a vacuum’s unit filter; a lone unit filter on one node is the constant cubic to 64 ε of a direct sum and holds no field-sized buffer; malformed layouts and a trial without densities are refused before allocation; a problem without the response names its rows at zero. The native suite, 80 tests, passes.

• Adapter (tests/test_dispersive.py): a symmetric mixture’s one product and its factor; krypton and xenon’s one polynomial group and its terms; the family by sector; the general route’s refusal; one evaluation of each evaluator and sector against numpy_ref within its bound.

• Memory. The ledger, tested: a lone symmetric response with the split-step absorber holds no field-sized buffer, and a mixture likewise; beside the in-stage collar one filtered field; the polynomial two filtered fields, two node slots, three products per rule and their spectra; beside Raman a node slot, its products and spectra; chunked nodes their packed projections. estimate_memory equals both compiled backends’ rows to the byte on six fused cases and a chunked one, and the Cartesian route’s admitted buffers bound numba_cpu’s holdings.

• Runs (pytest -n auto, solprop-kokkos 41ca10e installed as a wheel): the new module, 87 passed in 2 min; tests/backends, 3621 passed and 9 skipped (7 min 23 s); every other directory with src’s doctests, 10761 passed and 19 skipped, with one failure of test_convergence_records.py’s writer check, which needs tests/backends collected beside it and passes when it is (the per-directory split, not this slice). solprop-kokkos’s gate, scripts/check.sh, green once at the slice’s end in 14 min 48 s: Ruff, strict Pyright, the pinned clang-format, clang-tidy with strict warnings and the native tests (reference profile, Debug), the native tests under ASan and UBSan, the editable local build with the package’s suite (247 passed, 5 skipped, four threads), the distribution wheel and its clean import. ruff, basedpyright (no errors in the changed modules) and the public-tree scan on solprop’s changed files.

• Not run: no Linux or quiet-machine timing; no device build (slice 3c); solprop’s slow and coverage lanes and the docs build (the SEP’s end).

Q5 measured on Kokkos CPU

benchmarks/sep6/dispersive_cost.py --backend kokkos (the ledger entry of docs/dev/benchmarks.md): stage ratios to the constant cubic, seven interleaved rounds, medians, a load average of 17 to 34. One thread; the default pools in brackets.

case	symmetric, thg / no thg	polynomial, thg / no thg
one mode-averaged node	1.03 / 1.21 (1.21 / 1.35)	1.95 / 2.87 (1.43 / 1.86)
four modes, 33 nodes	1.06 / 1.02 (1.20 / 1.08)	2.06 / 3.65 (1.34 / 1.73)
He/Ar	1.02 / 1.00 (1.17 / 1.11)	3.18 / 6.19 (1.49 / 2.19)
Kr/Xe	1.03 / 1.00 (1.26 / 1.09)	1.83 / 3.46 (1.28 / 1.67)
air	1.03 / 0.97 (1.21 / 1.15)	3.87 / 8.32 (1.58 / 2.13)

• Met: the symmetric evaluator on every case at one thread, 0.97–1.21; the polynomial for one species and for krypton and xenon, 1.83–3.65, and on every mixture in the default pools.

• Missed: the symmetric one-node analytic sector in the default pools, 1.35, at a load average above 30, where every default-pool stage ran slower than at one thread; the polynomial on the mixtures of distinct leg bounds in the analytic sector at one thread, 6.19 (He/Ar) and 8.32 (air), as on numba_cpu (5.91 and 7.27): the roadmap’s item.

• Ten-step windows: 0.99–1.14 (symmetric), 1.15–5.11 (polynomial).

The rest of this report

• Sources checked: Part I §6 and §11; the Phase 0 to 2c reports; numba_cpu’s dispersive.py and bodies.py, numpy_ref’s pass, dispersive_cubic and mixture_filter; solprop-kokkos’s AGENTS.md, README.md and its stage pipeline. No new literature or material data.

• Limitations:

  • Kokkos runs the response on the fused node path only; the other routes refuse it.

  • The device build compiles the same source (functors over a device range) but has been neither built nor run here.

  • The analytic sector’s lone products are complex beside a real polarisation, so they hold their own products, embedded products and spectra (three complex fields), above Q5’s memory target, as on numba_cpu; so does the response beside Raman or plasma.

  • The memory estimate’s spatial count is the node budget’s, an upper bound of numba_cpu’s holdings rather than its rows.

  • Timings are same-process ratios on a loaded workstation; the default-pool ones are dominated by contention.

• Open questions:

  1. Resolved (John Travers, 2026-10-08): DISPERSIVE_KERR_FIELDS is accepted as a public name of solprop-kokkos.

  2. Resolved (John Travers, 2026-10-08): the collar at the nodes, before the projection, rather than on the projected products is accepted: the same operator, a different rounding.

  3. Resolved (John Travers, 2026-10-08): the symmetric one-node analytic sector’s 1.35 in the default pools is re-measured on a quiet machine before any optimisation is considered; no action now.

  4. Resolved (John Travers, 2026-10-08): the response’s memory beside Raman or plasma and in the analytic lone path, above Q5’s memory target, is accepted and recorded; no optimisation slice.

  5. Resolved (John Travers, 2026-10-08): the stale kernel schema of solprop-kokkos’s README.md was corrected on the branch (schema 24). The memory estimator’s design – the ledgers’ rows to the byte on the fused path, the node budget’s upper bound on the spatial routes until Kokkos’s own rows exist there – is accepted; slice 3b gives the spatial routes Kokkos’s rows.

• Handoff:

  • Slice 3b (the Cartesian free-space, line, planar and general modal routes on Kokkos CPU) takes numba_cpu’s per-chunk schemes (SpatialDispersive, GeneralDispersive): the response per chunk of columns or block of rows on the separable engine, and in the general route’s synthesis with each region’s cube and mixture. It removes those routes from _refuse_dispersive_route, extends the adapter’s layout to region keys, and gives dispersive_memory Kokkos’s spatial rows.

  • Slice 3c (the device build) compiles src/dispersive.cpp for CUDA and HIP (its product functors run over a device range, its other kernels are device lambdas, the factors cross through the staging ring) and qualifies the fused path with pytest -m gpu and device bounds; test_kokkos_dispersive_kerr.py skips on a device build until then.

Phase 3b — the remaining Kokkos CPU routes

• Scope: slice 3b of Phase 3 (§11), 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-3 in solprop and in solprop-kokkos (from solprop 5e11eeb and solprop-kokkos ceeb5ee, slice 3a’s heads). Kokkos CPU executes responses.dispersive_kerr on the Cartesian free-space and line routes, the hybrid planar route and the general modal route, with numba_cpu’s per-chunk structure, against numpy_ref and numba_cpu; with the fused path of slice 3a it runs the response on every route numpy_ref admits, and its adapter refuses none of them. A coupled wall flow beside the response, which slice 3a’s native code still refused, runs too. The memory estimate gives Kokkos’s own rows on these routes. The kernel schema stays 24; the device build is slice 3c’s.

• Delivered, solprop-kokkos:

  • include/solprop_kokkos/dispersive.hpp, src/dispersive.cpp: DispersiveWeights (the layout and the stage’s weights, every route’s; one cube per distinct group key for the polynomial), SeparableDispersive (the Cartesian, line and planar routes, chunk by chunk), GeneralDispersive (the general modal route, in its synthesis), the three-component bodies dispersive_isotropic_n and dispersive_conserving_n, and the shared filter, landing and weighing kernels; DispersiveResponse (the fused path) now holds a DispersiveWeights, its arithmetic unchanged. include/solprop_kokkos/slab.hpp: the planar kind’s slab rule in x (slab_contract), moved out of src/general.cpp for the response to share.

  • include/solprop_kokkos/core.hpp: Problem::dispersive_group_keys and dispersive_node_region (the general route’s). src/general.cpp: the response in the separable stage (evaluate_separable_stage), its densities, plans, ledger, path and close; src/general_modal.cpp, general_modal.hpp: GeneralModalSource owns the general route’s response, skips the regional contraction and its transform when no tensor or delayed response makes one, and writes or adds the response’s spectra ahead of the projection; src/bindings.cpp: the two new arrays.

  • src/solprop_kokkos/backend.py: _refuse_dispersive_route removed; the layout carries each group’s region and every node’s region on the general route, one cube per region, and the route’s real field family; the general node chunk counts the response’s buffers (general_dispersive_buffers), as numpy_ref and numba_cpu take it.

  • Tests: tests/native/test_dispersive.cpp (three new: the three-component contraction against the tensor sum and, for two components, the two-component body bit for bit; the cube layout; the regions’ refusals), tests/test_dispersive.py (the general route’s layout, both evaluators), tests/test_backend.py (the free-space route runs, its path and ledger, in place of its refusal); scripts/check.sh; CHANGELOG.md, README.md, the contract tuples’ docstrings.

• Delivered, solprop:

  • tests/backends/test_kokkos_dispersive_routes.py (133) and kokkos_dispersive_thread_probe.py’s route cases; test_dispersive_kerr_routes.py’s refusal test removed.

  • backends/memory.py: dispersive_memory gives Kokkos’s rows on the Cartesian, line and planar routes and on the general modal route (_separable_dispersive_memory, _general_dispersive_memory); tests/backends/test_memory.py.

  • benchmarks/sep6/dispersive_cost.py --backend kokkos on every case; the ledger entry of docs/dev/benchmarks.md; docs/dev/backends.md (§4, the route table, §11), docs/dev/kernel-contract.md, docs/guide/media/dispersive-kerr.md, the tutorial’s note on backends; the changelog.

Decisions

• numba_cpu’s per-chunk structure, natively (Phase 3a’s handoff). On the separable routes the response filters each chunk’s packed spectrum before the stage’s inverse transform, forms its products under the collar at the chunk’s nodes once the stage has put the bare field there – a free-space chunk’s temporal rows are its nodes; a planar block’s (p, y) rows are synthesised at its (x, y) nodes and the products projected back by the stage’s own slab rule – transforms them and lands their weighted spectra in the chunk’s polarisation spectrum after the stage’s forward transform, or in its place when alone. Free space’s inverse transform writes the filtered fields straight into the product blocks they become. On the general route the multipliers act on each chunk’s synthesised spectrum, the products are formed at each group’s region’s nodes and zero elsewhere, and their spectra land ahead of the projection, written rather than added when no regional tensor or delayed response forms the chunk’s polarisation (whose contraction and transform are then skipped).

• A lone symmetric response filters in place on every separable route. Kokkos’s in-stage absorber on these routes is a pass of its own that packs each chunk afresh, so nothing reads the bare field after the material pass; numba_cpu’s planar stage reads the block’s rows once for both, and there filters into a buffer of its own beside the in-stage collar. The operator is the same; Kokkos holds less.

• The general route keeps its synthesis’s 1 / n_to. Kokkos’s general source folds the inverse transform’s normalisation into the cubic rather than into the synthesis, so the response applies it to what it reads – the filtered spectrum and the bare signal – a power of two on every grid solprop builds, which moves no bit.

• One to three Cartesian components on the general route. The contraction sums the components the route resolves in order, which is numba_cpu’s three-component sum with the absent components zero.

• The coupled wall flow beside the response is admitted. Slice 3a’s native validation refused coupled_flow with the response, though numpy_ref and numba_cpu run it and Kokkos’s flow only carries each stage’s input around the fused stage source; with the refusal gone it agrees with the reference within the flow’s own ten-step bound.

• The estimate is exact on Kokkos. dispersive_memory(problem, "kokkos") returns the rows Kokkos’s ledger names on every route, to the byte, in the base group, the general route’s included (Kokkos’s general chunk group no longer adds them). numba_cpu’s spatial rows stay the node budget’s upper bound, and its general rows the chunk group’s, as accepted for slice 3a.

• The contract tuples are unchanged: the adapter reads regions and region_index, both listed already.

Deviations from Part I and the slice’s scope

• The planar route’s lone symmetric response filters in place beside the in-stage collar, where numba_cpu does not (above).

• The coupled wall flow’s admission is beyond the slice’s list of routes; it closes the one refusal of a combination numpy_ref admits that slice 3a left.

• Q5’s polynomial target is missed on the general modal route at one thread (below).

Verification

macOS (Apple silicon, 18 cores), Python 3.14.5, NumPy 2.5.2, optmat 0.4.0, pyFFTW 0.15.1 on FFTW 3.3.11; solprop-kokkos at 925df76 (the branch’s native code at c9bc4e7), the local profile (Homebrew Clang 23.1.2, Accelerate, native architecture), Kokkos 5.2.1, Kokkos-FFT 2.0.0, OpenMP, OMP_PROC_BIND=false; a load average of 11 to 16.

• Parity (test_kokkos_dispersive_routes.py; relative L2 of the stage source, or of the field after the steps; numba_cpu’s bounds, derived from the transforms and node reductions both backends perform, asserted positive):

  • free space and the line, both evaluators, field full and conserving, envelope conserving and full: one evaluation 2.1–4.6 ε against numpy_ref and 0.002–2.2 ε against numba_cpu, under 640–4500 ε; ten steps of the default integrator below 0.005 ε against numpy_ref and 0.65–1.5 ε against numba_cpu, under 93 000–663 000 ε; beside Raman 2.9–4.4 ε and 1.4–1.8 ε (ten steps below 0.001 ε and 0.8–1.3 ε); a helium–argon medium (the symmetric mixture filter rebuilt natively; the polynomial’s two groups), one evaluation and ten steps; a partial chunk; one Runge–Kutta step;

  • the planar route, both sectors: 3.4–5.8 ε and 2.4–2.9 ε against 1800–3500 ε, ten steps 0.16–0.28 ε and 0.75–1.2 ε; beside Raman 3.1–5.9 ε; beside the in-stage collar, in place; a partial block; one Runge–Kutta step;

  • the general route, a gas strand alone, beside a constant solid core and a helium–argon region, real and analytic forms, both evaluators: 0.54–1.6 ε and 0.3–0.95 ε against 560–1090 ε, ten steps below 0.6 ε and 0.53–0.96 ε; krypton and xenon one group and one cube; one Runge–Kutta step;

  • a coherent wall flow (a three-mode helium capillary, ten steps) beside each evaluator: 1.0e-15–1.1e-15 against the flow’s ten-step bound of 2.2e-13, where the response moves the field from the carrier policy’s by 2.2e-7;

  • the ten-step departures from numpy_ref sit far below one ε because these short windows are dominated by the linear propagation, which Kokkos’s separable and general engines form as the reference does; the one-evaluation comparisons are the sharp ones;

  • bitwise: two runs of one build on each route and evaluator, and ten steps at one thread and at four on the free-space, planar and general routes (the response’s transforms below the engine’s 8192-sample threshold run on one FFTW thread, as the routes’ own do).

• Memory (tested): dispersive_memory(problem, "kokkos") equals the native ledger to the byte on 30 cases – free space (both sectors, the envelope grid, beside Raman), the planar route (likewise) and the general route (both forms, the three strands), both evaluators – and every other response row is zero. A lone symmetric free-space response holds no field-sized buffer; the free-space polynomial writes its filtered fields into its product blocks (three products and their spectra, no filtered field); the planar polynomial two node slots.

• Native (tests/native/test_dispersive.cpp, 11 dispersive tests; the native suite 83): the three-component contraction against the tensor sum to 32 ε; the cube layout; the regions’ refusals before allocation.

• Runs: see “The gates” below.

Q5 measured on Kokkos CPU’s other routes

benchmarks/sep6/dispersive_cost.py --backend kokkos --cases free line planar general (the ledger entry of docs/dev/benchmarks.md): stage ratios to the constant cubic, seven interleaved rounds, medians, a load average of 12 to 15. One thread; the default pools in brackets.

case	symmetric, thg / no thg	polynomial, thg / no thg
Cartesian, 16 × 16	0.99 / 1.08 (1.10 / 1.07)	1.59 / 2.27 (1.36 / 1.59)
line, 64	1.02 / 1.14 (1.00 / 1.08)	1.52 / 2.08 (1.16 / 1.35)
planar	0.96 / 1.02 (1.39 / 1.61)	2.12 / 3.56 (2.15 / 2.86)
general modal, real / analytic	1.04 / 0.81 (1.15 / 1.15)	4.96 / 4.15 (2.64 / 2.86)

• Met: the symmetric evaluator on every route at one thread, 0.81–1.14, and in the default pools but on the planar route; the polynomial on the Cartesian, line and planar routes in every pool (1.16–3.56), and on the general route in the default pools.

• Missed: the symmetric planar route in the default pools, 1.39 and 1.61, where every planar stage ran two and a half times slower than at one thread (contention); the polynomial on the general route at one thread, 4.96 and 4.15, where numba_cpu measured 2.32 and 1.44: alone, the response replaces the regional contraction but adds five batched transforms of the chunk’s rows (two filtered fields, three products) to the route’s own two, on a 256-sample grid where the transforms are most of the stage (open question 1).

• Ten-step windows: 0.80–1.13 (symmetric), 1.16–4.75 (polynomial).

The gates

Run at the end of the slice, after its report was interrupted, on the heads solprop-kokkos c5c000b (the slice’s code, with a lint fix: an over-long docstring line and an unused import the refusal’s removal left) and solprop 460e0c9:

• scripts/check.sh in solprop-kokkos: green in 8 min 0 s — Python format, lint and strict typing, C++ format, clang-tidy and the native tests, the native tests under ASan and UBSan, the editable build’s package suite (249 passed, 5 skipped), the distribution wheel and its clean import. Its first two runs failed on the lint above and on a dependency fetch two concurrent builds in one build directory had collided on; neither touched the code.

• solprop, against a wheel of c5c000b (local profile): tests/backends 3751 passed, 9 skipped; tests/nonlinear and tests/translate 2199 passed, 3 skipped; tests/driver 1452 passed; test_seps, test_sol, test_structure_figures 400 passed, 1 skipped.

• Not run: solprop’s full gate, slow and coverage lanes (once, at the end of the SEP); the device build (slice 3c).

The rest of this report

• Sources checked: Part I §6 and §11; the Phase 2b, 2c and 3a reports; numba_cpu’s dispersive_spatial.py, dispersive_general.py, dispersive.py and general_modal.py; numpy_ref’s free, planar and general passes; solprop-kokkos’s AGENTS.md, README.md, its separable stage and general modal source. No new literature or material data.

• Limitations:

  • The device build compiles the same source (the new functors run over a device range, the slab rule is the stage’s team kernel) but has been neither built nor run here.

  • The general route’s polynomial misses Q5 at one thread (above).

  • The parity cases are numba_cpu’s: the planar route on a field grid only, one polarisation component on the separable routes; plasma and envelope plasma beside the response on these routes are executed (the stage pipeline is the fused path’s) but not compared.

  • A process that has run both Kokkos and numba_cpu ends in a segmentation fault at interpreter exit, with the carrier policy as with the response and on slice 3a’s fused path alike; the runs themselves complete and agree. It predates this slice.

  • Timings are same-process ratios on a loaded workstation; no Linux or quiet-machine timing.

• Open questions:

  1. Resolved (John Travers, 2026-10-08): the general route’s polynomial at one thread (4.96 and 4.15) is accepted as the cost of the route’s transforms; it meets Q5 in the default pools. Folding the cube’s transform into the route’s own stays an optimisation idea.

  2. Resolved: the buffer layout is each backend’s to choose where the operator is the same.

  3. Resolved: the coupled wall flow beside the response is admitted.

  4. Resolved: the crash at interpreter exit is an issue of its own, outside SEP 6.

• Handoff to slice 3c (device qualification). The run, on a rented GPU after its plan is reviewed, must cover:

  • Builds: the cuda preset (CUDA, HYBRID GEMM over cuBLAS, HOPPER90, OpenMP off) and the hip preset (AMD_GFX942, HYBRID over rocBLAS), each with the native tests; on CUDA also cuda-custom (CUSTOM GEMM, no TPL BLAS), which exercises the fused path’s synthesis and projection GEMMs without a vendor BLAS. The wheel of each build installed into solprop’s environment and check_build() passed.

  • Code paths: every native kernel of src/dispersive.cpp on the device range (over_nodes’s two-dimensional policy): SelfProduct, PairProducts, GeneralCube, GeneralPairs; the filter, landing, weighing, offset and mixture kernels; the staging ring’s host-to-device copies of the factors; the slab rule’s team kernel on the response’s subviews; and the transforms on cuFFT and rocFFT (the response’s c2r plans write over their input, which the filter rewrites every stage).

  • Tests: test_kokkos_dispersive_kerr.py and test_kokkos_dispersive_routes.py, which skip on a device build today, run under device bounds: one evaluation, one Runge–Kutta step and ten steps of the fused path’s multimode capillary (both evaluators, field and envelope, both sectors, the embedded rule’s lazy finish), a He/Ar mixture of varying composition (the rebuild), nitrogen’s Raman beside it; the Cartesian free-space and line routes (both evaluators, the four sector cases, beside Raman, a partial chunk); the planar route (both sectors, a partial block); the general route (the gas strand alone and beside the solid core, both forms); the coherent wall flow. The device bound adds the device projection’s summation order (docs/dev/backends.md §11) to the CPU bound; repeatability is bitwise run to run, thread invariance does not apply. The ledger tests and dispersive_memory’s equality hold unchanged on a device.

  • Measurements: Q5’s stage and ten-step ratios on the device for the fused-path cases and the four routes of this slice, and the response’s device memory (memory_ledger()), labelled per device.

Phase 3c — device qualification

• Scope: slice 3c of Phase 3 (§11), 2026-10-08, run by Opus 5.5 (AI) for John Travers on the heads of feature/sep-0006-dispersive-nonlinearity-3 (solprop 1bfbd5c, solprop-kokkos c5c000b), on one AMD Instinct device (CDNA 3, gfx942) in a 12-minute session, and on one NVIDIA Hopper device (HOPPER90) in a 32-minute session at solprop 3f7cd57 (the same code with this report). The device builds compile the same source as the CPU build; no code changed on a device.

• Delivered:

  • tests/accelerator_bounds.py: dispersive_device_bound, the dispersive comparisons’ device envelope — the predeclared fft_bound per transform the stage makes, with the CPU bound’s count of transforms and node reductions and its factor of two, the larger of the CPU and device bounds holding.

  • tests/backends/test_kokkos_dispersive_kerr.py, test_kokkos_dispersive_routes.py: run on a device build under that envelope (they skipped there); the two thread-count tests stay host-only (a device build is held run to run).

  • benchmarks/sep6/data/device/hip/: the build record, the test counts and every cost row; the ledger entry of docs/dev/benchmarks.md; backends.md §11 and the manual’s backend notes.

• Verification (on the device):

  • The build: hip preset, HYBRID GEMM over rocBLAS, rocFFT, Kokkos 5.2.1, Kokkos-FFT 2.0.0, Kokkos Kernels 5.2.0, ROCm 7.14 prerelease, Release, AMD Clang 23; native tests passed (ctest).

  • solprop-kokkos’s suite: 254 tests, 0 failures, 13 skipped.

  • solprop’s dispersive device tests (test_kokkos_dispersive_kerr.py, test_kokkos_dispersive_routes.py, test_kokkos_gpu.py, test_memory.py): 316 passed, 2 skipped (the host-only thread tests) — one evaluation, one step and ten steps on every route against numpy_ref and numba_cpu under the device envelope, both evaluators and sectors, field and envelope grids, the mixtures (the symmetric filter rebuilt on the device; the polynomial’s groups), Raman, plasma and the coupled wall flow beside the response, bitwise repeatability run to run, and the memory estimate equal to the device ledger.

  • pytest -m "gpu and not slow": 2 passed.

  • CUDA (cuda preset, HYBRID GEMM over cuBLAS, cuFFT, GCC 13): the native tests passed; solprop-kokkos’s suite 254 tests, 0 failures, 13 skipped; solprop’s dispersive device tests 316 passed, 2 skipped (the host-only thread tests); pytest -m "gpu and not slow" 2 passed; and a CUSTOM-GEMM build (the fused path’s synthesis and projection without a vendor GEMM) passed the two dispersive modules, 218 passed, 2 skipped.

• Q5 on the devices: every case meets it on both. AMD: the symmetric evaluator at 0.86–1.30 of the constant cubic’s stage, the polynomial at 1.02–3.53, including air (2.06–2.10), which missed on the host; the planar route’s symmetric response sits at the target (1.29–1.30). Hopper: 0.95–1.28 and 1.01–3.28 (air 1.50–1.55). The tables are the ledger entries.

• Deviations: none from the plan’s correctness scope. The CUDA session’s first build stopped on a missing timing utility in the device image, installed before the build was rerun; no code changed.

• Limitations: one AMD generation on a prerelease ROCm stack and one Hopper device; the device departures are held to the envelope, not recorded per case; no device sanitizer was run (ROCm ships none; CUDA’s was not used).

• Open questions: none.

• Handoff: Phase 3 is complete on CPU and on one device generation; it merges in solprop and solprop-kokkos together. Phase 4 (§11) follows: the full-size comparisons of the demonstration cases (§8; the helium RDW per mode count, Phase 1c), application guidance, and the end-of-SEP gate and lanes before Final.

Phase 4 — scientific qualification and closing scope

• Scope: Phase 4 of §11, 2026-10-08, implemented by Opus 5.5 (AI) for John Travers on feature/sep-0006-dispersive-nonlinearity-4 (solprop only; solprop-kokkos unchanged at main 78add4f, installed from it). The full-size demonstration comparisons of §8 as worked examples, the helium RDW’s sensitivity by mode count (Phase 1c’s open question 1), the application guidance, a recommendation on Q6, the closing scope of SEP 0 §6 short of Final, and the end-of-SEP gate and lanes. No package behaviour changed; the lanes’ fixes are listed below.

• Delivered:

  • docs/examples/processes/plot_07_rdw_dispersive_kerr.py and its generated page: the M2 case (1.5 m, He 0.8 bar to vacuum, 7.5 fs, 275 µJ, 800 nm, plasma on) under the carrier, symmetric and polynomial policies with four modes — the 100–150 nm band along the fibre, the pump’s peak power, the exit spectrum with the bounded and best-effort bands shaded from the manifest — then with two and one modes, and without plasma; its docstring states that the VUV legs make it a sensitivity study, not a qualified prediction.

  • docs/examples/processes/plot_08_duv_fwm_dispersive_kerr.py and its page, §8’s optional argon example: 1 m of a 50 µm capillary, 10 µJ pump at 400 nm and seed at 800 nm, a 19-point pressure scan around the linear phase matching under the three policies, and the pump’s depletion in photons at each one’s best pressure.

  • benchmarks/sep6/rdw_mode_count.py and benchmarks/sep6/data/rdw_mode_count.json: the nine M2 runs (three policies × one, two, four modes) at the default and a hundred times tighter step tolerance, on M2’s grid and a wider one (70 nm to 6 µm over 600 fs), 36 runs.

  • benchmarks/sep6/fit_uncertainty.py: §4.7’s symmetric errors beside the fit’s own one-sigma dispersion uncertainty (below).

  • docs/guide/media/dispersive-kerr.md: the stage cost on every backend (numba_cpu, Kokkos CPU, the two device builds) in one table; the full-size RDW figures; a new section When the carrier is enough, and which evaluator to choose (the measured demonstration results, guidance by observable, which evaluator, what the regions mean for a result, the backends); Reading what a run recorded as its own section; a closing link to this SEP.

  • Current pages: materials.md, general-modal.md (two places), luna.md and solver.md no longer say that Kokkos refuses the response or that the reference evaluates per species (it evaluates per symmetric group since Phase 2c); materials.md ends with a link to this SEP beside SEP 3’s; the guide’s backend table (docs/guide/index.md) gains the response’s row; the processes gallery’s README.rst names the two new examples.

  • docs/dev/roadmap.md: the SEP’s entry restated; new informal items Beyond the scalar dispersive electronic response (tensor dispersion, silica, the GNLSE route, the xenon and CO bounded quartets) and The constant cubic’s number-conserving rule beyond an octave (Phase 0’s open question 2); the RDW sensitivity added to Resonant gas susceptibilities.

  • CHANGELOG.md: a Phase 4 entry citing SEP 6; the stale “Kokkos refuses it” sentences of the Phase 1 and 2 entries removed; the device qualification noted.

The demonstration results

All on the default backend, Kokkos CPU (solprop-kokkos 78add4f, the local profile, default thread pools), macOS on Apple silicon, Python 3.14, optmat 0.4.0, at a load average of 15 to 20.

Helium RDW (M2), the 100–150 nm band at the exit (µJ; the change is the carrier’s against the polynomial’s; centres in nm, carrier / symmetric / polynomial):

Modes	Carrier	Symmetric	Polynomial	Carrier vs polynomial	Symmetric vs polynomial	Centres
1	6.44	9.95	9.90	−35.0 %	+0.43 %	135.99 / 132.80 / 132.85
2	10.83	14.12	14.10	−23.2 %	+0.14 %	135.22 / 132.75 / 132.80
4	10.35	13.54	13.52	−23.5 %	+0.13 %	134.55 / 132.18 / 132.23

• The active band is supported from 151.9 nm, bounded from 120.3 to 151.9 nm and best effort from 81.8 to 120.3 nm: the RDW band straddles the last two.

• With four modes the pump’s peak power, which oscillates along the second half of the fibre with a period of about 7 cm in step under every policy, is 4 % higher under the carrier policy (106 against 102 GW at its largest), and 8 % less energy leaves the fibre (142.1 against 155.0 µJ). Without plasma (carrier and polynomial only): 128 against 118 GW, the same 220.3 µJ at the exit, and 12.29 against 14.75 µJ in the band (−17 %). The plasma therefore carries part of the electronic response’s difference: the lower peak ionises less.

• Accepted steps rise by a fifth under the dispersive policies (1870 against 2254 and 2236 with four modes; a third with two, 7 % with one).

• Convergence (rdw_mode_count.py): at rtol=1e-8, atol=1e-12 every band energy moves by 1–2 parts in 10⁶ and every centre by less than 0.01 nm; on the wider grid the band energies move by 0.4–0.9 % and the centres by 0.03–0.06 nm, and the polynomial’s excess over the carrier, 53.8 / 30.2 / 30.7 % for one, two and four modes on M2’s grid, is 53.7 / 29.8 / 30.4 %. The differences are the models’.

• Run times: the worked example (nine runs with plasma, two without) 1 min 29 s in poe docs-examples; the 36 benchmark runs 7 min 11 s, 5.5–26 s each.

The mode-count sensitivity (Phase 1c’s open question 1). At full size the dispersive policies give the band 54 % more energy than the carrier policy with one mode and 30–31 % more with two or four; the two- and four-mode figures agree. Phase 1c’s reduced case (1 m, 1.2 bar, 100 nm to 1.5 µm) gave about 60 % with one mode and 9 % with two: the one-mode sensitivity is the same, the two-mode one is three times larger at full size. A mode count is part of the observable’s definition: the sensitivity is to be reported with it, and one mode overstates it for this case. The physical reason the mode set matters this much was not isolated.

Argon DUV FWM, 50 µm, 1 m, the linear phase matching of HE₁₁ at 209.4 mbar:

	Carrier	Symmetric	Polynomial
best pressure (scan step 5.2 mbar)	204.1 mbar	193.7 mbar	193.7 mbar
largest signal along the fibre there	3.33 µJ	3.29 µJ	3.27 µJ
the same at 204.1 mbar; at 209.4 mbar	3.33; 3.22 µJ	3.01; 2.68 µJ	3.01; 2.68 µJ
signal at the 1 m exit, at the own best pressure	2.37 µJ	2.69 µJ	2.69 µJ

• The carrier policy places the best pressure 10 mbar (5 %) high, overstates the signal by 11 % at its own best pressure and by 20 % at the linear phase matching, and agrees on the best yield to 2.5 %. The symmetric evaluator is within 0.4 % of the polynomial at the best pressure and 1.1 % at worst across the scan.

• At each best pressure the signal peaks near 0.5 m with 0.22 signal photons per launched pump photon (44 % of the pump’s photons used) and about a quarter of the pump band’s energy left; beyond that it falls.

• Why it adds to the tutorial: a fibre twice as long (the signal’s turn-over and decline), the pressure scan (the operating point moves; the best yield does not) and the pump depletion in photons. The tutorial’s half metre at the linear pressure shows only the 20 % overestimate.

• Run time: 5 min 36 s in poe docs-examples (60 runs).

The fit’s own uncertainty (fit_uncertainty.py; Phase 1c deferred this map to Phase 4). On §4.7’s quartets, the one-sigma uncertainty of each quartet’s dispersion \(\gamma(q)/\gamma_0\) from the marginal covariance of optmat’s \((A, B, B')\) (Birge-scaled), against the symmetric error, in per cent, median / maximum, and the fraction of quartets whose symmetric error exceeds it:

Gas	Symmetric error	One sigma	Error above sigma
He	0.006 / 0.124	0.015 / 0.109	0.26
Ne	0.007 / 0.153	0.141 / 0.274	0.00
Ar	0.053 / 0.913	0.162 / 0.366	0.21
Kr	0.115 / 1.714	1.066 / 3.197	0.00
Xe	0.260 / 3.212	0.481 / 2.682	0.32
H2, D2	0.047 / 0.718	0.063 / 0.499	0.42
N2	0.047 / 0.837	0.209 / 0.437	0.10
O2	0.183 / 2.908	0.500 / 1.415	0.23
CH4	0.133 / 1.831	0.404 / 1.353	0.17
SF6	0.105 / 2.141	0.491 / 1.019	0.11
N2O	0.048 / 0.656	2.265 / 5.109	0.00

The symmetric factorisation’s typical error is below the fit’s statistical uncertainty of the dispersion in every gas; its largest errors, at the widest-spread quartets, exceed it in a tenth to two fifths of the quartets for most gases. The figures depend on the anchor: against the pump’s own Kerr value rather than \(\gamma_0\) the correlated part cancels and the uncertainty is several times smaller. A covariance is not a bound on the truncation or on borrowed priors (§4.8). Published in the manual at John Travers’s decision (open question 4), relative to \(\gamma_0\) with the anchor stated.

The guidance written

docs/guide/media/dispersive-kerr.md, When the carrier is enough, and which evaluator to choose:

• By observable: self-phase modulation near the carrier is exact under every policy, and the carrier policy is adequate where the Kerr diagonal varies little over the occupied band (12 % from 2 µm to 400 nm in helium, 37 % in argon, 87 % in xenon, §4.7); a band generated far from the carrier (a dispersive wave, an FWM signal, UV from a near-infrared pump) needs a dispersive policy; a phase-matched wavelength moves by a few nanometres; an operating point moves by several per cent while the best yield hardly changes.

• Which evaluator: "symmetric-kerr" for production and scans (the cost of the carrier policy, within half a per cent of the polynomial on the RDW and about one on the argon scan); "gas-polynomial" as the reference, for checks in the most dispersive gases and for mixtures sharing a leg bound.

• The regions for a result: supported is exact relative to the fit; bounded is an estimate read as a lower bound; best effort is a sensitivity, reported with the carrier result beside it, never as a correction.

• Backends: every backend and every route but the GNLSE; device builds qualified.

Q6 — a recommendation for John Travers (no default changed)

Recommendation: keep "carrier-kerr" as the default ("recommended" unchanged), and recommend "symmetric-kerr" in the manual for broadband, multi-colour and UV-generating gas runs, as now written. Revisit when a resonant VUV response of helium exists or when per-gas nondegenerate measurements qualify the bounded region.

• For a switch: inside the data the dispersive policies are exact relative to the fit and the carrier approximation’s errors are real (argon: the operating point 5 % off, the signal 11–20 % too high at a given pressure); the symmetric evaluator costs 0.8–1.35 times the constant cubic’s stage on every backend, and a whole four-mode RDW run 1.35 times on Kokkos CPU here (one timing on a loaded machine), most of it a fifth more accepted steps; §4.10 makes it never less sound than the carrier.

• Against a switch now:

  1. The dispersive policies refuse what the carrier policy admits — a supplied species without optmat’s model, a constant KerrResponse declared beside the selection, the GNLSE route. As the default they would turn those runs into refusals, or make "recommended" mean different laws by route and species, which §5 (no automatic model switching) and §9 rule out.

  2. Every gas run’s results change — the RDW energy by 30 %, every adaptive step history — including the worked examples, tutorials and the comparisons with stored witnesses; reproducing an earlier result would need electronic="carrier-kerr".

  3. The largest effects found are in best-effort regions (helium’s VUV), where the dispersive result is a sensitivity, not a better prediction; as the default it would present a held constant as the expected answer.

• Options: (A) keep the carrier default (recommended); (B) "recommended" becomes "symmetric-kerr" for every gas with optmat’s model, with an explicit rule for supplied species, a declared KerrResponse and the GNLSE route; (C) select by the run’s band — excluded by §9.

The gates

Run once, sequentially, at solprop d441363 (this branch’s examples, guidance and benchmarks committed), solprop-kokkos 78add4f installed, SOLPROP_KOKKOS_THREADS=1, OMP_PROC_BIND=false, on a workstation at a load average of 15 to 21:

• poe check (13 min 38 s): ruff, the formatter, basedpyright, numpydoc and codespell clean; the default lane 14 527 passed, 28 skipped and 1 failed: tests/test_distribution.py::test_loadgroup_reuses_module_fixtures_and_spreads_independent_cases, whose subprocess starts two xdist workers under a 45 s timeout and timed out under the load. It passes alone and in the targeted run below; it was not changed. The sequence stopped there, so the gate’s docs step did not run in it; the docs builds of poe docs-examples had found one nitpick warning, a :func: role on the private _route_bytes in dispersive_memory’s docstring (Phase 3a), fixed as a literal (d441363, before the lanes).

• poe test-slow (9 min 29 s): 137 passed, 1 skipped, 1 failed: tests/driver/test_stepindex_gnlse.py::test_stepindex_strand_propagates_on_both_cpu_backends, one sample of 2048 at 2.1e-7 of the largest against its bound of 2e-7 (numpy_ref against numba_cpu over fifteen centimetres of a step-index GNLSE strand, a route SEP 6 does not touch). Rerun alone through poe test-slow on that module it passes (2 passed, 36 s). The likely cause is numba_cpu’s FFTW planning under FFTW_MEASURE, which picks its algorithms by timing and so its rounding by the machine’s load; not changed here (open question 5).

• poe cov (12 min 53 s): 99.83 %, 46 lines and branches uncovered, and the same test_distribution timeout. All but three lines were SEP 6’s: error paths and rare branches its phases added and never ran through the lane. Fixed in 96ebb05 with tests: DispersiveKerrModel’s malformed coefficients, a model without a support record and a record of another model; DispersiveKerrTerms’s field validation, a problem’s leaf on the wrong route, an unknown population, an undeclared general region, a carrier flag against its sector, a dispersive response beside a TDSE gas or beside a general guide’s own channels, the node buffers without the leaf; numpy_ref’s pass and general source refusing mismatched densities and carriers; a general node chunk holding no gas node, on both CPU backends; the general route’s region notice; numba_cpu’s embedded rule finished once, and absent on the radial route; GasKerr’s display. One line was dead: the constant Kerr components’ own thg check, which compile_responses has preceded with the electronic responses’ since Phase 1a; it is removed. The other two lines were SEP 12’s, the TDSE build record of an installed solprop-tdse distribution (not installed here), now tested with a stand-in distribution.

• The Kokkos-selected slow lane (SOLPROP_TEST_BACKEND=kokkos poe test-slow, 8 min 34 s): 139 passed.

• The fixes were verified by targeted runs, not a second lane run (on John Travers’s instruction a rerun of the lanes was stopped): ruff and basedpyright on the changed files; pytest -n auto on the eleven affected modules with the failing test_distribution case, 549 passed; a coverage run of the eight dispersive and TDSE modules showing every listed gap covered but the general route’s region notice and undeclared region, whose two tests were added after it and pass.

• Documentation: poe docs-examples built both worked examples (its build reports two unreadable tutorial images, an artefact of that mode, which executes no tutorial); with this report, tests/test_seps.py (127 passed, 1 skipped) and poe docs (every tutorial executed, no warning) pass.

Deviations from Part I and the phase’s scope

• Branched from main at 03d046e, Phase 3’s merge 73eb312 plus a numba-cache fix merged since, so that the lanes run on the tree that will be merged.

• A companion worked example, not an extension of plot_00: nine runs, a mode-count study and a plasma check would bury the canonical example and the README figure it draws.

• The argon example’s geometry is the tutorial’s (50 µm, 10 µJ each), not Durfee et al.’s; one mode only.

• The symmetric evaluator was not run without plasma (only the carrier and the polynomial).

• Fit-uncertainty maps (§7, deferred by Phase 1c) are published as a table in the manual, not drawn as maps.

The rest of this report

• Sources checked: Part I §2, §4.7, §7, §8, §11, §12; every Part II report; optmat 0.4.0’s HyperpolarizabilityModel (parameters, covariance, reduced chi-squared, gamma_sigma_au’s Birge scaling) read from the installed package. Durfee et al. [9] for the process only. No new literature.

• Limitations:

  • Timings are from one loaded workstation; the run times above are wall times of the examples, not benchmarks.

  • The RDW comparison is a sensitivity of an observable to unqualified VUV legs; the argon comparison is exact relative to optmat’s fit but carries the fit’s own uncertainty and one mode.

  • The pressure scan resolves the best pressure to one step, 5.2 mbar.

  • Why the RDW’s sensitivity depends on the mode set was not analysed.

  • plot_00_vuv_dispersive_wave was not re-run (its source is unchanged): its page still shows its output from before SEP 16 (9.86 µJ in 100–150 nm), where the companion’s carrier run with the same physics gives 10.35 µJ.

• Open questions (every earlier one is now answered, on the roadmap or below):

  1. Resolved (John Travers, 2026-10-08): Q6 — the carrier policy stays the default; the manual recommends "symmetric-kerr" for broadband or UV-generating gas runs.

  2. Resolved: no output-schema bump (Phase 1a’s open question 3).

  3. Resolved: a constant KerrResponse beside a dispersive selection is refused on every route (Phase 1a’s open question 4).

  4. Resolved: the fit-uncertainty comparison is published in the manual, relative to \(\gamma_0\), with the anchor’s effect stated.

  5. Resolved: the two load-sensitive tests were made load-independent rather than widened: the step-index strand test plans numba_cpu’s transforms with FFTW_ESTIMATE (6.9e-9 against its unchanged 2e-7 bound), and test_distribution’s subprocess and rendezvous limits, which are hang guards, are long enough that a loaded machine’s slow worker start cannot trip them.

  • Answered: Phase 1a’s open question 2 (Q5’s targets are for the compiled stages; numpy_ref is the oracle, measured in Phases 2 and 3); Phase 1c’s open question 1 (above); Phase 2a’s open question 5 (the planar-hybrid tutorial runs on main under the step controller). On the roadmap: Phase 0’s open question 2 (the constant cubic’s \(\Pi_+\)); §12 Q2’s xenon and CO quartets; the tensor, silica and GNLSE scope; the resonant VUV response; the polynomial’s mixture cost. Outside solprop: §12 Q2’s O2 spin-forbidden states (optmat’s). Outside the SEP: Phase 3b’s open question 4, the crash at interpreter exit after both compiled backends ran; a minimal probe here (a Kokkos and a numba_cpu run in one process) exited cleanly.

• What remains to set Final: John Travers’s decisions on the questions above; then the Final pull request of SEP 0 §6 — status, the Manual pages field, git mv to archive/, the index row and toctree entry moved, the relative links fixed (the manual pages link ../dev/seps/sep-0006-…), the roadmap’s SEP entry removed (its informal items stay), and the CHANGELOG line (it cites SEP 6).

• Handoff: Phase 4 merges into solprop alone; solprop-kokkos is unchanged.

Part III — History

• 2026-09-27: Drafted with two proposed dispersive evaluators, gas quartet comparisons, exploratory cost measurements and an explicit broadband harmonic-selection qualification question. Renumbered from 4 to 6 before initial merge to avoid concurrently assigned SEPs.

• 2026-09-28: Review revision. Corrected the gas-fit count (twelve of optmat’s fifteen, solprop’s inventory), stated the intensity convention and optmat’s \(n_2\) form, mapped \(S_2/S_4\) to optmat’s \(W_2/W_4\), added the symmetric band cut-off rule, the harmonic-leg domain table, the mixture-curve filter for the symmetric model, the output-only median and 95th-percentile columns, and the matching test rows. All §4.7 and §4.9 numbers were reproduced independently from the stored fits.

• 2026-09-29: Recorded the 2026-09-28 review by Fable 5.1 (AI) in the new review fields. Review status is In review: a human review is still needed before acceptance.

• 2026-10-07: Final review by Opus 5.5 (AI) and review by John Travers (human); review status Reviewed. Re-audited against solprop b70f2fb, optmat v0.3.0 and solprop-kokkos fa5d57c (SEP 3 final, kernel schema 23, output schema 15, Multistep() the default integrator, SEP 12’s TDSE response final). At the reviewers’ request: the gas scope follows optmat’s fifteen hyperpolarizability fits rather than a solprop list, with CO2, CO and O3 added to §4.7–§4.8; the argon DUV FWM and helium VUV RDW (M2) demonstration cases were added with their coefficient table and run through qualification, the tutorial and the manual; and the solprop-kokkos implementation is required, with no exception. Also: §5 written against the delivered MaterialSelection.electronic, the numerical band separated from the material support (§4.5), composition with the TDSE response stated, and §9’s persisted record made concrete. Every §4.7 survey row, FWM row and the §4.9 field-grid diagnostic were reproduced exactly on the current tree. Same day, after optmat’s analysis of where its fits are supported (optmat a9e5dbf, OEP 5 draft 0d5a0ab): added §4.10. Every leg is bounded at \(\sqrt{3S_{2,\max}/4}\), which is exact within the fitted sums. Runs are never refused for the frequencies their field occupies, as helium runs reaching below 80 nm require. The band is labelled supported, bounded or best effort against each gas’s resonance band edge. Q2 is answered in substance, with its remaining items listed. The demonstration-case table gained the bounded values and band-edge column, with a paragraph on how pump dynamics and VUV emission share one response. John Travers confirmed the policy “label, never refuse” for every gas, O3 and the polyatomics included.

• 2026-10-07: Accepted by John Travers, whose review was the human review recorded above. Part I is frozen. Phase 1 needs an optmat release of the support record proposed by optmat’s OEP 5.

• 2026-10-07: Phase 0 delivered: the operators’ field, envelope and vector forms in both harmonic sectors, an independent triplet-sum oracle with the algebra tests, and the material record reproduced through optmat’s public models. John Travers accepted its decisions on Q1, Q3, Q4 and Q5; Part I’s “lower bound”, band-edge and polyatomic statements are qualified in the report after optmat’s OEP 5 review. Status In progress.

• 2026-10-07: Phase 1, slice 1a delivered: both dispersive policies on numpy_ref for the single-mode, multimode and radial routes, kernel schema 24, the manifest’s dispersive law and regions, per-species mixture channels, and refusals by name on the other routes, on numba_cpu and on Kokkos.

• 2026-10-07: Phase 1, slice 1b delivered: numpy_ref evaluates the dispersive response on the free-space, line, planar and general modal routes, a general modal region declaring its gas fill (GasKerr) under the run’s electronic policy; numba_cpu and Kokkos refuse it by name.

• 2026-10-07: Phase 1, slice 1c delivered: the manual chapter on the frequency-dependent electronic response, the tutorial comparing the three policies on the argon DUV FWM and helium RDW cases, and their ten-step demonstration tests on numpy_ref.

• 2026-10-08: Phase 1 merged in solprop and solprop-kokkos (kernel schema 24). John Travers accepted GasKerr, the unversioned regions field, three-component evaluate and the 40 s tutorial.

• 2026-10-08: Phase 2, slice 2a delivered: numba_cpu executes the dispersive response on its fused node path (single-mode, multimode, radial; field and envelope; both evaluators and sectors; mixtures; beside Raman and plasma), the multipliers acting on the modal columns, with the Q5 measurements; the free-space, planar and general modal routes stay refused by name for slice 2b.

• 2026-10-08: Phase 2, slice 2b delivered: the collar applied to the dispersive products before the output multipliers (G[w (G E)^3]) in the reference and numba_cpu, and the node chunk counting the response’s buffers, both decided by John Travers that day; numba_cpu on the Cartesian free-space, line, planar and general modal routes, its refusal removed; the polynomial’s species of equal leg bound grouped, an exact rewriting; the symmetric mixture-curve filter measured for a decision; the tutorial on numba_cpu; Q5 measured again on every route.

• 2026-10-08: Phase 2 merged. Phase 2, slice 2c delivered: the symmetric evaluator of a mixture takes one filter from the mixture’s own Kerr diagonal (§6), on numpy_ref and numba_cpu and every route, rebuilt as the composition changes along z and recorded with its composition in the manifest and the resume key, with the exact per-species sum its test reference; the general modal route’s node chunk counts the response’s buffers; both decided by John Travers that day. Q5 met for every symmetric mixture.

• 2026-10-08: Phase 2 complete: slice 2c merged (the symmetric evaluator’s mixture filter, the general route’s node budget); its open questions decided by John Travers. Phase 3 (Kokkos) begins.

• 2026-10-08: Phase 3, slice 3a delivered: Kokkos CPU executes the dispersive response on its fused node path (single-mode, multimode, radial; field and envelope; both evaluators and sectors; mixtures under the symmetric mixture filter; beside Kerr, Raman and plasma) with numba_cpu’s structure, against numpy_ref and numba_cpu; the memory estimate counts the response’s buffers on the fused and spatial routes (Phase 2c’s decision 3). The free-space, line, planar and general modal routes stay refused by name on Kokkos for slice 3b; the device build is slice 3c’s.

• 2026-10-08: Phase 3, slice 3a’s open questions decided by John Travers (all five accepted). Slice 3b delivered: Kokkos CPU executes the dispersive response on the Cartesian free-space and line routes, the planar route and the general modal route with numba_cpu’s per-chunk structure, and beside a coupled wall flow, against numpy_ref and numba_cpu; Kokkos refuses none of the response’s routes, and the memory estimate gives its rows on each to the byte. The device build is slice 3c’s.

• 2026-10-08: Phase 3, slice 3b’s open questions decided by John Travers; slice 3c qualified the device builds on one AMD CDNA 3 and one NVIDIA Hopper device: every dispersive test passes under the predeclared device envelope, and Q5 is met in every case on both.

• 2026-10-08: Phase 4 delivered: the helium RDW (M2) and the argon DUV FWM run at full size under the three policies as worked examples; the RDW’s sensitivity by mode count (54 % with one mode, 30–31 % with two or four, converged in step and grid); the application guidance in the manual; a recommendation on Q6 (keep the carrier default) for John Travers’s decision; the closing scope short of Final; the end-of-SEP lanes, run once, their coverage gaps fixed and verified by targeted runs.

• 2026-10-08: Phase 4 merged with John Travers’s decisions (Q6: the carrier policy stays the default; the fit-uncertainty comparison published). Final, approved by John Travers; moved to the archive. Its remaining work is on the roadmap.