SEP 17 — Broadband two-photon continuum response beside strong-field ionisation#


SEP	17
Title	Broadband two-photon continuum response beside strong-field ionisation
Author	John Travers
Drafting agent	Opus 5.5 (2026-10-08); GPT-6 (review revisions, 2026-10-08; substantial revision, 2026-10-12)
Status	Accepted
Review status	Reviewed
Reviewers	John Travers (human); Opus 5.5 (AI); GPT-6 (AI)
Type	Feature
Created	2026-10-08
Resolution	Re-accepted 2026-10-12 by John Travers after the independent review and the three follow-up corrections recorded in Part III. Phase 0 qualification remains required before production implementation in Phase 1. The earlier acceptance and return to Draft remain in the history.
Requires	SEP 3 (material data); SEP 6 (electronic response composition); optmat OEP 6’s implemented continuum data
Supersedes	—
Superseded-by	—
Implementation	—
Manual pages	—

Part I — Proposal#

Revision and review scope. This is the revised design of 2026-10-12. The header’s review status and reviewers refer to this Part I, reviewed on 2026-10-12 (Part III). The earlier human and AI reviews, acceptance and numerical checks remain in Part III as history; they do not qualify the new continuum response. Implementation has not begun.

1. Abstract#

Propose an opt-in, scalar, field-grid response for direct two-VUV-photon ionisation in Ar and He. A positive distribution of effective continuum states is driven by the squared analytic VUV field. Its weights come from optmat’s existing cross sections. The same interaction determines a retarded atomic response, its polarisation and its energy exchange, rather than assigning a window-calibrated local rate and a separate loss current. The first deliverable is prescribed-field diagnostics and propagation with this continuum optical response, including its dispersive part, with final ionisation-yield diagnostics. It does not identify transient continuum occupation with a free-electron density. Additional plasma feedback is a separate qualification gate, not silently omitted from a supposedly complete two-photon model. Success means reproducing OEP 6’s integrated weak-field limits, conserving the declared energy, converging the continuum and source projection, and establishing an atomic and material validity domain relevant to the RDW test. No new optmat quantity is required for this first approximation; section 4.9 records the material gaps and the triggers for a coupled OEP. The continuum approximation and its dispersive matching require review before production implementation.

2. Motivation and scope#

The maintainer’s question is whether two-photon absorption of a VUV resonant dispersive wave (RDW) can explain its suppression near the two-photon ionisation onset in a gas-filled capillary driven by a self-compressing 800 nm pump. Pump-driven ionisation can blue-shift the pump and detune RDW generation. That mechanism and attenuation after generation can coexist. Distinguish departure from a phase-matching curve from the shortest observable RDW across optimised conditions; report the centroid, blue edge and detection limit separately. The input duration is a dataset parameter, not fixed by this proposal; use the generated local VUV field, not the input pump duration, for atomic tests.

The sequence is: replay existing fields to estimate loss and final yield; add the continuum optical response if those estimates could matter; add extra plasma feedback only if its effect cannot be bounded as negligible and its timing has been qualified. A small excitation probability per atom can still produce appreciable propagation loss through many atoms. Conversely, small direct loss does not prove negligible plasma feedback. A substantial prescribed-field correction requires a coupled propagation calculation; it is not itself a prediction of the attenuated output.

The existing strong-field rate is evaluated at a declared pump carrier. Applied to the total field, VUV oscillations can change its nonlinear average without representing absorption of VUV photon pairs. Removing that content from the rate and adding a continuum response are separate changes: both need controlled comparisons. Real pump-assisted channels and dressing are also separate physics, not recovered by the split. The immediate objective is the Ar/He RDW experiment; four-wave-mixing sources and broader gas propagation remain possible later applications.

First scope: pure Ar or He, fixed linear polarisation, field grids and the single-mode capillary route with a resolved transverse field quadrature. Apply the atom at physical field nodes and project back to the mode; a peak-power or mean-field substitution does not reproduce a fourth-power yield. The first-scope response is, however, homogeneous of degree three in the local field with linear selectors and a fixed neutral profile. For a frequency-independent mode shape \(F(r)\) it therefore factorises exactly: the node response is \(F^3\) times one modal response, and its projection is the Kerr overlap \(\int N(r)F^4\,dA\) (section 4.4). The radial rule is still needed for the pump plasma, and for any later saturating closure. The electronic excitation is perturbative at each atom, although the propagated VUV may change substantially. A pump-ionisation baseline is allowed only when its neutral depletion is negligible over the relevant VUV interaction or is a fixed initial neutral fraction for that interaction. A varying depletion cannot be inserted into an integrated diagnostic or this stationary-coefficient response without a new derivation.

Deferred: additional two-photon plasma feedback until section 4.7’s gate; strong depletion or saturation; mixed pump–VUV amplitudes and pump dressing; bound resonances and their (2+1) ionisation; mixtures, other gases, multiple optical modes, envelope grids, vector fields and other geometries. OEP 8’s material coefficients can support a separate bound-response SEP, particularly the He 1s2s feature. They are not a dependency of the direct continuum model. Atomic TDSE calculations are validators, not required inside every optical right-hand-side evaluation.

3. Context#

The earlier version of this proposal was accepted with Q3, Q4, Q13 and Q18 as qualification blockers. Its window-wide rate calibration reproduced integrated yield and energy but did not establish ionisation timing or the spectral work of its current. This revision replaces that candidate with an explicit weak-excitation continuum and narrows the first delivery to Ar/He optical response and final-yield diagnostics. Shared depletion and additional VUV plasma are conditional follow-up work, not promised by the initial optical model. The old question numbers are retained in section 12, with their changed resolutions and scope made explicit. There is one implementation track and no separate rate-based two-photon response to enable beside it.

The existing plasma response supplies pump-ionisation and free-electron currents. SEP 18 provides band-exact running integrals, but does not supply a physical ionisation-time history. The proposed optical continuum supplies a polarisation directly; it does not need to pass through the plasma current’s running integral. Q18 still requires the correct source factor and discrete work. References in the archived SEP 18 to this proposal’s former section 4.9 describe the earlier design; the current treatment is in sections 4.3 and 4.6 and Q18.

SEP 12 provides a TDSE comparison route; its native response must be used without replacing its absorptive cubic by a real Kerr term. Its calibrated material and numerical limitations remain separate from this SEP’s model error. For structural atomic comparisons use native scale one and source_filter="none": rescaling optical current does not rescale microscopic ionisation yield. Use an atomic model’s own cross section to test the response’s structure, and compare that cross section with optmat separately. The in-tree helium TDSE route is not an argon validator.

optmat OEP 6 supplies two_photon_ionisation_model, cross_section, pair_cross_section, thresholds, region masks, uncertainties and identities. OEP 8 and its review explicitly leave propagation polarisation, electron accounting and energy balance in solprop. The optmat adapter and material identities follow SEP 3; no cross-section table is copied into solprop. These APIs must be present in the optmat version pinned for implementation; the current optmat floor is not evidence that all of them are installed.

4. Theory#

4.1 Fields, units and the known integrated constraint#

Use the optics convention, \(\widetilde A(\omega)=\int A(t)e^{i\omega t}dt\). At one physical node let \(A=E_{V,a}\) be the analytic VUV field in V/m, with real field \(E_V=\operatorname{Re}A\); optmat’s positive-frequency field is \(E_V^+=A/2\). Let \(N_g\) be the initially available neutral density in m⁻³, fixed during the atomic calculation. Local-field corrections are not introduced. The cross-section convention uses vacuum-equivalent local-field intensity \(I_0=\epsilon_0c|A|^2/2\), not modal power or the dispersive Poynting flux. The latter is still used for the propagated field’s physical energy.

In the pair-sum approximation of OEP 6, define

\[ S(\Omega)=\int A(t)^2e^{i\Omega t}dt,\qquad k(\Omega)=\frac{\epsilon_0^2c^2\sigma^{(2)}(\Omega/2)}{\hbar^2\Omega^2}. \]

The two-photon matrix element is assumed independent of frequency split at fixed \(\Omega\). Its validity needs tests of the actual single-photon support and splits; a supported mean photon energy alone is insufficient. Reference [5] supplies an amplitude-calculation methodology, but its two-photon calculation uses equal photon energies, with intermediate and final energies \(E_0+\hbar\omega\) and \(E_0+2\hbar\omega\). It does not validate unequal-photon-energy pairs at fixed sum; that requires separate nondegenerate calculations or independent pulse comparisons. The total yield and energy per initially neutral atom are

\[ P_\infty=\frac{1}{2\pi}\int k(\Omega)|S(\Omega)|^2d\Omega,\qquad U_\infty=\frac{1}{2\pi}\int\hbar\Omega k(\Omega)|S(\Omega)|^2d\Omega. \tag{1} \]

All continuum integrals are over positive angular frequencies above the appropriate threshold. \(\sigma^{(2)}\) has units m⁴ s; \(k\) has units m⁴ V⁻⁴ s⁻¹, and \(S\) has units V² m⁻² s. Thus \(P_\infty\) is dimensionless and \(U_\infty\) is in J. For a long monochromatic pulse, (1) gives \(P_\infty/T=\sigma^{(2)}(I_0/\hbar\omega)^2\). The pulse energy removed per length is the transverse integral of \(N_gU_\infty\), in J/m. Equation (1) is not a saturated probability law.

4.2 The minimal effective continuum#

Choose a declared finite continuum domain \(\mathcal D\) on which material data or explicitly labelled completion models are available. With positive quadrature weights \(w_j\) in s⁻¹ and nodes \(\Omega_j\), set

\[ g_j=\sqrt{\frac{w_j k(\Omega_j)}{2\pi}},\qquad \dot b_j=-i\Omega_j b_j+i g_j A^2,\qquad b_j(t_0)=0. \tag{2} \]

Here \(b_j\) is dimensionless and \(g_j\) has units m² V⁻² s⁻¹. The ground amplitude is fixed at one: these are weak-excitation auxiliary equations, not a saturation model. One effective channel per energy suffices for the declared separable approximation. Its scalar coupling phase can be absorbed into \(b_j\); no measured scattering phase is being inferred. This simplification does not recover channel interference between different frequency splits, angular distributions or electron trajectories.

Solving (2) gives

\[ b_j(t)=i g_j\int_{t_0}^t e^{-i\Omega_j(t-s)}A(s)^2ds. \tag{3} \]

After the field vanishes, \(|b_j|^2=g_j^2|S(\Omega_j)|^2\). Therefore \(\sum_j|b_j|^2\) and \(\sum_j\hbar\Omega_j|b_j|^2\) converge to (1) on \(\mathcal D\). This proves the integrated normalisation, not physical transient accuracy.

Equation (2) is retarded in the specified analytic drive. A Fourier band selector and analytic-signal construction are themselves nonlocal in time; do not call the whole filtered real-field model microscopically causal. Test sensitivity to the selector and exclude apparent advance caused by its tails from an atomic-memory claim. There is no window-wide yield coefficient that changes an earlier atomic response when only a later part of the prescribed drive is changed.

4.3 Polarisation and work from the same interaction#

A classical complex-coordinate Hamiltonian, per initially neutral atom, for (2) is

\[ \mathcal H=\hbar\sum_j\Omega_j|b_j|^2 -\hbar\left(Q^* A^2+Q A^{*2}\right),\qquad Q=\sum_j g_j b_j. \tag{4} \]

Indeed \(i\hbar\dot b_j=\partial\mathcal H/\partial b_j^*\). With \(E_V=\operatorname{Re}A\), the analytic dipole consistent with analytic work normalisation is

\[ d_a=-4\frac{\partial\mathcal H}{\partial A^*} =8\hbar A^*Q,\qquad \frac{d\mathcal H}{dt}=-\tfrac12\operatorname{Re}(d_a\dot A^*). \tag{5} \]

The partial derivative holds the atomic coordinates fixed. Integrating by parts, with field and interaction vanishing at both endpoints, gives

\[ \tfrac12\operatorname{Re}\int A^*\dot d_a\,dt =\mathcal H(t_1)-\mathcal H(t_0) =\sum_j\hbar\Omega_j|b_j(t_1)|^2\ge0. \tag{6} \]

These identities are exact for the auxiliary model. Instantaneous work can be negative through reversible exchange; do not clip it. Positive completed-pulse work is the passivity claim, not pointwise dissipation. Endpoint interaction terms must be included for an unfinished pulse. The proposed physical polarisation is \(P_V=N_g\operatorname{Re}\Pi_+d_a\), where \(\Pi_+\) retains positive frequencies without analytic-signal doubling. This projection preserves completed-window work against the positive-frequency \(A\); derive the finite-window endpoint and discrete residuals before implementation. No extra two-photon ionisation-loss current is added: (5) already pays the full absorbed pair energy, including ionisation and photoelectron kinetic energy.

Equations (2)–(6) define an effective completion of the measured on-shell response. They are not asserted to be the unique off-shell atomic Hamiltonian. The continuum coordinates have no spatial electron information. Their completed-pulse energy identity alone cannot qualify a plasma model. Coupling through \(A^2\) keeps only the sum-frequency pathway that is resonant with the continuum. The counter-rotating (\(\Omega+2\omega\)) and difference-frequency (\(|A|^2\)) terms of a real-field quartic response are omitted. They are nonresonant parts of the existing electronic response, so their omission is part of the matching problem of section 4.5, not an extra loss.

4.4 Finite bandwidth, memory and numerical representation#

Eliminating \(b_j\) gives \(Q=iR*A^2\) with the retarded kernel \(R(t)=\Theta(t)\sum_j g_j^2e^{-i\Omega_jt}\). In the continuum limit this is an integral over \(k/(2\pi)\). Its reactive and absorptive parts are linked by the same retarded construction. Near-threshold memory is retained; replacing it by a carrier coefficient is only a separately tested limit.

A finite undamped quadrature can recur. Resolve its energy spacing over the entire atomic time window and demonstrate that recurrence occurs outside the window; for a uniform spacing the recurrence scale is \(2\pi/\Delta\Omega\). Converge both quadrature spacing and window length independently. Do not use an arbitrary damping width to hide recurrence: Lorentzian broadening can create artificial subthreshold absorption. A rotating frame and exact homogeneous evolution with converged integration of the drive can avoid resolving the high continuum frequency with a general-purpose ODE solver. First implement the explicit auxiliary equations as an oracle. A zero-padded, linear retarded convolution may replace them only after time-origin, padding, endpoint and work tests; a circular FFT convolution with its initial state wrapped around is wrong.

The model is cheap relative to an atomic TDSE, not necessarily to an instantaneous cubic. With \(N_c\) continuum nodes, \(N_t\) fine time samples and a chunk of \(N_x\) physical field nodes, the direct cost is \(O(N_xN_cN_t)\) and a streamed implementation needs \(O(N_x(N_c+N_t))\) storage. Recurrence-free resolution needs \(N_c\gtrsim(\Omega_{\max}-\Omega_{\min})T/2\pi\) for a window \(T\): a 15 eV domain and a 1 ps window give \(N_c\approx3.6\times10^3\). With \(N_t\sim3\times10^4\) and sixteen radial nodes that is about \(2\times10^9\) complex updates per right-hand-side evaluation, before convergence margins. The direct form is therefore an oracle, not a plausible production kernel for propagation.

Two analytical reductions are expected to carry production. First, \(Q=iR*A^2\) is exactly a linear retarded convolution before sampling and truncation. Adapt the doubled-grid padding and window-convergence approach of the Raman translation (solprop.translate.raman); its existing arrays and real-transform path are for real kernels. Here both \(R\) and the drive \(A^2\) are complex and require corresponding complex arrays and transforms, not direct use of the real Raman kernel interface. The kernel can be sampled from the continuum integral directly, without discrete continuum nodes or their recurrence. Its long tail still needs convergence checks. A collar-shaped tail taper changes that kernel and can alter its absorptive spectrum; it is not an exact reduction or a guarantee of passivity. Qualify its optical work, passivity and window dependence against the untapered auxiliary oracle, and enlarge the window until the relevant delays are unaffected by the taper. Second, the transverse factorisation of section 2 makes the single-mode response one modal evaluation scaled by \(F^3\) instead of \(N_x\) node evaluations. The per-node path stays as a test of that equality. Benchmark measured \(N_c\), accuracy and transform costs before the choice is fixed in Phase 1. No pole fit or new dependency is required for the first oracle.

If the selected material model or declared completion has a finite, nonzero limit of \(k\) immediately above threshold and zero below, it has a step. That step gives \(R(t)\) a \(1/t\) tail and a logarithmic singularity in the stationary monochromatic reactive coefficient. A nonzero step is a conditional model assumption here: the stored Ar/He curves do not determine the response throughout their threshold gaps. A finite-duration pulse drives the time-domain convolution, which remains finite; it does not acquire an infinite polarisation at threshold. Define the memory time operationally as the kernel truncation beyond which the admitted finite-pulse results converge. Converge quadrature, window and kernel sampling against the chosen threshold structure explicitly.

4.5 Material support and the dispersive matching problem#

Physical zero below threshold is distinct from unavailable and unsupported_detail. Report all three, using optmat’s region masks. The cross section controls integrated loss at occupied pair energies, but unoccupied continuum energies can still influence the reactive response. Consequently a small missing pair-power fraction does not bound the dispersive error or the missing absorption without additional information.

For the prescribed-field diagnostic, report the retained integral and missing support separately. For propagation require an explicit continuum completion record naming the energy domain, outer taper, gap assumptions, material identity and uncertainty variants. A zero outside a finite domain is a declared truncated-model assumption, never an optmat physical value. Do not silently fill a threshold gap, continue a constant cross section to infinity, or label a truncated response fully qualified. Vary the outer domain and taper independently of quadrature and test their effect on the pulse.

The continuum polarisation includes a real third-order contribution. Adding it to the existing electronic response risks counting some dispersion twice. The proposed first matching rule is a real, conservative subtraction of the continuum model’s monochromatic reactive coefficient at a declared VUV reference frequency, with the existing background kept at that reference. Its coefficient is computed from the chosen model, not fitted to the observed cut-off. Derive the subtraction from a local quartic Hamiltonian and apply it through the same field and source selectors, so its completed-pulse work is zero. Phase 0 must verify its factor, reference sensitivity and comparison with the existing electronic model. The latter is already best effort in this VUV region; matching to it does not turn it into validated dispersion. If predictions depend materially on this choice, the first model cannot settle the experimental mechanism. Do not remove the entire reactive kernel by hand and retain the absorption while calling the result the causal model of (2).

The threshold is where the matching is least secure. To fix the sign and limit, consider \(k(\Omega)=k_0>0\) on the finite pair-frequency interval \([a,b]\) and zero elsewhere. Before the conservative matching subtraction, the real coefficient of \(|A|^2A\) in the stationary monochromatic dipole \(d_a\) is

\[ C_{\mathrm{re}}(\omega) =\frac{4\hbar}{\pi}\operatorname{PV}\int_a^b \frac{k_0}{\Omega-2\omega}\,d\Omega =\frac{4\hbar k_0}{\pi}\ln\left|\frac{b-2\omega}{a-2\omega}\right|, \qquad 2\omega\ne a,b. \]

It grows positively as \(2\omega\) approaches the lower edge \(a\). This is the undamped, stationary limit of the declared step model, not a divergence of the finite-pulse response. In the atom, bound Rydberg structure below threshold and, in Ar, autoionising structure between the ionic fine-structure thresholds are outside this model (Q11) and can affect matching in the RDW-relevant band. Absorption is constrained by optmat’s selected calculated cross sections only on their qualified support: missing and unresolved threshold regions remain uncertainties in absorption as well as dispersion.

Phase 0 also runs a labelled absorptive-only sensitivity variant, replacing the retarded kernel spectrum by \(\widetilde R_{\mathrm{abs}}(\Omega)=k(\Omega)/2\), with \(Q_{\mathrm{abs}}=iR_{\mathrm{abs}}*A^2\). It is a comparison, not the causal model and not an admitted production optical mode. If the experimental conclusion differs between the two, the reactive threshold response is an open model dependence and is reported as such. Agreement does not validate a shared missing-data completion.

4.6 Band split, source projection and pump baseline#

Choose a real, even spectral partition \(H(\omega)\) with \(0\le H\le1\): zero for \(|\omega|\le\omega_c-\Delta\), a Planck transition to one at \(|\omega|=\omega_c\), and one above it. Define

\[ E_V=\mathcal F^{-1}[H\widetilde E],\qquad E_p=E-E_V,\qquad A=\operatorname{analytic}(E_V). \]

Use the fine-grid endpoint and Nyquist rules of solprop.fourier.analytic_signal. The bands are real and sum to the original field. The proposed cut is \(\hbar\omega_c=\min(E_1,E_2/2)\) from optmat’s one-polarisation electronic support record, where \(E_1\) and \(E_2\) are its first one- and two-photon excitation energies. This is SEP 6’s boundary of qualified electronic dispersion, not the ionisation threshold. Read the levels, thresholds and support-record identity from optmat; do not hard-code a wavelength. The transition lies below the cut; Q2 fixes and qualifies its width.

A pair with one photon below the cut can exceed the ionisation threshold only if its partner exceeds \(I_p/\hbar-\omega_c\). Report the actual single-photon and pair support against this limit. The split cannot justify ignoring all mixed pairs, pump dressing, or higher-order pump-assisted channels. Content remaining in the pump band is not automatically within the strong-field rate’s validity, either. The earlier mixture rule is outside the revised first scope and needs a later qualification.

If \(A\) is synthesised through a spectral selector \(B\), return the continuum source through \(B^\dagger\). In the real-field work inner product, \(\langle E,B^\dagger J_V\rangle=\langle BE,J_V\rangle\); filtering only the drive loses this identity. The same requirement applies to the pump loss-current branch [6] when its rate sees a selected pump field. Source projection onto the single optical mode must be the adjoint of physical-field synthesis in the declared transverse quadrature. Density is applied once. Measure any source outside the retained spectral and modal space, and separate its work from collars and linear attenuation.

Derive the internal-sign implementation from the conventions’ named boundary helpers; (5) is in the optics convention. Reuse the polarisation source factor for \(P_V\). A direct-current implementation must use the existing current source factor, not integrate a VUV current by a trapezoidal prefix and apply an unrelated prefactor. Qualify the actual discrete work, spectral crop and time endpoints, not only (6).

Keep three comparisons distinct: the legacy total-field pump rate; the declared split pump baseline; and that same split baseline plus continuum optical response. The change caused by removing VUV content from a pump-carrier rate must not be attributed to two-photon absorption. A split-only baseline is a labelled diagnostic, not a complete recommended model. Use exactly the same split in the pump-only and combined tests. The existing pump electrons still drive the plasma current through the total local field. No VUV two-photon energy is charged to the pump loss current. Retain the strong-field rate’s own energy per event and pre-ionisation semantics, checking its table range and peak-weighted energy against the filtered pump field: its peak need not be bounded by the original field’s peak. Record pump event energy, continuum optical work and free-electron work separately. A future shared-depletion model must count each electron once and use the same discrete event counts for population and energy; the initial weak-excitation model does not claim to implement that closure.

4.7 Additional electrons: final yield is not a birth-time law#

\(P_c(t)=\sum_j|b_j(t)|^2\) is nonnegative, but can decrease and includes off-resonant, virtual excitation while the field is present. Even a subthreshold drive can have \(P_c(t)>0\) and \(P_c(\infty)=0\). Therefore neither \(N_gP_c(t)\) nor a clipped positive \(\dot P_c\) is an admissible default free-electron density/rate. A spectral factor \(|H|^2=k\) with a guessed phase and \(|h*A^2|^2\) also does not establish the electron birth history or its compatibility with (5).

The first implementation supplies final added-electron density \(N_gP_c(\infty)\) as a diagnostic only. Its optical feedback includes the continuum polarisation; the VUV-produced electrons do not yet enter the Drude current. Label every such propagation result continuum optical response; additional plasma omitted. This omission is acceptable for the experimental inference only after its possible effect has been bounded. Vary diagnostic electron histories over physically plausible birth intervals as a sensitivity experiment, including temporal overlap with the pump; such scans do not constitute a derived or generally rigorous bound on all histories. If sensitivity is material, stop short of selecting an experimental mechanism.

Gate for a later plasma-enabled phase: identify an outgoing-electron or other independently justified timing model; show no free electrons for reversible subthreshold excitation; compare the predicted current/work as well as final yield with converged atomic calculations on VUV-only and pump-plus-VUV fields. A delayed absorber flux is not automatically the birth rate at the atom. Any chosen time coarse-graining or escape-time prescription and its sensitivity must be declared. A new atomic calculation may be needed; this is not information in \(\sigma^{(2)}\). Only then specify shared neutral depletion and the free-electron current and re-review the added scope. Do not exponentiate (1), invent a positive rate, or count the ionisation energy again in the plasma loss. Failure of this gate leaves a useful optical-response model, not a completed test of all two-photon feedback.

4.8 Weak-excitation and omitted-physics qualification#

Track both final yield and the maximum auxiliary occupation; neither alone establishes weak atomic dressing. Test intensity scaling, spectral split sensitivity, chirp and pump overlap against independent atoms. Start with the existing, fixed initial neutral fraction; require pump depletion over the atomic memory time to be negligible for the stationary response. A slowly varying multiplier on (5) can change its energy identity and is not an authorised shortcut. Strong depletion, extra charge states and stimulated recombination are outside the first qualification.

Bound resonances, higher-order VUV processes and pump-assisted channels can remain important even when direct one-pump-plus-one-VUV ionisation is energetically closed. The continuum-only result does not bound their contribution. Use optmat’s existing material records, and the Ar calculations they curate [3, 4], with atomic comparisons to assess their relevance. The Ar calculations in optmat’s OEP 8 review do not validate all near-threshold wavelengths, and helium validation does not qualify argon. The current He and Ar support intervals are read from optmat at translation and recorded, not copied as permanent numerical limits in solprop.

4.9 Does this need a coupled optmat OEP?#

Decision proposed: no new optmat OEP is required for the initial model in this SEP. The auxiliary weights, spectral completion choices, dipole and energy identities are solprop’s propagation approximation. They do not create a new measured material quantity. optmat continues to supply its cross sections and other material data.

Input or missing information	Present source	Consequence
Degenerate \(\sigma^{(2)}\), thresholds, masks, uncertainty and identity for Ar/He	Implemented OEP 6	Enough to construct and normalise (2) on supported domains; update the dependency pin before implementation
Pump rates and energy per event; density and electronic response	Existing optmat interfaces used by SEP 3/6/17	Reuse; do not transcribe into solprop
Bound levels and their coefficients	OEP 6 records and proposed OEP 8	Separate bound-response work; not needed to implement the direct continuum
Cross sections in the gaps immediately above threshold, especially Ar’s ionic fine-structure region	Incomplete OEP 6 support	Additional curation/calculation may be essential to a threshold claim; do not replace by zero
Cross sections outside the chosen domain and real-response matching information	Partial data; no uniquely fixed completion	Recorded solprop model sensitivity; refuse an accuracy claim if it dominates
Nondegenerate amplitudes or channel-coherence information	Not supplied by the degenerate curve	Validate the separable approximation first; a new material representation would require a coupled OEP if needed
Electron birth/current during overlapping pulses	Not encoded by a total cross section	solprop response derivation and atomic validation, not a new optmat plasma-polarisation API

New sources, errata and added points for the existing cross-section quantity follow optmat’s ordinary curation and register rules, not a new OEP merely because the data are missing. A new OEP becomes necessary if qualification demands a new quantity family/schema, a nondegenerate material amplitude or overlap model, a new fitted material model/default, or another major optmat interface change. Its scope must name that additional material contract; the causal field response remains in a solprop SEP. Do not draft a placeholder OEP promising a plasma polarisation or an unspecified complex response. Reassess this decision at the Phase 0 gate, recording the evidence. This distinguishes sufficiency for building an explicit approximation from sufficiency for a physically qualified prediction at every wavelength of the experiment.

5. Proposed interface#

Names are a proposal, not implemented API. Use one declaration with explicit modes:

continuum = TwoPhotonContinuum(
    species="Ar",                       # or "He"
    material="optmat",
    split=BandSplit(...),
    completion=ContinuumDomain(...),     # explicit support/completion and matching record
    coupling="diagnostic",              # or "optical" after Phase 0 qualification
)

coupling is not called mode, which in solprop names a waveguide mode. diagnostic consumes physical fields or saved fields with their documented power-to-field conversion and transverse mode normalisation. It reports (1), support fractions and whether the pulse completed. It changes neither the field nor the plasma. optical adds (5) and its declared conservative matching subtraction to the existing response combination, with pump ionisation optional. It reports final continuum yield and work, but no additional Drude density. There is no plasma=True option until the later gate has supplied and reviewed a concrete model.

ContinuumDomain specifies angular-frequency endpoints and quadrature resolution, outer taper, unsupported-region treatment and matching reference, all with SI units. Numerical tolerances are separate from material/model uncertainty. User completion variants are recorded as assumptions with identities, never silently promoted to optmat recommended values. Default physical mode is off; no default arbitrary gap filling. Refuse incompatible routes, polarisation, species, nonfinite/negative material values, nonpositive quadrature weights, invalid domains, a backend without the response and simultaneous native TDSE or another two-photon optical response for the same atom.

Missing spectral coverage must not be hidden by a successful run. Diagnostic mode can report a retained integral and incomplete; optical mode needs an explicit completion policy or refuses the material construction. Dynamically occupied unsupported physics is reported and prevents a qualified interpretation; it is not converted to a silently clipped source. Define empty-denominator and empty-band diagnostics without division by zero. Scope refusals do not turn small support fractions into error guarantees.

Record material/source digests, optmat version, split, completion and matching policy, continuum quadrature, model version, qualification label, omitted plasma flag and initial neutral fraction in the run manifest and resume identity. Statistics separate computed atomic work, projected source work, final continuum yield, endpoint residuals, missing-data exposure and numerical boundary loss. Atomic auxiliaries reset at the start of the time window on every optical RHS call, including rejected optical steps; there is no hidden history across propagation distance or across RHS retries.

6. Implementation strategies#

Option	Advantage	Limitation	Recommendation
Carrier-evaluated local rate/current	Smallest cost	Cannot resolve threshold-straddling pair spectra	Scale check only
Earlier window-calibrated rate	Correct integrated weak-field totals	Unqualified timing and source-work distribution	Comparison only; replaced by the continuum candidate
Explicit effective continuum, (2)–(6)	Retarded auxiliary dynamics and one work identity; finite model is inspectable	Continuum resolution, reactive completion and electron-timing limits; \(O(N_xN_cN_t)\) per evaluation	Reference oracle; production only if the convolution fails its qualification
Retarded convolution of the same model	Raman-like doubled-grid kernel; no continuum-node recurrence; cost independent of \(N_c\)	Kernel-tail, padding and source-work qualification; same physical limits	Expected production route, after oracle parity (section 4.4)
Full atomic TDSE	Independent dynamics, within its atomic approximation	Much greater cost and atom-dependent errors	Selected validation runs

Physics-layer declarations and the optmat adapter are separate from translation’s quadrature, coupling and selector arrays. Backends contain numerical loops and no material selection. Add a typed immutable continuum leaf to the kernel contract and bump SCHEMA_VERSION on implementation; do not choose a future version number now. The polarisation/current source factors already exist. Reuse the disabled path exactly, with no continuum buffers or transforms when off. The driver’s step/retry semantics are unchanged. Diagnostics and source projections must share the physical quadrature.

Implement numpy_ref and its material-independent oracle first, then numba_cpu, then Kokkos CPU. Unsupported backends refuse by name. Qualify GPU separately under the standing backend policy. Avoid materialising the continuum-by-time trajectory in production; chunk physical nodes and stream the auxiliary state. No core dependency addition is proposed. A future reduced kernel cannot discard threshold memory merely to meet a speed target.

7. Verification and validation#

The distinction is between exact identities of the chosen approximation and independent validation against atoms. Tolerances are set by convergence and source uncertainty, not by accepting the first residual obtained.

Normalisation: long, weak monochromatic pulses reproduce optmat’s rate and absorbed pair energy. Compare direct quadrature of (1) with independent integration of (2) for Gaussian, chirped, two-colour and delayed-pulse drives, including pairs on either side of threshold. Keep coherent cross terms.
Energy: compare (6), atomic final energy and the actual projected optical source work; refine time, continuum energy and transverse quadrature separately. Test incomplete endpoints, empty bands, real-field zeros, broad spectra, selector transitions, DC/Nyquist rules, positive-frequency projection and spectral cropping. Verify the matching subtraction does zero completed-pulse work.
Retardation: hold an earlier analytic drive fixed while changing a later pulse; earlier auxiliary states must agree. Separately quantify band-selector tails.
Continuum convergence: vary the outer domain/taper as a model study, spacing as numerical quadrature, and the window as an endpoint/recurrence study. Never conflate convergence of a truncated continuum with completeness of the material response. Resolve a declared threshold step separately: check the stationary reactive coefficient against its signed principal-value logarithm away from the step edges, and verify a finite, converged response for pulses centred across the threshold.
Convolution parity: the complex doubled-grid convolution matches the auxiliary oracle on the same drive. Delayed pulses whose separation enters the kernel taper expose its error; enlarge the window and converge that error separately from sampling. Check optical work and passivity as well as waveform error; padding alone does not qualify a tapered kernel.
Transverse factorisation: on a single mode, per-node evaluation and the scaled modal evaluation agree to rounding, with and without a fixed neutral profile.
Threshold sensitivity: compare the causal response with the labelled absorptive-only variant on the replayed and propagated RDW cases (section 4.5).
Weak response: yield scales as field amplitude to the fourth power, dipole to the third; compare narrowing bandwidth with the local attenuation law. Field depletion along distance may be finite while single-atom excitation remains perturbative.
Independent atom: compare VUV-only pulses first, then add delayed and overlapping pump fields. Compare total absorbed work, final yield and complex nonlinear current or dipole where available. Use the atomic model’s own cross section for a structural comparison; separately compare it with optmat’s material values. Do not calibrate the absorptive response with an unrelated Kerr scale. Helium correlated calculations and argon calculations require separate evidence; an SAE discrepancy is not a numerical tolerance to absorb. The existing optmat checks at sampled energies do not settle the near-threshold case.
Electron gate: demonstrate the virtual-population counterexample before trying any plasma closure. Final yield agreement alone does not pass section 4.7.
Propagation: use about ten steps or seconds-long short segments for qualification, including no-pump, split-pump baseline and combined optical response. Compare energy, spectrum and per-channel work, not only peak wavelength. Full experimental scans belong in a worked example or research run, never the test suite.
Backend parity: match NumPy, Numba and Kokkos CPU on identical descriptors and short segments, including disabled-path equality and repeated/rejected RHS calls.

Drafting check, not atomic validation: independent DOP853 integration of a three-node synthetic continuum in dimensionless units verified the factor in (5). With \(\hbar=1\), \(\Omega=(3.2,4.0,5.1)\), \(g=(0.006,0.008,0.005)\) and \(A=\exp(-t^2/2-2it+0.13it^2)\) on \([-8,8]\), relative/absolute solver tolerances \(10^{-11}/10^{-13}\), the final occupation is \(0.000318831632448\). Independent Simpson integration of the Fourier amplitudes agrees to \(6.8\times10^{-12}\) relative, and \(-\tfrac12\operatorname{Re}\int d_a\dot A^*dt\) agrees with final continuum energy to \(8.6\times10^{-12}\), at both 6001 and 12001 time samples. These synthetic numbers are reproducible formula checks, not material data or production test tolerances. No atomic validation or RDW propagation was run for this draft.

8. Documentation#

Add docs/model/two-photon-continuum.md with the derivation, material/solver boundary, field normalisation, selectors, support/completion rules, optical-work identity, convergence instructions and explicit limits on electron timing. Link it from the nonlinear, ionisation, material and running pages. Document each public declaration and manifest field. A short tutorial compares the diagnostic with the optical response in a supported band; it must not claim a threshold prediction from incomplete data. A worked example replays a declared RDW dataset and then performs the controlled pump/continuum comparisons, with the actual input pulse and geometry. Publish all omitted-physics labels alongside its spectra, and link the final SEP from its manual.

9. Compatibility and impact#

Additive and opt-in. No existing material choice, default response, convention or simulation result changes when disabled. The new leaf changes the kernel schema and requires backend admission and resume compatibility checks. Record the response and model assumptions in output metadata. Update the manual and changelog on implementation, not merely because this draft exists. Deliberate changes to the treatment of VUV content in the pump rate are documented on the Luna differences page with the baseline tests. No code or data in optmat changes as part of this draft; release/pin work precedes the adapter implementation. The first scope now excludes Ne/Xe, mixtures, mode-averaged and multimode routes that the earlier design intended to support. Those remain on the roadmap. SEP 2 remains independent: its narrowband attenuation law provides a limiting check, not a second optical response to combine with this one.

10. Alternatives considered#

The former window-calibrated rate and separate real loss current are not the selected candidate in this revision. Matching integrated yield does not establish loss timing or spectral work. The previous review’s two-tone current residual and pump-source counterexample remain regression cases for Q4/Q18; they are preserved in Part III. Likewise, a quotient using the total field to force instantaneous ionisation-loss work can be singular at a total-field zero while the pump rate is nonzero. Adjoint source coupling is the proposed route, with its discrete qualification still required.

A positive rate obtained by taking a square root of a cross section is not enough: its phase, timing and source work are unspecified. A carrier coefficient cannot provide the threshold response of a broadband pulse. A full nondegenerate, channel-resolved amplitude model may eventually be necessary but is not a prerequisite for testing the explicitly separable approximation. A separate optmat OEP solely to reproduce its existing cross sections would add process without material content. Conversely, the absence of a new OEP must not be used to hide missing material data or transfer the propagation closure into optmat. A fitted cut-off or arbitrary damping chosen to match the experiment would not discriminate the hypotheses and is excluded.

11. Phases#

Retain the qualification-first numbering: Phase 0 precedes production implementation in Phase 1. Each implementation phase uses its own feature/sep-0017-… branch and appends a report to Part II with decisions, checks, source evidence, limitations and handoff. The current Part I must be reviewed and re-accepted before implementation.

Phase 0 — Qualification and prescribed-field diagnostic. Derive and review the continuum and dipole normalisation, selectors, discrete work and conservative matching term. Build the synthetic oracle and research replay diagnostic; audit the actual optmat pin, material coverage and experimental pulse definitions. Compare selected atoms, assess omitted channels and document the first admitted domain. Resolve the production gates in Q1–Q4, Q6, Q9, Q13 and Q16–Q18; reassess Q20 and benchmark the execution routes of Q21. A useful diagnostic can be delivered even if the optical-response gate fails, but that does not complete the SEP or authorise production optical feedback.
Phase 1 — NumPy optical response. Implement the immutable declarations, adapter and seam, physical-node response, manifest and diagnostic outputs; tests, manual and changelog. Include the conservative matching term and expressly omit additional VUV plasma. Qualify short propagation segments against the oracle and split baseline.
Phase 2 — Numba CPU. Compiled loops, shared descriptor admission, parity, coverage and short performance ledger. Unsupported backends refuse by name.
Phase 3 — Kokkos CPU. Native implementation and parity, descriptor admission and the checks required by both repositories. GPU qualification remains separate.
Phase 4 — Scientific example and completion. Representative RDW replay and a worked example with the declared pump/continuum comparisons. Record failures of material or atomic qualification plainly. Complete the repositories’ required final gates and CPU parity, document the admitted domain and what the optical-only model can establish, and move unresolved plasma and omitted physics to the roadmap.

Conditional plasma extension: if added-electron feedback could alter the scientific conclusion, derive and review section 4.7’s timing/current model in a revision before acceptance or a later SEP after Part I is frozen. Identify any new optmat material contract at that point and draft its coupled OEP. Full VUV plasma feedback and strong shared depletion are not claimed delivered by the optical-only phases. Ne/Xe, mixtures, mode-averaged and multimode configurations are also follow-up work, each with its own material, source-projection and atomic qualification.

12. Open questions#

Q1–Q18 retain their subjects and identities from the earlier review. This revision changes the proposed response and some earlier scope decisions; old review references in Part III refer to the text reviewed at that time. Q19–Q21 make the new gates explicit.

Q1 — optmat release and references (Phase 0; production gate). Confirm the pinned release supplies OEP 6’s required APIs, the selected Ar/He source identities and uncertainty/region semantics. Update the pin before adapter implementation.
Q2 — Cut, width and mixtures (Phase 0). Qualify section 4.6’s material-derived cut and transition width against actual fields. Mixtures are now deferred; the earlier common-cut rule is not an admitted configuration in the first model.
Q3 — Broadband response (Phase 0; production gate). Replace the local-rate candidate with (2)–(6). Verify its normalisation and weak-excitation domain; validate the frequency-split approximation separately for Ar and He. An exact integrated identity is not evidence for physical transient accuracy or a saturation law.
Q4 — Source coupling (Phase 0; production gate). Derive and qualify the adjoint selectors for both continuum polarisation and pump loss, including physical-field synthesis and modal projection. Retain the earlier counterexamples. Positive work against a selected band alone is insufficient; coordinate with Q18.
Q5 — Default (unchanged). Opt-in. No new default is proposed by this revision.
Q6 — Missing material coverage (Phase 0; production gate). Read actual pair and single-photon support, especially near Ar’s threshold. Diagnostic mode may report an incomplete retained integral; optical mode requires a declared completion or refuses construction. Dynamically occupied gaps are reported and block a qualified inference, not silently clipped. A small missing share is no accuracy guarantee. This replaces the earlier warning-only construction policy.
Q7 — Continuum without a pump rate (revised scope). Permit VUV-only diagnostics and optical response for qualification and supported applications. A pump rate is optional. A split-only baseline is a labelled comparison, not a complete model.
Q8 — Envelope grids (deferred; roadmap). Refused in the first implementation.
Q9 — SEP 6 and reactive matching (reopened; Phase 0 production gate). The continuum now includes a real response. Derive and review the conservative reference subtraction, test reference/domain sensitivity and avoid double counting with the electronic background. Its best-effort VUV label is not upgraded by matching. Report the conditional monochromatic logarithm, finite-pulse convergence and the absorptive-only comparison; if the conclusion depends on them, the reactive response near threshold stays open. Shared threshold data gaps also limit both variants.
Q10 — SEP 2 (independent). Use its narrowband attenuation limit as a check. Its solid coefficient and proposed local current are not this gas continuum model.
Q11 — Bound resonances and (2+1) ionisation (deferred; roadmap). optmat OEP 8 supplies proposed material work; a separate solprop response is still needed. Bound coefficients do not complete this continuum model or justify omitting bound physics.
Q12 — Polarisation (deferred; roadmap). Fixed linear polarisation only; vector fields and other polarisation states need material and response qualification.
Q13 — Timing and coherence (Phase 0; production gate). The window-wide rate is replaced, but the new model must pass delayed-pulse, coherent cross-term, selector-tail and memory tests. Hold the earlier prescribed analytic drive fixed when testing retardation. Do not infer electron birth times from the auxiliary occupation (Q19).
Q14 — Other species and routes (deferred; roadmap). Ne/Xe, mixtures, mode-averaged and multimode cases, free-space, planar and general modal routes need separate admission and qualification. The first route is a single-mode capillary with resolved physical-node quadrature. For the continuum response alone a mode-averaged single mode is exact (section 2); its exclusion is a scope choice because its pump plasma is not.
Q15 — Interface, statistics and grid (Phase 1). Finalise section 5’s declarations (the coupling keyword avoids the waveguide sense of mode), manifest identities, numerical admission, zero-denominator semantics and separate atomic/source work, final yield and support diagnostics. Rejected optical steps must not leave hidden atomic history.
Q16 — Omitted VUV and pump-assisted channels (Phase 0; then roadmap). Assess three-photon ionisation, bound excitation and pump dressing/assistance in the actual fields, using available optmat quantities and atomic comparisons. A significant unmodelled channel prevents attribution; its implementation is outside this scope.
Q17 — Atomic validation (Phase 0; production gate). Establish the frequency splits, intensities, durations and pump overlaps for each species. Compare complex current/dipole, work and final yield, using the atom’s own cross section for structural tests. Helium is not an argon validator. No fixed 6 or 10 fs duration substitutes for the generated local VUV pulse; a missing suitable reference remains an evidence gap.
Q18 — VUV work and discrete source (Phase 0; production gate). Qualify (6) under positive-frequency projection, adjoint selectors, finite endpoints, spectral crop, transverse quadrature and the correct source factor, with density applied once. The direct polarisation avoids an unnecessary current integral, but not these work tests. The pump branch uses the existing band-rule plasma path. Set tolerances by convergence before production implementation.
Q19 — Added plasma (conditional extension). Is its effect negligible for the selected experiment? If not, derive and validate an electron-timing/current model before feeding extra electrons into the plasma or adding shared depletion. Neither transient occupation nor a clipped derivative is a valid default. Optical-only results cannot close a materially plasma-dependent mechanism question.
Q20 — Coupled optmat proposal (reassess at Phase 0). None for the initial cross-section-based approximation. Existing-quantity gaps are curation work. New nondegenerate/channel-coherence representations, fitted material models or major material interfaces require a named coupled OEP before their implementation.
Q21 — Cost and continuum compression (Phase 0 benchmark; Phase 1 choice). Select recurrence-free quadrature for the oracle and benchmark it against the doubled-grid kernel convolution and the transverse factorisation of section 4.4. The direct auxiliary form is expected to be too costly for propagation. Do not broaden or discard spectral structure for speed.

13. References#

optmat, OEP 6 — Two-photon ionisation and absorption of the noble gases, Final; its integrated pair formula and current data/model manual.
optmat, OEP 8 — The bound two-photon resonances of the noble gases for propagation, Draft, Reviewed; see its entries in optmat’s review register.
C. McKenna and H. W. van der Hart, “Multiphoton ionization cross sections of neon and argon,” J. Phys. B 37, 457–470 (2004). doi:10.1088/0953-4075/37/2/013.
I. D. Petrov, B. M. Lagutin, V. L. Sukhorukov, A. Knie and A. Ehresmann, “Correlation and polarization effects in two-photon photoionization of Ar,” Phys. Rev. A 93, 033408 (2016). doi:10.1103/PhysRevA.93.033408.
A. Mihelič and M. Horvat, “Calculation of multiphoton ionization amplitudes and cross sections of few-electron atoms,” Phys. Rev. A 103, 043108 (2021). doi:10.1103/PhysRevA.103.043108.
M. Geissler, G. Tempea, A. Scrinzi, M. Schnürer, F. Krausz and T. Brabec, “Light propagation in field-ionizing media: extreme nonlinear optics,” Phys. Rev. Lett. 83, 2930 (1999). doi:10.1103/PhysRevLett.83.2930.

Source audit. [1–2] and optmat’s curated source records for [3–5] were read for this draft. The authors’ accessible preprint of [5], arXiv:2103.16422, was checked for its amplitude/cross-section method and, on follow-up review, section II.2’s equal-photon-energy formulation. It does not supply the required nondegenerate validation or validate the reduced transient model proposed here. Primary papers [3–4] and [6] were not independently rechecked in full for this draft; their bibliographic details and roles follow the curated records and the earlier version of this proposal. No numerical material value was transcribed. Equations (2)–(6) are the explicit candidate derived here from [1]’s integrated constraint, with the synthetic check in section 7; they are not attributed to a source as a published atomic model.

Part II — Implementation record#

Empty until Phase 0. No implementation phase has been completed.

Part III — History#

Historical scope. The reviews and acceptance below concern the earlier Part I. Their section numbers, question resolutions and validation results describe that version. They are preserved as evidence, not applied to the substantially revised Part I above. The revised Part I was reviewed on 2026-10-12 (the last section).

2026-10-08: Drafted, paired with optmat’s OEP 6 (Accepted the same day), with its relation to SEP 2 (Draft) and SEP 6 (Final) worked out.
2026-10-08: GPT-6 completed the AI review and revised Part I. John Travers stated that his human review is not yet complete; only the AI reviewer is recorded, so review status was In review under SEP 0 and lifecycle status remained Draft.
2026-10-08: John Travers confirmed completion of his human review and accepted the SEP in the review discussion. Review status is Reviewed and lifecycle status is Accepted. Part I is frozen on acceptance; Q3, Q4, Q13 and Q18 remain explicit Phase 0 prerequisites requiring resolution and review before Phase 1.
2026-10-09: Opus 5.5 checked the review revisions against the code and numerically: the two-tone residual (8/33), the pump-current counterexample (−1/8), the adjoint-filter work identity and the cumulative trapezoid’s multiplier, \((\theta/2)\cot(\theta/2)\) of the exact one at \(\theta=\omega\Delta t\), which at the default ffac = 6 is 0.907 at \(\omega_{\max}\) for one integral and 0.82 for the two of the Drude current. That factor also applies to the existing field-grid plasma path, independently of this SEP. The drafting-agent field now names the review revisions’ agent. Part I is unchanged.

AI review, 2026-10-08#

The review covers all of Part I against accepted optmat OEP 6, SEP 0 and the current plasma, Fourier, TDSE and kernel contracts. Findings are corrected in the proposal or assigned to explicit Phase 0 decisions; no replacement physical closure is adopted by this review.

Energy and passivity: the proposed real current does not generally pay the pair-energy target, and positive work against a filtered band does not prove positive work against the propagated field. Added analytic counterexamples and an adjoint-filter candidate; reopened Q4 and added Q18 as prerequisites for Phase 1.
Timing and coherence: the window calibration is nonlocal in time; separated coherent pulses retain weighted spectral interference. Gating changes the physical functional and cannot automatically repair timing. Q3 and Q13 require a reviewed closure and qualified regime before implementation.
Discrete propagation: cumulative trapezoidal integration differs from the exact current multiplier inside the active band. Discrete channel attribution, source work, cropping, collars and density factors need an explicit consistent accounting.
OEP 6 coverage: zero in a missing interval is a downstream omission, not an optmat value. Incomplete pair power supplies no relative loss bound; a propagated solution is not a lower bound. Pair-mean support also cannot qualify individual photon energies or nondegenerate amplitudes.
TDSE and admission: macroscopic current scaling does not scale microscopic ionisation yield. Comparisons use unfiltered native scale one. Added empty-band and zero-yield semantics, actual-bin grid admission, mixture-split consistency and separate event-energy and work diagnostics.

Evidence and limits. Independently checked the analytic-signal normalisation against OEP 6’s \(E^+\) formula and re-derived the work and filtering identities. A 65,536-point periodic numerical check reproduces the two-tone residual \(8/33\) and the pump-current counterexample’s work \(-1/8\) against the total field, to floating-point precision. Independent Bessel quadrature and SI constants reproduce §4.11’s transverse factor 0.565536 and initial energy-loss coefficient 0.635747 m⁻¹; the ripple factors are 1.863965 and 29.788855. Checked optmat’s proposal and data-page coverage, and inspected solprop’s actual plasma integrals and TDSE source assembly. No cross-section dataset or source coefficient was changed. The primary atomic-data papers were not independently re-audited here; their evidence remains optmat’s curated records and OEP 6’s source audit. No TDSE, backend implementation or propagation qualification was performed.

Verification of this revision. git diff --check, codespell on this SEP and its roadmap entry, pytest tests/test_seps.py (134 passed, one expected process-SEP skip), and poe docs with Sphinx warnings treated as errors. Commands ran through uv in the existing project environment, with synchronisation disabled and a writable temporary cache. The analytic counterexamples, the monochromatic rate normalisation, the adjoint-filter work identity and the trapezoidal Fourier multiplier were checked numerically. The full package, slow and coverage lanes were not run: this revision changes proposal documentation only.

Substantial revision, 2026-10-12#

John Travers authorised consolidation into a revised SEP 17 and its return to Draft under SEP 0. The earlier acceptance and reviews above remain part of the record.
Replaced the proposed window-calibrated local rate and separate VUV loss current with the explicit weak-excitation continuum candidate in Part I. The first delivery is prescribed-field diagnostics and continuum optical feedback with final yield; additional plasma needs a separately derived and validated timing/current model.
Narrowed initial admission to pure Ar/He, fixed linear polarisation and a single-mode capillary with physical-node quadrature. Preserved the broader species, mixtures, modal and omitted-physics work on the roadmap.
Retained Q1–Q18 identities, revised their recommendations where the response changes them, and added Q19–Q21 for additional plasma, material-contract reassessment and continuum cost. Kept the Phase 0 qualification gate before Phase 1 implementation.
Assessed optmat’s existing material inputs: no new OEP for the initial approximation; data gaps and a future new material contract remain explicit. No optmat data changed.
Status Draft; current Part I Not reviewed. The previous reviewers remain recorded in the dated history; neither their prior approval nor this drafting work counts as review of the revised design.

AI review of the revision, 2026-10-12#

Opus 5.5 read the whole of the revised Part I against SEP 0’s acceptance criteria, optmat OEP 6 and solprop’s modal, Raman and Fourier code. Findings were edited into Part I; none needed the drafting agent’s clarification.

Checked and correct. Equation (1) agrees with OEP 6’s \(K|S_2|^2\) under \(A=2E^+\), and its monochromatic limit gives \(\sigma^{(2)}(I_0/\hbar\omega)^2\). The solution (3), the Hamiltonian derivative of (4), the factor four in (5), the work identity (6) and the positive-frequency projection of \(P_V\) were re-derived. An independent DOP853 run reproduced section 7’s drafting check: final occupation \(3.18831632448\times10^{-4}\), Fourier quadrature to \(10^{-11}\) and the work identity to \(10^{-11}\) relative.
Transverse factorisation (section 2, Q14). The claim that an effective area cannot reproduce the fourth-power response was too strong for this model. With linear selectors, a fixed neutral profile and the frequency-independent Marcatili \(\mathrm{HE}_{11}\) shape, the response factorises exactly and projects through the Kerr overlap. Recorded as an exact reduction and a test; the radial rule stays for the pump plasma.
Cost and execution route (sections 4.4, 6, Q21). Added a cost estimate showing the direct auxiliary form is about \(10^9\) complex updates per right-hand side for a representative capillary run. Recommended the existing Raman doubled-grid convolution machinery as the expected production route, with the auxiliary equations as its oracle.
Threshold structure (sections 4.4, 4.5, Q9). A finite cross section at threshold makes the kernel decay as \(1/t\) and the reactive coefficient diverge logarithmically at the two-photon onset, the RDW-relevant band, where the omitted Rydberg and autoionising structure matters most. Added an operational memory time, threshold convergence tests and a labelled absorptive-only sensitivity comparison.
Omitted pathways (section 4.3). Stated that \(A^2\) coupling keeps only the resonant sum-frequency pathway; the nonresonant terms belong to the matching problem.
Interface and references. Renamed the proposed mode= keyword to coupling=, avoiding the waveguide sense of mode. References [3–6] were listed but uncited; each is now cited where it is used.

Evidence and limits. The derivations and the three-node check were redone independently; no atomic calculation, kernel benchmark or propagation was run. The cost figure is an estimate from the stated sizes, to be replaced by Phase 0’s benchmark. The primary atomic-data papers were not re-audited.

2026-10-12: Opus 5.5 completed the AI review of the revised Part I (above). The human review is recorded for John Travers, who asked on 2026-10-12 for the review status to be set to Reviewed. Review status Reviewed; lifecycle status remains Draft pending re-acceptance.

Follow-up AI review and re-acceptance, 2026-10-12#

GPT-6 reviewed the revised Part I and the independent review changes, checking the material support records, Raman and modal implementation, and the cited primary amplitude paper. The three corrections requested by John Travers were applied:

Threshold structure and evidence. Made the nonzero threshold step conditional, derived the signed principal-value logarithm for a finite constant-\(k\) interval, and distinguished the stationary monochromatic singularity from finite-pulse response. Corrected the description of calculated cross sections and retained missing/unresolved threshold data as limits on absorption and dispersion alike.
Amplitude citation. Checked [5], section II.2, against its equal-photon-energy equations. It supports an amplitude method, not the nondegenerate validation that this proposal still requires.
Convolution implementation. Distinguished the exact convolution identity from sampled and tapered execution. Raman supplies padding/window methodology; the continuum requires complex kernel/drive arrays and transforms. Made taper-induced changes to optical work, passivity and convergence explicit in the qualification.

Evidence and limits. Independent principal-value quadrature matched the signed logarithm at six frequencies on either side of the lower edge, with maximum absolute error \(1.6\times10^{-14}\) in a dimensionless constant-k example. A separate synthetic Fourier-grid calculation checked the absorptive-only work identity (relative error \(2.2\times10^{-16}\)) and transverse factorisation with a fixed nonuniform neutral profile (\(1.4\times10^{-16}\)). These are algebra checks, not atomic validation or production tolerances. No new atomic, kernel-performance or propagation calculation was run. No material number was transcribed; other primary atomic-data papers were not re-audited by this follow-up review.

2026-10-12: GPT-6 completed the follow-up AI review after the corrections above; John Travers instructed that SEP 17 be Accepted. The earlier human and Opus 5.5 reviews remain recorded. Status Accepted; review status Reviewed. Phase 0’s scientific and numerical qualification gates remain prerequisites for production implementation; acceptance does not claim they have passed. Part I is frozen.