SEP 27 — An edge-free field-grid plasma source: hold the current at the temporal collar#


SEP	27
Title	An edge-free field-grid plasma source: hold the current at the temporal collar
Author	John Travers
Drafting agent	Opus 5.5 (2026-10-19)
Status	Final
Review status	Reviewed
Reviewers	John Travers (human); GPT-6 (AI)
Type	Change
Created	2026-10-19
Resolution	Accepted by John Travers, 2026-10-19; Final, 2026-10-20, under the recorded closing scope amendment
Requires	SEP 18 (the plasma column’s running-integral rule); SEP 26 (the held-current identity, the analytic collar derivative and the steady-current certificate)
Supersedes	—
Superseded-by	—
Implementation	Phases 1–5 complete on `feature/sep-0027-phase-5`; native backend on `feature/sep-0027-phase-3`; locally committed, awaiting integration
Manual pages	Field plasma, grids, Luna differences, output

Part I — Proposal#

1. Abstract#

The field-grid plasma response (PlasmaResponse, the Geissler-type rate model) multiplies its polarisation \(P=\int J\,dt\) by the fine temporal collar \(g\) before the transform. After the pulse the free-electron current is steady, so \(P\) keeps growing and the collar switches it off. In current variables the source then contains the collar’s switch-off, which no electron emits: a steady current radiates nothing at \(\omega\neq0\). SEP 26 removed the same artefact from the TDSE source and measured it on the rate route offline: 3.4–7.4% of the 90 nm–1.6 µm source and 8.4–18% of 0.9–1.6 µm at SEP 20’s late L6 planes, and about \(1.5\times10^{-5}\) at 90–200 nm.

This SEP replaces the gated polarisation, as the only behaviour, by the transform of a current held at its collar-weighted value. Every time variation of \(J\) inside the window is kept, and only the switch-off is removed. The source is formed directly from the sampled current in frequency,

\[ A_J=\frac{\mathrm{DFT}[gJ]}{i\omega}-\frac{\mathrm{DFT}[g'J]}{(i\omega)^2} \quad\text{on the active bins}, \]

which is exact to rounding on resolved collars, under either running-integral rule. The column’s third running integral, the one that formed \(P\), is no longer needed. There is no public option and no compatibility path (§9). The truncating form survives only as a private, labelled, non-resumable comparison hook for validation.

Success means:

the analytic and fixed-field tests close within budgets derived in advance;
SEP 26’s fixed-field measurement is reproduced by the production form;
numpy_ref, numba_cpu and Kokkos CPU agree within SEP 25 budgets;
propagated before/after comparisons are reported with their integration-error bounds and controls.

2. Motivation and scope#

The problem. Most of the maintainer’s propagation runs, and the parents and baselines of SEP 12, SEP 20 and SEP 26, use the field-grid rate plasma. Its near-infrared source carries a spurious collar term whose size grows with the plasma’s residual drift and with the delay between ionisation and the trailing taper. On the run_short case (60 fs requested window, 21 fs taper on a 2048-sample, 50 as lattice) the effect is large. SEP 26 Phase 2 measured it on fixed saved fields of the L6 rate run (reference 5):

plane	band (90 nm–1.6 µm)	0.9–1.6 µm	0.9–1.2 µm	1.2–1.6 µm	90–200 nm
3.969 m	3.37%	8.40%	3.24%	18.7%	\(1.8\times10^{-5}\)
4.05 m	7.36%	18.3%	7.15%	38.2%	\(1.4\times10^{-5}\)

These are \(\lVert Q_{\rm route}-Q_{\rm held}\rVert_B/\lVert Q_{\rm route}\rVert_B\) in the assembled source units of §4.4, with \(Q_{\rm held}\) built as \(A_J\). The running-integral closure residual was \(3\)–\(10\times10^{-7}\) and the collar’s certificate \(1.8\times10^{-14}\). This was a fixed-field source comparison: no propagation was run.

In scope.

The held source on numpy_ref, numba_cpu and the Kokkos CPU build, on every route that already admits the field-grid plasma (§3), for one or several channels, either running-integral rule and every admitted mix with Kerr and Raman.
Its discrete definition, certificate, rounding budget and work identity; the seam and saved-output changes.
Validation: analytic tests, a fixed-field replay of SEP 26 §4.7, propagated before/after comparisons, backend parity and the effect on existing witnesses and benchmarks.

Out of scope.

Kerr, Raman and dispersive-Kerr gating, which stay as they are (§4.2).
The envelope plasma (EnvelopePlasmaResponse, §4.2) and the TDSE source.
A field plasma on the general modal route, which refuses it today and keeps refusing it.
Collisions, recombination, an explicit drift-current model and THz emission below the active band (§10).
The window and collar defaults (grids) and the field’s absorbing collar (SEP 21).
General GPU performance optimisation. The implementation includes a CUDA build check and bounded HIP device qualification under the maintainer’s revised decision below.

3. Context#

Today’s column (solprop.backends.numpy_ref.passes). plasma_scan forms the exposure and the free current \(J_{\rm free}=(e^2/m_e)\int fE\,dt\). plasma_current adds \(J_{\rm loss}=I_pW(1-f)E/|E|^2\) and returns \(P=\rho\int(J_{\rm free}+J_{\rm loss})\,dt\). plasma_channels_polarisation does the same for several channels, with the weights inside the shared integrals. All three integrals use responses.plasma_integral, the band rule by default or the trapezoid (SEP 18, field plasma).
The collar. nonlinear_polarisation sums the plasma, Kerr and Raman polarisations and multiplies the total by grids.t_window_fine; the dispersive Kerr applies the collar inside, to its products. The total is projected, transformed and cropped, then multiplied by the prefactor \(p\propto-i\omega/(2\varepsilon_0c\,{\rm Re}\,n_{\rm eff})\) (crop scale included) and the spectral collar \(W\). The fine collar is the grid’s temporal Collar, with its zeros at the lattice ends and each taper at least 5% of the realised window. Collar.window_derivative (SEP 26) gives its analytic derivative.
Routes that admit the field plasma.
- Modal and radial, through _over_nodes: synthesis, column, time-domain projection and one transform per modal column.
- Cartesian, through _nonlinear_free: one temporal transform per node, then the transverse FFT. Line free space is the same path with one \(y\) sample, and is covered with it.
- Planar, through _nonlinear_slab: \(n_pn_y\) temporal transforms each way.
- General modal refuses the gas and plasma keywords beside a provider. That is unchanged.
Compiled columns. numba_cpu has a fused single-pass trapezoid column and a multipass band column; solprop-kokkos has a host team column, a tiled device column and trapezoid columns. Each forms the three running integrals and collars the total with Kerr.
PlasmaResponse.evaluate returns the physical, ungated \(P=I_R[J]\) and serves as the physics oracle of the column’s currents.
Prior work. SEP 18 §4.8 noted that the collar adds \(w'P\) to \(\partial_t(wP)\). SEP 26 §4.7 defined the comparison in current variables. Its Phase 3 moved the question to the roadmap item that this SEP answers.

4. Theory#

Transforms use the internal FFT convention \(\tilde f(\omega)=\int f e^{-i\omega t}dt\) (conventions §2), so \(\mathcal F[f']=i\omega\mathcal F[f]\). Write \(q=i\omega\). \(J\) is the total plasma current density, densities and channel weights included. \(t_a\) is the lattice start, where \(J=0\); \(t_\pm\) denote the tapers’ times. \(z\) is reserved for the propagation coordinate.

4.1 What today’s source contains, and what holding removes#

The propagated source is proportional to \(q\,\mathcal F[gP]\). Since \(gP\) vanishes at both lattice ends,

\[ q\,\mathcal F[gP]=\mathcal F[gJ]+\mathcal F[g'P],\qquad \int g'P\,dt=-\int gJ\,dt . \]

The first term switches the current off with the collar. The second is a current pulse, the switch-off of the polarisation. Neither is emitted by an electron. The held source is SEP 26’s form applied to \(J\):

\[ \boxed{\;S_{\rm held}=\frac{\mathcal F[gJ']}{q}=\mathcal F[gJ]-\frac{\mathcal F[g'J]}{q}\;} \qquad(\omega\neq0), \]

with polarisation-equivalent \(S_{\rm held}/q\). The term removed from today’s source, in current units, is \(\mathcal F[g'P]+\mathcal F[g'J]/q\). \(J\) is never differentiated, and \(g'\) is analytic (SEP 26 §4.2).

The continuation. Integrating by parts, with \(J(t_a)=0\), the held current is

\[ J_{\rm h}(t)=\int_{t_a}^{t}gJ'\,ds=g(t)J(t)-\int_{t_a}^{t}g'J\,ds . \]

General case. On the plateau it is \(J(t)-c_-\), with \(c_-=\int_{\rm lead}g'J\,dt\). After the window it is held at \(H=-\int_{\rm lead+trail}g'J\,dt\).
Empty leading taper (\(g'J=0\) there, the usual case): \(J_{\rm h}=J\) on the plateau, and \(H=-\int_{\rm trail}g'J\,dt\), which is \(J_\infty\) when \(J\) is steady through the trailing taper.

The leading taper can be occupied by pre-ionised electrons, or by ionisation in a field that reaches it; admission does not constrain the field there. \(\mathcal F[J_{\rm h}]=S_{\rm held}\) at every \(\omega\neq0\), and \(\int J_{\rm h}\) grows at slope \(H\), the collisionless plasma’s own continuation.

Three cases:

A steady drift: the hold removes both switch-offs. Holding \(P\) instead gives \(\mathcal F[gJ]\), which still switches the current off.
No drift but a residual polarisation (\(J\to0\), \(\int J\,dt\neq0\)): today’s form radiates the polarisation’s switch-off; the held form has no edge.
Variation inside the trailing taper is kept as \(gJ'\). Variation after the window is dropped: that is the window’s finite-memory error.

Which part dominates. On the trailing taper, \(|g'J|/|\omega g'P|\sim1/(\omega\,\Delta t_c)\), with \(\Delta t_c\) the delay from ionisation to the taper: about 0.02 in run_short’s near infrared. SEP 26’s measurement agrees. The polarisation edge alone matches the full route-minus-held figure to \(7\times10^{-6}\) of the band source at 3.969 m.

4.2 Which terms are affected#

The Drude current is held. After the pulse it is the Brunel drift \(J_\infty=(e^2/m_e)\int fE\,dt\) (references 1, 2), generally nonzero.
The ionisation-loss current \(\propto W(|E|)\) is exactly zero below the rate table’s field_min. It is held with \(J\), contributes to \(H\) only where the field reaches a taper, and its in-window variation is kept.
Kerr \(P_K=\varepsilon_0\rho\chi^{(3)}E^3\) is local. Its collar edge is bounded by \(|g'|\max_{\rm taper}|E|^3\), of order \((E_{\rm taper}/E_{\rm peak})^3\approx10^{-9}\) for run_short’s edge field (SEP 18 §4.4). A local response has nothing to hold, and where the field reaches the taper the collar is acting as the absorbing boundary. Kerr keeps \(g\).
Raman \(P_R=\varepsilon_0\rho E\,(R*E^2)\) vanishes with \(E\), and keeps \(g\). The dispersive Kerr keeps its inner collar.
The envelope plasma enters as the gated drive \(gJ'\) times \(-1/\omega^2\), the held structure. With shock=True this is exactly \(\mathcal F[gJ']/q^2\). With shock=False its factors are frozen at \(\omega_0\), the carrier approximation of the same structure. It is unchanged.
TDSE: admit_tdse still refuses a plasma beside a TDSE response.

4.3 Every current exactly once#

\(J_{\rm free}\) and \(J_{\rm loss}\) of every channel enter once, through one shared \(J\).
The held value adds no current: at \(\omega\neq0\) a steady \(H\) contributes nothing.
The removed terms are the collar’s switch-offs.
Kerr and Raman enter once, collared as today. The linear index stays in the operator, and pre-ionised electrons stay in \(J_{\rm free}\).
No coexisting response contains a plasma current (the envelope plasma is envelope-grid only; TDSE is refused).

4.4 The discrete held source#

Lattice, as in SEP 26 §4.4: \(t_n=(n-N/2)\Delta t\), the raw DFT \(x_k=\sum_nx_ne^{-2\pi ikn/N}\), and \(\Delta t\,x_k=(-1)^k\sum_m\mathcal F[x](\omega_k+m\omega_s)\) for smooth \(x\) supported in one period. Let \(X=\mathrm{DFT}[gJ]\) and \(Z=\mathrm{DFT}[g'J]\), nodal and per component, with \(J\) the column’s current from its two remaining running integrals.

Definition. On the active bins \(A\), the plasma’s polarisation spectrum is

\[ \boxed{\;A_J=\frac{X}{q}-\frac{Z}{q^2}\;} \]

and zero elsewhere. The non-plasma part keeps today’s \(V_{\rm nl}=\mathrm{DFT}[g(P_K+P_R)]\), plus the dispersive Kerr. The assembled stage source is

\[ Q=p\,W\,\mathrm{crop}\Big[\textstyle\sum_{\rm nodes}\Pi\,(V_{\rm nl}+A_J)\Big], \]

with \(\Pi\) the route’s projection weights. Every bound and relative figure in §7 is carried through \(\Pi\), the crop, \(p\) and \(W\), and divided by a norm in these same units.

Accuracy. On a resolved collar \(A_J\) is the continuous held form, apart from the sampled products’ alias and rounding. Its third integral is exact division, so the route’s integration rule now governs only the exposure and free-current integrals.
Ordering is free. Projection commutes with the temporal DFT. A route may project \(g(P_K+P_R)\), \(gJ\) and \(g'J\) and transform the modal columns, or transform per node, combine with the weights \(1\), \(1/q\) and \(-1/q^2\), and project in frequency. The cheaper order is chosen per route (§6).
A decomposition check, not a definition. \(A_P=V-Y/q-Z/q^2\), with \(V=\mathrm{DFT}[gP]\) and \(Y=\mathrm{DFT}[g'P]\), subtracts the edge from today’s sampled polarisation. It differs from \(A_J\) by \(C/q\), where \(C=qV-X-Y\) is the running-integral closure residual that SEP 26 measured at \(3\)–\(10\times10^{-7}\).

Drafting probe. The review’s table, against the analytic held step (run_short lattice, \(J=\tfrac H2[1+\operatorname{erf}((t+5\,{\rm fs})/1\,{\rm fs})]\); band norm, relative to the analytic polarisation):

rule	today’s gated	\(A_P\)	in-column (§10)	\(A_J\)
band	2.56	\(7.0\times10^{-8}\)	\(7.0\times10^{-8}\)	\(5.3\times10^{-16}\)
trapezoid	2.56	\(1.6\times10^{-4}\)	\(1.6\times10^{-4}\)	\(5.3\times10^{-16}\)

4.5 The collar certificate#

The steady-current identity on the sampled collar is certified per configuration at translation, as in SEP 26 §4.4, normalised by the full transform:

\[ \kappa_{\rm ss}=\frac{\big\lVert\mathrm{DFT}[g]-\mathrm{DFT}[g']/q\big\rVert_A}{\lVert\mathrm{DFT}[g]\rVert_2}\le10^{-10}. \]

It is recorded in the run metadata. The same normalisation is now applied to the TDSE certificate by a pending bug fix (Q3). Drafting probe:

grid (band, requested window)	\(N\)	taper samples	band-relative ratio	\(\kappa_{\rm ss}\)
90 nm–1.6 µm, 60 fs (run_short)	2048	424	\(1.8\times10^{-14}\)	\(7.9\times10^{-17}\)
90 nm–1.6 µm, 100 fs	2048	102	\(3.7\times10^{-14}\)	\(2.0\times10^{-15}\)
200 nm–2 µm, 40 fs	512	76	\(7.7\times10^{-12}\)	\(3.4\times10^{-13}\)
400 nm–2 µm, 1 ps	8192	1848	\(1.4\times10^{-7}\)	\(5.9\times10^{-17}\)
synthetic tapers of ¼–40 samples, run_short lattice	2048	—	1.0 … \(1.2\times10^{-8}\)	\(9.3\times10^{-3}\) … \(7.9\times10^{-10}\) (refused)

What it certifies. \(\kappa_{\rm ss}\) qualifies that one collar identity: a steady current is held, not switched off. It does not certify aliasing of arbitrary sampled products \(gJ\) or \(g'J\). Those carry the column’s existing sampling assumptions, as \(\mathrm{DFT}[gP]\) does today, and are budgeted per test (§7).

4.6 Rounding budget#

The budget is absolute and operand-based (SEP 25, SEP 26 §4.5). Errors are propagated through the named operators per active bin:

The current. \(\delta J\) is the column’s existing bound on its two running integrals (SEP 25 §4.6), unchanged.
The products.
- \(\delta(gJ)\le|g|\,\delta J+\varepsilon(1+c_g)|gJ|\);
- \(\delta(g'J)\le|g'|\,\delta J+(19+3|\zeta|)\varepsilon|g'J|\) (SEP 26 Phase 1, deviation 2).
Projection with absolute weights adds \(n_{\rm nodes}\varepsilon\) of the absolute projected operands.
The FFT gives \(\sqrt N\lVert\delta x\rVert_2+c_F\varepsilon\log_2N\sqrt N\lVert x\rVert_2\) per bin, with \(c_F=8\) qualified on the supported smooth sizes as SEP 25 requires.
Division. \(1/q\) and \(1/q^2\) scale by \(1/\omega_k\) and \(1/\omega_k^2\), adding \(2\varepsilon\) relative; one subtraction adds \(\varepsilon(|X/q|+|Z/q^2|)\).
Assembly. Crop, \(p\) and \(W\) add a few \(\varepsilon\) relative, and scale everything by \(|p|W\).

This is SEP 26 §4.5’s budget with a second division, applied to a current that carries its own column bound.

4.7 The work identity#

For one real nodal column, the band-limited work is

\[ \mathcal W[S]=\frac{\Delta t}{N}\sum_{k\in A}w_k\,\mathrm{Re}\big(\overline{E_k}\,S_k\big), \qquad w_0=w_{N/2}=1,\ w_k=2\ \text{otherwise}. \]

By Parseval, \(\Delta t\sum_na_nb_n=\frac{\Delta t}{N}\sum_{k=0}^{N/2}w_k\mathrm{Re}(\overline{a_k}b_k)\). Take the held source in current units, \(S=qA_J=X-Z/q\). The periodic antiderivative of \(g'J\) on the plateau is \(c_--m-(n/N)\Sigma\), with \(\Sigma=\Delta t\sum g'J=-H\) and \(m\) its mean. Then, exactly up to rounding,

\[ \mathcal W[qA_J]=\Delta t\sum_nE_nJ_n \;+\;\underbrace{\Delta t\sum_nE_n(g_n-1)J_n}_{\text{field on a taper}} \;-\;\underbrace{\Delta t\sum_nE_n\big(I[g'J]_n-c_-\big)}_{\text{field on the trailing taper}} \;-\;H M_1\;-\;O_{\rm out}, \]

with these terms:

\(M_1=\Delta t\sum_n(n/N)E_n\), the field’s first moment over the period;
\(O_{\rm out}=\frac{\Delta t}{N}\sum_{k\notin A}w_k\mathrm{Re}\big(\overline{E_k}\,\mathrm{DFT}[gJ-\mathcal J[g'J]]_k\big)\), the out-of-band part, where \(\mathcal J\) is the periodic antiderivative that \(1/q\) represents (DC included, so the term \((m-c_-)\Delta t\sum E_n\) is folded in).

\(HM_1\) is the window term of a held current represented on a periodic lattice; it falls as \(1/T\). For a band-limited field confined to the plateau it is small, because \(\sum_n nE_n\) follows \(\partial_\omega\widehat E\) at zero frequency. The taper terms are not small when the field overlaps a taper, and no bound from the field’s out-of-band fraction alone controls them: the review’s counterexample is such a case. When the field does not overlap the tapers, the identity is \(\mathcal W[qA_J]=\Delta t\sum EJ-HM_1-O_{\rm out}\). A drafting probe on run_short (a plateau field and a stepped current) closed it to \(3\times10^{-15}\) of the work; there \(HM_1\) was itself only \(5\times10^{-16}\) of it. SEP 18’s population and loss-work check is separate and unchanged.

5. Proposed interface#

No public option. Every field-grid PlasmaResponse uses the held source. Signatures are unchanged. Translation refuses a configuration with \(\kappa_{\rm ss}>10^{-10}\), naming the collar’s taper samples and the remedy (a wider window).
PlasmaResponse.evaluate keeps its meaning: the physical, ungated \(P=I_R[J]\), the physics oracle of the column’s currents. Its docstring says that propagation assembles the held source (§4.4) from the same current. The assembled-source oracle is a separate private function of the reference backend, used by the tests.
A private comparison hook. numpy_ref’s pass takes a private switch that reproduces today’s \(\mathrm{DFT}[gP]\), for validation only.
- A run using it carries a private benchmark label in its metadata and in its hashed configuration, so it can never share a resume key with a production run.
- Its output refuses to resume, and is written only by the benchmark scripts, to their own directories.
- It is not public API, and it is not implemented in the compiled backends.
Run metadata. metadata()["plasma_source"] records \(\kappa_{\rm ss}\), its limit and the collar’s taper samples, computed once at translation.
No in-run edge diagnostic. Edge fractions and \(\rho_-=\max_{\rm lead}|J|/\max|J|\) are an offline analysis of saved fields (benchmarks/sep26/rate_plasma.py, generalised).

output = prop_capillary(..., plasma=True)   # unchanged call; the source is now held

6. Implementation strategies#

Recommended: \(A_J\) in frequency.

Column. The column outputs the plasma’s \(gJ\) and \(g'J\) beside \(g(P_K+P_R)\), instead of a collared total, and drops its third running integral.
Transforms. The route transforms the three arrays and combines them with the weights \(1\), \(1/q\) and \(-1/q^2\), or projects first, whichever is cheaper.
Cost per stage evaluation, for \(n_c\) field components:

route	today	with \(A_J\)	per-node column
single-mode modal	1 projection, \(n_c\) modal transforms	3 projections, \(3n_c\) modal transforms (projections negligible)	one scan fewer
radial (\(n_m=n_{\rm nodes}\))	1 projection GEMM, \(n_mn_c\) transforms	per-node transforms (\(3n_{\rm nodes}n_c\)), combined, then one frequency-domain projection on the cropped bins; projecting three arrays would triple the dominant GEMM and is not used	one scan fewer
Cartesian and line	\(n_{\rm nodes}n_c\) per-node transforms	\(3n_{\rm nodes}n_c\) per-node transforms, combined before the transverse FFT	one scan fewer
planar	\(n_pn_yn_c\) row transforms	\(3n_pn_yn_c\), combined before \(k_y\)	one scan fewer

The net change is \(+2\) real transforms and \(-1\) filter-and-scan per node and component. Indications:

In numpy_ref on run_short’s lattice, one band-rule running integral of 33 rows costs about four real FFTs of the same rows (440 against 104 µs), so the change is a net saving there.
In compiled code a scan costs about 10–15 flops per sample, and a real transform about \(1.25\log_2N\approx14\) flops per sample at \(N=2048\), so the estimate runs from roughly neutral (modal) to about \(+20\%\) of a nonlinear evaluation (Cartesian, where the per-node transforms already weigh most beside the column).

The option kept for later, B, adds a scan instead: an estimated 10–25% of the plasma column. Phase 1 measures both.

Memory. Two more nodal rows per concurrent node, and two more modal or spectral arrays where a route projects first. Both are in memory_ledger().

Kept as a later optimisation (B, the first draft’s in-column form). The held polarisation can be built in time with the route’s running integral, closed periodically. The review showed what it needs:

periodic, not point-reflected, filters for the new integrals (R1);
an additive edge residual \(R(1-R)Z/q^2\) (R3);
support recomputed after closure (R10);
a longer rounding recurrence (R7).

It is admitted later only if it matches \(A_J\) within a derived budget on every supported grid. The single-extra-transform variant \(V-\mathrm{DFT}[g''P+2g'J]/q^2\) is an \(A_P\)-type form carrying \(C/q\), and is likewise not primary.

Layers.

Physics: the PlasmaResponse docstring.
Translation: the fine collar derivative and the certificate.
Backends: the column outputs and the route assembly.
Driver: the metadata, the benchmark label and the resume refusal.

Seam and saved output.

grids gains t_window_fine_derivative, and the column’s output contract changes, so the kernel SCHEMA_VERSION takes its next free number.
The output SCHEMA_VERSION also takes its next number. RunMetadata.configuration_hash digests it, so every file written before this SEP has a different resume key and its resume is refused. That is the concrete rejection boundary for old production files.
A concurrent cleanup moves the output schema to 21 and the TDSE descriptors to v5. The implementer takes the next free numbers at implementation, and tests the refusal.

Backends.

Consistency. Phases 1–3 are reviewed separately and integrated into main together, so the backends never disagree.
numpy_ref (Phase 1): plasma_current and plasma_channels_polarisation return \(gJ\) and \(g'J\); each route assembles \(A_J\).
numba_cpu (Phase 2): both columns write the two rows and drop the third integral. The Raman finish collars only the non-plasma part.
solprop-kokkos (Phase 3): the same in the host team, tiled device and trapezoid columns, plus the derivative table on the device and the route assembly. Device builds compile the same code; their qualification follows the standing policy.

Interaction with the stepper.

Each stage evaluation is a pure function of the stage field, as today. No state crosses stages, rejections or restarts, and no evaluation is added.
No allocation per stage.
The error norm sees changed stage derivatives. Step counts are measured, not gated (§7.3).

Dependencies. None.

7. Verification and validation#

Tolerances are derived before evaluation and stated beside each assertion.

Exact equality holds only for masks, accounting, lossless copies and the structural zero.
“Gated” means the private hook’s output.
Matched comparisons use §4.6 plus the named discretisation terms.
Relative figures are in §4.4’s assembled units, divided by a norm in the same units.

7.1 Analytic and unit tests (default suite, fast)#

Each runs on a given \(J\) through the assembled-source oracle, then through the column from a field.

T1 Steady current. \(J\equiv H\): per bin \(|A_J|\le|H|\,|\mathrm{DFT}[g]-\mathrm{DFT}[g']/q|_k/\omega_k+\) budget, with the band norm \(\le\kappa_{\rm ss}\lVert\mathrm{DFT}[g]\rVert_2|H|/\omega_{\rm lo}+\) budget. The gated source is reported as nonzero.
T2 Post-pulse drift. The erf rise of §4.4, then steady. The oracle is \((-1)^kHe^{-\omega_k^2\sigma^2/4}e^{-i\omega_kt_c}/(\Delta t\,q_k^2)\). Each per-bin term is additive and absolute, so bins with negligible physical signal are covered:
- the rise’s mass outside the plateau, \(|H|\operatorname{erfc}(d/\sigma)/(\Delta t\,\omega_k^2)\);
- the steady tail’s collar residual, \(|H|\,|\mathrm{DFT}[g]-\mathrm{DFT}[g']/q|_k/\omega_k\);
- the product aliases \(|H|e^{-(\omega_N\sigma)^2/4}/(\Delta t\,\omega_k^2)\);
- the budget. Both rules, broad and narrow rises, asymmetric placement. A band floor may be used for relative reporting only.
T3 Zero. \(J\equiv0\) gives exactly zero, held and gated.
T4 A known ionisation history (free current only). The fraction is prescribed, \(f=f_0+\tfrac{\Delta f}{2}[1+\operatorname{erf}((t-t_i)/\sigma_i)]\), with \(E=-\partial_tA\) and \(A=A_0e^{-t^2/\tau^2}\sin(\omega_0t+\varphi)\). Then \(J_\infty=+(e^2/m_e)\Delta f\,(A*\phi_{\sigma_i})(t_i)\), which the review checked by quadrature. The loss current is set to zero, since the held source is linear in \(J\); the loss current is covered by T1, T6 and the column runs.
- (a) The removed term equals the analytic switch-off of \(J_\infty\) and of the residual polarisation on the trailing taper, within the budget and T2’s terms.
- (b) At a zero of \(A*\phi_{\sigma_i}\): \(J_\infty=0\) but \(\int J\,dt\neq0\). The held source has no edge, and the gated one has the polarisation’s switch-off.
- (c) Across \(\varphi\), the removed term is the sum of the two parts, each linear in its own amplitude (\(J_\infty\) and the residual polarisation).
T5 Kerr and Raman unaffected.
- Held minus gated is independent of \(\chi^{(3)}\) and the Raman amplitude, within the budget.
- Held and gated agree within the budget only when both \(J\) and \(\int J\,dt\) vanish before the trailing taper.
- A compact current with \(\int J\,dt\neq0\) shows the residual-polarisation edge removed.
T6 Decomposition check. On real column currents (optmat He, a few-cycle ionising pulse; one, two and three components where admitted; one and two channels; pre-ionisation): \(\lVert A_P-A_J-C/q\rVert\le\) budget. \(\lVert C/q\rVert\) is reported.
T7 Certificate. Collars of ¼–40 samples are refused; the §4.5 grids are admitted; the recorded \(\kappa_{\rm ss}\) agrees with an independent recomputation within \(10\varepsilon\log_2N\) absolute.
T8 Work (§4.7).
- Gated: with a field confined to the plateau, \(|\mathcal W[qA_J]-\Delta t\sum EJ+HM_1+O_{\rm out}|\le\) budget.
- Reported, not gated: one case with a field on the trailing taper, showing both taper terms.
T9 Leading taper. With pre-ionisation and a field on the leading taper, the plateau offset \(c_-\) and the final value \(H=-\int_{\rm lead+trail}g'J\,dt\) are recovered from the held source, against §4.1, within the budget. \(\rho_-\) is recorded.
Seam and output.
- The previous kernel schema’s consumer refuses the problem.
- A file at the previous output schema refuses to resume.
- A benchmark-labelled gated output refuses to resume, and never shares a production resume key.
- The memory ledger lists the new rows.
Hot-path benchmark (benchmark lane, bounded). One nonlinear evaluation on run_short’s lattice, a radial and a Cartesian case: \(A_J\), gated and B’s prototype, with warmup, interleaved repeats and a distribution. A ledger entry, with the hardware in the ledger.

7.2 Fixed-field replay (offline, minutes)#

rate_plasma.py gains the production source. It reads the same hashed L6 planes with the same rule, nodes, density and crop.

(i) The production \(Q\) against SEP 26’s \(Q_{\rm held}\). That is the same \(A_J\) computation through different code, so the bound is the §4.6 budget in assembled units.
(ii) The route-minus-held figures of §2, including \(1.77\times10^{-5}\) at 90–200 nm, are reproduced within the same budget relative to \(\lVert Q_{\rm route}\rVert_B\).
(iii) The gated hook reproduces \(Q_{\rm route}\) within the column budget.
\(\lVert C/q\rVert_B\) is reported per plane.

7.3 Propagated before/after comparisons (CPU execution; no GPU)#

Full-length exception. The run_short rate parent runs on numpy_ref in about a core-minute. Maintainer decision (John Travers, 2026-10-19): the full-length 0–4.05 m campaign below is an explicit exception to SEP 0 §7 and AGENTS.md’s qualification limit, because these runs are fast. Backend qualification and hot-path timings stay bounded (§7.1, §7.4).

Runs (both forms; gated through a benchmark-script flag):

(a) The run_short rate case, 0–4.05 m, 51 planes, on an rtol ladder from \(10^{-2}\), tightened by factors of about 3 until the residual bound below is resolved.
(b) The SEP 20 L6 rate segment, 3.969–4.05 m, launched from the same parent plane, and from a held parent’s plane.
(c) A window control: (a) at the converged level with a 120 fs requested window.

Every run records atol, the maximum step and the save-plane policy: fixed common planes, identical for both forms.

Convergence. For each form, \(d_j\) is the difference between successive levels on the metric below. The tightest level’s residual is bounded from the last three levels, by \(d_{\rm last}\,r/(1-r)\) with \(r=d_{\rm last}/d_{\rm prev}\), if \(r<1\) and the three are consistent; otherwise tighten further. No order is assumed from the rtol ratios. A form difference below the sum of the two forms’ residual bounds is unresolved.

Metric. At common planes, \(\lVert a-b\rVert_B/\lVert b\rVert_B\) on complex spectra (\(b\) gated) over the full band, 90–200 nm, 0.9–1.2, 1.2–1.6 and 0.9–1.6 µm. Also energy, peak power, 90–200 nm energy, steps accepted and rejected, evaluations and wall time.

Matched-input controls. At every tenth saved plane, both sources are evaluated on the same saved field (§7.2). This separates the direct change from the propagated one.

Window control. Both grids’ spectra are converted to the continuous-transform normalisation (\(\Delta t\) included, the \((-1)^k\) origin removed, the interface amplitude convention). Band energies are compared, with complex spectra only on coinciding bins. The launch is rebuilt from the same pulse definition on each grid.

Investigation threshold. A 90–200 nm difference above 0.5% at the converged level is investigated with the matched-input controls and the ladder, not failed.

Hypotheses, not pass conditions. No sign, size or attenuation is a pass condition.

The 0.9–1.6 µm spectra change at the percent level at the late planes.
The UV changes little, although the source record’s UV difference is nonzero and propagation can amplify it.
The held form’s window sensitivity is lower than the gated form’s.

7.4 Backend parity#

numba_cpu and Kokkos CPU are held to numpy_ref, with SEP 25’s budgets and §4.6:

one evaluation, one step, eight steps, the statistics and a ten-step run;
both rules, every component count each route admits, one and two channels, pre-ionisation;
plasma alone, with Kerr, with Raman, and with both, so that the changed collar placement is exercised;
on the modal, radial, Cartesian, line and planar routes;
the general modal refusal unchanged.

7.5 Existing witnesses, benchmarks and records#

Every field-grid plasma result changes. No test is weakened.

Luna plasma witnesses compare against the gated hook, like with like, at their existing tolerances. The held form is reported against them as the deliberate difference.
Other stored plasma witnesses are regenerated from the held form, with provenance.
The SEP 18 qualification is replayed. Its third-integral comparison becomes exact division and is restated.
Benchmark cases with a plasma are re-measured, with new ledger entries.
The TDSE campaigns’ rate parents and baselines remain historical evidence. Phase 4 reports how the L6 gates move against a held rate baseline.
Galleries. Tutorials re-run on every build. Worked examples with a field-grid plasma are re-run before Final.

8. Documentation#

Field plasma: the identity, the continuation, \(A_J\), the certificate, the work identity, evaluate’s meaning, and a tested worked comparison against the truncation.
Grids: what the collar does to a memory current.
Differences from Luna: the held form.
Kernel contract: the schema row and the column’s outputs.
Output: the output schema bump.
Docstrings, the changelog, the roadmap and benchmarks/sep27/.

9. Compatibility and impact#

No backward compatibility (maintainer’s decision, 2026-10-19). solprop is unreleased, so the held source simply replaces the truncation as the only behaviour, with no public option, no dual path and no resume of old files. The output and kernel schema bumps make old files refuse to resume (§6).
What changes. Every field-grid plasma result; scripts are unchanged.
The maintainer’s existing results are evidence of the truncating form. §7.3’s hypotheses are not predictions.
Luna. The held form is a deliberate difference.

10. Alternatives considered#

B, in the column (§6): no extra transform, but boundary, additive-edge, support and rounding complications. Kept as a later optimisation, admitted only against \(A_J\).
\(A_P\), subtraction from today’s polarisation: carries \(C/q\). Kept as a check.
A wider collar. The in-band edge falls super-algebraically with the taper width. On a unit step current in 0.9–1.6 µm it is 1.36 at 21 fs, \(7.7\times10^{-2}\) at 40 fs, \(2.2\times10^{-3}\) at 80 fs and \(1.2\times10^{-5}\) at 160 fs, for 4–8 times the lattice. It changes every grid and leaves the edge in principle.
Subtracting the steady current explicitly needs an arbitrary turn-on. For a collisionless model it gives nothing the hold does not. Reconsider with collisions (SEP 5).
Holding the polarisation still switches the current off.
Leaving it, documented: percent-level near-infrared source errors and a window dependence in the most-used routes.
A public option is not needed for unreleased software.

11. Phases#

Phase 1: numpy_ref and the seam (feature/sep-0027-phase-1).
- The column outputs, the \(A_J\) assembly on every admitted route, the certificate and metadata.
- The private hook and its label, and both schema bumps.
- T1–T9, the seam and output tests, the hot-path benchmark, the manual and the changelog.
Phase 2: numba_cpu (stacked). Both columns and every route; §7.4 parity; a ledger entry.
Phase 3: Kokkos CPU (solprop-kokkos branch of the same name). The columns and assembly; §7.4 parity; the Kokkos lanes; a CUDA/HIP compile-only check where available; a ledger entry. Phases 1–3 integrate into main together.
Phase 4: validation, which may start once Phase 1 is reviewed.
- The benchmark-script flag for the gated runs.
- §7.2, then §7.3 on CPU execution, under 20 core-hours, and §7.5.
Phase 5: the final gate and Final. The worked examples and the Luna page; the full gate, the slow, coverage and Kokkos CPU lanes.

Blocking conditions.

Acceptance: responses to the reviews.
Integration of Phases 1–3: T1–T9 and §7.4 pass on all three backends.
Final: §7.2 passes, every §7.3 investigation is closed, every phase is reported, the lanes pass, and the manual is updated. A finding that the held form is wrong returns the SEP to Draft.

Cost. Phase 1 is about SEP 26 Phase 1’s size. Phases 2 and 3 are each a column-output change plus route assembly. Compute is minutes for §7.2 and under 20 CPU core-hours for §7.3.

Risks and integration responsibility.

SEP 24 Phase 3 and SEP 17 touch the same column files. The SEP’s implementer integrates with any concurrent SEP at merge, rebasing onto whichever lands first and re-running the affected parity tests. SEP 17’s continuum electrons, if they enter \(J\), are held with the rest.
Every field-grid plasma result changes at once.
The per-node routes pay two extra transforms (§6). If Phase 1’s benchmark shows this unacceptable, B is pursued under its admission condition.

12. Open questions#

Resolved by the maintainer on 2026-10-19: Q1 integrate Phases 1–3 together; Q4 every admitted field-grid route; Q5 re-run the field-grid worked examples before Final; Q6 originally required compile-only device coverage. On acceptance, the maintainer revised Q6: verify the A40 CUDA build, then run one HIP qualification campaign on an AMD GPU, with a USD 20 rental cap. CPU qualification and physics validation precede the rental.

Q2 Implementation — resolved 2026-10-19: \(A_J\) (the maintainer confirmed the changed recommendation, and that the integration rule then governs only the exposure and free-current integrals, §4.4). Background: The maintainer accepted B, in the column. The review showed B’s discrete identity is conditional on a boundary extension the admitted grids do not all satisfy (R1, \(2.6\times10^{-9}\) on a 512-point grid), and that it needs an additive edge term, support recomputation and a longer rounding recurrence (R3, R7, R10). \(A_J\) is exact to \(5\times10^{-16}\) under both rules (R2). Its cost is two extra real transforms per node, set against one scan saved: a net saving in numpy_ref, and an estimated range from neutral (modal) to about \(+20\%\) of a nonlinear evaluation (Cartesian) in compiled code, to be measured in Phase 1. Recommendation: \(A_J\) as the definition and implementation; B only as a later, budget-admitted optimisation.
Q3 Certificate normalisation. Resolved: the full transform. The earlier recommendation of a roadmap item for the TDSE certificate is superseded by the bug fix itself (cleanup branch refactor/remove-compat-paths, commit f6b76751, pending merge).

13. References#

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
F. Brunel, “Harmonic generation due to plasma effects in a gas undergoing multiphoton ionization in the high-intensity limit,” J. Opt. Soc. Am. B 7, 521–526 (1990). doi:10.1364/JOSAB.7.000521
D. J. A. McKechan, C. Robinson and B. S. Sathyaprakash, “A tapering window for time-domain templates and simulated signals in the detection of gravitational waves from coalescing compact binaries,” Class. Quantum Grav. 27, 084020 (2010), eq. (7). doi:10.1088/0264-9381/27/8/084020
N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM (2002), ch. 24. doi:10.1137/1.9780898718027
SEP 26 Phase 2’s fixed-field rate-plasma record, benchmarks/sep26/evidence/rate-plasma.json in the solprop repository, written by benchmarks/sep26/rate_plasma.py, with input hashes from benchmarks/sep20/evidence/manifest.json (phase4.l6_runs).

Part II — Implementation record#

Acceptance and execution agreement — 2026-10-19#

John Travers accepted the revised direct-current formulation and authorised all five phases. Part I is frozen with the revised Q6 device qualification above. SEP 28 implementation waits for this physics change and its measured baseline. Phases 1–3 remain an integration unit; subsequent upstream changes are integrated by ordinary merges, without rewriting branch history.

Approved T2 correction — 2026-10-19#

John Travers approved replacing T2’s steady-tail allowance by independent sampling errors for its two products. The equation \(A_J\) and the \(10^{-10}\) translation certificate are unchanged. Part I’s original T2 bound is superseded by this correction.

Counterexample. On the admitted 90 nm–1.6 µm, 60 fs smooth grid (\(N=1200\), \(\kappa_{\rm ss}=5.3291\times10^{-12}\)), the erf rise at 3 fs with width 2 fs exceeds the original allowance by a factor of 33 at one bin. Its largest absolute source discrepancy is \(1.6435\times10^{-24}\) in the raw polarisation-equivalent units of a unit current. Sampling the same continuous products 8, 16 and 32 times more finely reduces that discrepancy to \(2.98\), \(2.70\) and \(2.70\times10^{-29}\). The admitted 512-sample power-of-two grid also violates the old allowance. The 360-sample smooth 200 nm–2 µm grid fails admission (\(1.312\times10^{-6}\)) and is now an explicit refusal case, not an analytic-source qualification case.

Reason. The two edges’ discretisation errors can cancel in \(\operatorname{DFT}[g]-\operatorname{DFT}[g']/q\). A current occupying only the trailing edge has no corresponding leading-edge error to cancel. The global steady-current certificate therefore cannot bound that per-bin product error.

Replacement. Let \(X_c\) and \(Z_c\) be the continuous transforms of \(gJ\) and \(g'J\), in the same raw-DFT scaling and phase origin as \(X\) and \(Z\). Compute them independently by adaptive quadrature, splitting at the analytic collar boundaries and the rise. With absolute quadrature errors \(e_X,e_Z\), use

\[ \frac{|X-X_c|+e_X}{|\omega|} +\frac{|Z-Z_c|+e_Z}{\omega^2} +\text{analytic rise-tail allowance}+\text{rounding budget}. \]

The two sampling terms cannot cancel. They are computed before the production assembly, without inspecting its result; weak physical bins get the same absolute error accounting. tests/backends/test_held_plasma.py implements the independent quadrature and retains the original erf oracle and operand-based rounding terms. The quadrature reports a numerical error estimate, not an interval-arithmetic proof. The initial 36 analytic/certificate/identity checks pass; these are not the complete phase or propagated-physics qualification.

Phase 1 implementation checkpoint — 2026-10-19#

Delivered. The NumPy source separates collared non-plasma polarisation from current, projects all three streams on modal and planar routes, and combines nodal spectra before the single radial projection or transverse FFT. Both running integral rules stop at current in production. The physical evaluate oracle and its old pass helpers retain ungated polarisation for independent physical tests. This is a deliberate implementation detail differing from §6’s proposed helper renaming: nonlinear_sources owns the new production result instead.

Kernel schema 33 carries the analytic collar derivative; output schema 22 changes the equation identity. The certificate uses the full fine transform, including DC, and caches only scalar evidence with a weak reference to its immutable grid. Its measured residual is execution evidence, outside the configuration hash; source form, limit and taper counts are hashed. The private gated context is NumPy-only, labels its configuration and execution, and refuses continuation. The unported Numba backend refuses field plasma by name in this checkpoint. Kokkos’s old schema is incompatible. Phases 1–3 remain an integration unit.

Review findings addressed. The one-channel convenience property is absent on mixtures, so admission, source allocation, assembly and metadata now test the channel tuple. A full mixture-source regression checks this independently of the Kerr-separation test. Existing spatial test fixtures with a 150 fs request had only 30–31 taper samples and residual 9.763e-9; their window is now 200 fs, and the old fixture remains an explicit admission-refusal test. The threshold is unchanged.

Verification so far. Short checks cover analytic drift and zero currents, independent product-sampling bounds, prescribed ionisation and residual polarisation, work and decomposition identities, occupied leading tapers, every NumPy geometry and both rules, scratch reuse, Kerr/Raman separation, one/two channels and components, and pre-ionisation. The B comparison prototype is checked against its periodic multiplier including the additive edge residual. Focused source type-checking reports zero errors; lint and formatting pass. The complete focused slice, full-project type check and exclusive timing comparison are queued on the CPU qualification machine; this checkpoint is not a completed phase report.

Fixed-field replay. benchmarks/sep27/rate_replay.py verifies the three original L6 input hashes, then calls production assembly and the private gated hook. Both saved planes reproduce SEP 26’s offline figures. At 3.969 m, the full-band norm ratio is 0.03373296046656 and the 90–200 nm ratio is 1.770797114535e-5. At 4.05 m, they are 0.07361588320044 and 1.447237841658e-5. These are matched-input source ratios, not propagated output differences or convergence results. The unchanged SEP 26 script and historical TDSE evidence remain intact.

Sources and limitations. The identities were re-derived from the accepted SEP and tested against analytic Gaussian/erf currents, adaptive quadrature and the original fixed-field calculation. No new physical constant or material coefficient was transcribed; the existing optmat rate tables supply the real-column tests. The cited papers were not independently re-read for this checkpoint. Quadrature error estimates are numerical estimates, not interval proofs. Propagated tolerance ladders, full backend ladders, performance conclusions, examples and the final gates remain outstanding. No rental has been provisioned or charged.

Phase 2 implementation checkpoint — 2026-10-19#

Delivered. Both compiled columns now stop at the physical current. The band column uses five histories instead of six and omits its outer filter and scan. The fused trapezoid column writes every current sample separately from the collared Kerr/Raman polarisation. HeldWorkspace owns both products, projected streams, FFTW plans, embedded quadrature streams and frequency assembly. Modal and planar routes project before their temporal transforms; radial routes transform, combine and crop per node before one frequency projection; Cartesian and line routes assemble before the transverse transform. All persistent arrays appear in memory_ledger(), and node admission charges mixed-response histories together. No production equation or public call signature changed in this phase. The old rejected scan/tile prototypes retain their physical-polarisation oracle in test support, independently of the changed production column.

Approved rate-bound correction. John Travers approved replacing the old fixed 16-epsilon rate allowance after the first-sample current exposed its missing conditioning term. A stress input described as below threshold actually has a tiny nonzero rate. Its first loss current differed by approximately 7e-14 relative between the array and compiled evaluations on the CPU qualification machine; its historical integrated polarisation was exactly zero at that sample. The input and production rate implementation are retained unchanged.

The revised allowance is derived from the operands before evaluating the column. For the real field norm, gamma(8) bounds relative rounding. Assuming one-epsilon log and exp, the logarithm’s absolute error is at most gamma(8) + eps*abs(log(E)). Forming the local coordinate adds gamma(3) times the sum of its logarithm, lower knot and interval-offset magnitudes. With r = abs(local) + local_error, the cubic’s error is bounded by its derivative magnitude at r times local_error, plus gamma(6) times its absolute Horner polynomial. expm1 maps this to relative rate error. If interval selection can straddle a knot, both stored polynomials and their rounding bounds are retained; no exact continuity of their rounded coefficients is assumed. The stress set stays away from the table’s discontinuous lower clip. These are explicit libm accuracy assumptions, not vendor proofs. The current-chain test carries this rate allowance through exposure, population, free current, loss and density.

Verification so far. The first focused compiled slice passed 265 tests. The expanded slice passed 551 tests, including every admitted geometry under both rules, complete masked rows, partial chunks, scratch reuse, ledger agreement, NumPy analytic tests, spatial ladders and the historical prototypes. Its four remaining failures were the rate-bound cases above; their corrected rerun and broader propagation/statistics/low-storage ladders are pending. Source lint and focused type checks pass. The two remaining under-resolved spatial fixtures and the synthetic mixture fixture now use resolved windows; their former settings remain covered by certificate refusal. The admission limit remains 1e-10.

Limitations and handoff. This is an implementation checkpoint, not completed backend qualification. Exclusive alternating source timings, the extended ladders, Kokkos implementation, full physics campaign and final gates remain outstanding. The unchanged baseline has been captured before candidate measurements. No GPU rental has started. Phase 3 receives the current-stream contract at kernel schema 33 and retains the established envelope-plasma path.

Phase 4 preparation — 2026-10-19#

The production-assembly replay has passed on the CPU qualification machine; benchmarks/sep27/evidence/rate-replay.json retains input hashes, source certificate, both saved-plane comparisons and sanitized source provenance. It independently reproduces the earlier offline ratios within the operand-based rounding budgets. This is fixed-field evidence, not a propagation claim.

physics.py launches fresh labeled runs, saves common-plane continuous-normalized spectra, energy, peak power, UV energy, work counts and matched-input source comparisons. A short run exercises both source forms. campaign.py holds an exclusive campaign lock, charges actual child CPU time including failed workers, and applies a hard CPU limit before each child starts. An interrupted unaccounted worker prevents further launches. The shared allowance is 72,000 CPU seconds; exhaustion is recorded as unresolved. No full-length physics run had started at this checkpoint. convergence.py retains three-level differences and their observed contraction; noncontracting sequences remain explicitly unresolved. The accounting and residual-formula checks pass without launching a full run.

Historical witnesses and phase-2 follow-up — 2026-10-19#

The historical Luna plasma comparisons now construct and run through the private NumPy gated hook, keeping their existing spectral bounds and stored arrays. Non-plasma comparisons and production backend parity retain their original routes. The SEP 12/20 parent reconstructions explicitly label the historical equation; their private identity adapter refuses a held-current output. This adapter does not permit any output to resume under another schema or equation.

A frozen Luna witness supplies its own historical collar. Its derivative does not satisfy the held-current certificate (residual 6.703e-5), so the private gated hook records only its equation and non-resumable identity. It makes no claim to the held-current certificate. Production admission remains unchanged at 1e-10, and tests assert refusal both before and after leaving the private context.

The corrected rate bound passed its original stress cases. The expanded Numba slice passed 236 tests and found one further under-resolved low-storage fixture; its field-plasma window is now 200 fs. The follow-up collar, low-storage and Luna slice passed all 60 tests. The combined documentation, historical reconstruction, benchmark-controller and kernel slice passed 626 tests with one expected process-SEP skip. A full-source type check found one NumPy scalar needing an explicit conversion to float in a test; the focused corrected type check passed. These are slice checks, not the final full gate, slow lane or coverage qualification.

Phase 3 implementation checkpoint — 2026-10-19#

Delivered. Kokkos kernel schema 33 carries the analytic collar derivative and rejects incompatible source contracts before allocation. The host band column, tiled device column and trapezoid column write physical current separately from the collared non-plasma response. Persistent real streams and transform plans form the two current products. Modal routes project before transforming; planar routes contract each complete transverse row first; Cartesian and line routes combine before the transverse transform. Radial routes transform the three nodal streams, combine and crop, then use one real matrix product on interleaved complex values. Every inactive bin and every padded final-chunk row is written explicitly.

Review and accounting. The obsolete preloaded-polarisation template path was removed: no compiled production path retains the outer plasma integral. Native and Python ledgers agree for all new buffers, including embedded projections, radial packing and the native owning copy of the derivative. Translation now charges the tiled component histories and reserves complete planar rows. Charging planar projected streams per physical node would incorrectly reject an otherwise admissible default-budget problem; they are charged per projected modal column. Regression tests cover both the minimum-row refusal and rounded chunk admission.

The approved rate bound also revealed that rounding a clipped zero upward to the smallest subnormal invented loss current at zero field. The bound now preserves the rate table’s exact lower clip; production rate evaluation is unchanged.

Verification so far. The native debug suite passes with an independent eight-sample pre-ionised-current/DFT oracle, complete-row reuse and malformed derivative/active-frequency refusal cases. Strict native-adapter typing reports zero errors. The source/ledger slice passed 116 tests; the native Python response slice passed 107 tests. A further 82-test source, planar and low-storage slice passed 76 and refused six old 150 fs planar fixtures at the unchanged certificate limit. Those fixtures now request 200 fs and their rerun is pending. The final native gate, complete CPU lanes, timing campaign and CUDA/HIP qualification remain outstanding. No device support claim is inferred from the host build.

Sources and handoff. The native oracle re-derives the sampled current from a cosine field and discrete trapezoid, then evaluates both DFT terms independently. No physical constants or literature coefficients were transcribed. Phases 1–3 remain one integration unit. The unchanged baseline and candidate use identical pinned numerical dependencies for exclusive timing. SEP 28 remains deferred.

Phase 4 physics results and qualification follow-up — 2026-10-19#

The full-length campaign completed 120 fresh workers, charging 2,510.49 CPU seconds (0.6974 core-hours) including controller and worker overhead, well within the 20 core-hour allowance. The immutable run records, actual work counts, matched-input comparisons, input/output hashes and complete solver source manifest are collected in physics.json. The collection verifies every record against its ledger hash. Its own postprocessing used a further 2.91 CPU seconds. No historical TDSE output was rewritten.

All five complex-spectrum bands have bounded last-three-level residuals on every saved plane, for both parents and both L6 input families. The 60 fs and 120 fs parents finish at rtol=7.7435e-11; L6 finishes at rtol=5.6450e-8. At 4.05 m the held-minus-gated UV energy change is -0.1596090% (60 fs), -0.0672139% (120 fs), -0.0008128% (historical L6 parent), and -0.0008209% (held L6 parent). Their summed relative residual estimates are respectively 1.91e-10, 3.50e-10, 6.91e-8 and 7.79e-8. These bounds estimate truncation residuals through the observed geometric contraction; they are not rigorous bounds on every physical or sampling error.

The endpoint full-band complex-spectrum changes are 0.23565% and 0.01852% for the two parent windows. On the same 60 fs exit field, the matched-input plasma source changes by about 7.31% over the full band but only 4.47e-6 relative in the UV. Source differences and propagated differences are distinct measurements, and neither an expected sign nor an expected magnitude was used to accept a result.

Window investigation. The 120 fs control changes the native UV band-energy sum by -1.10997% for the gated source and -1.01846% for the held source. An exact decomposition retains both sums: approximately -0.8843 and -0.8854 percentage points respectively come from the changed frequency quadrature nodes and edge cells. The changes evaluated on the common coarse lattice are -0.22566% and -0.13309%. This explains most of the above-0.5% discrepancy without changing the energy definition, interpolating spectra or applying a fitted correction. The remaining common-bin change is a genuine window sensitivity; the finite-window model is not claimed to be window independent.

Precision review. At a few intermediate parent planes, scalar differences at roughly 1e-12 relative no longer contract independently. Every such raw sequence remains marked unbounded in the report; endpoint observables and complex-spectrum ladders are resolved. John Travers questioned the unnecessary precision at this point. Further tightening was stopped because it adds no useful discrimination to the reported physics changes. A proposed conversion of spectral residuals into scalar Cauchy–Schwarz bounds was not adopted; this record does not silently replace those scalar sequences or claim that all scalar residuals are bounded.

Backend follow-up. The corrected planar fixtures pass the complete 82-test source, planar and low-storage slice. The native Python response slice passes 107 tests, and forcing four band tiles on the CPU passes the 17 current-source and ledger tests. The interpreted-kernel coverage pass passes 522 tests. The full default, compiled coverage and slow lanes are still in progress and have exposed further failures. Native static analysis found nine redundant host view copies in the new projection path; these were replaced by aliases, with the device lambdas retaining value capture. Its affected unit and the remaining native gate are being rerun. CUDA/HIP verification and the exclusive compiled timing campaign remain pending; the SEP is not Final.

Qualification review — 2026-10-19#

The full default lane exposed old schema expectations, historical witness fixtures that had not entered the private gated context, and plasma fixtures whose temporal collars fail the new certificate. These fixtures now use admitted collars; the certificate limit and the numerical parity bounds are unchanged. The memory admission helper preserves the original transform when its chunk is unchanged, restoring mixture-update sharing. Explicit tests cover historical memory accounting, the driver resume refusal, and partial-chunk accumulation of embedded current streams.

The embedded-current regression retains an ionising analytic-rate input whose trailing current is small after cancellation. Its first assertion incorrectly scaled the column error by that tail. Applying the already accepted §4.6 recurrence instead carries the unchanged full-current allowance through the collar products and absolute projection weights. The test also observes the current before those operations and checks its allowance independently. The diagnostic measured current errors below 0.004 of that allowance and projected errors below 0.00014 of the propagated allowance; no production equation or tolerance coefficient changed.

The existing circular-plasma loss-work check already meets its independently derived analytic bound at each of its three temporal resolutions. Removing the third running integral makes its old ratio of absolute work errors unsuitable for establishing the trapezoid order: the absolute error also includes fixed-band projection. On review, the maintainer approved removing only this redundant order assertion. The analytic work checks, their three grids and bounds, SEP 18 population checks, and independent running-integral convergence tests remain unchanged. No replacement precision ladder or relaxed work tolerance is introduced.

The two failed slow-lane records are solprop’s own M2 order and quadrature-alarm studies, not comparisons to Luna’s stored fields. Their physics assertions pass; their old numerical records describe the gated equation. They have been regenerated with an explicit held-current identity and source provenance; all four study tests pass. The harness now selects its recorded backend explicitly, including when an optional backend is installed. The accuracy page quotes the refreshed records. Historical Luna and TDSE witnesses retain their existing data and bounds.

Concurrent integration — 2026-10-19#

Merged SEP 23 phase 2 through d59ed0bf by an ordinary topic-branch merge. Its Numba far-field statistics leave the held-current source equation and transform ordering unchanged. The shared memory estimate retains both the backend-specific held-current buffers and the far-field partials, and the interpreted coverage lane includes both new kernel test modules. Kernel schema 33 and output schema 22 remain the integrated versions; SEP 23 phase 2 does not change either contract. The merged SEP 28 documentation records future work only; SEP 28 implementation stays deferred. The final CPU and device checks qualify this combined source tree.

CPU qualification and CUDA compilation review — 2026-10-19#

The integrated source pair has completed its CPU qualification. The qualification record retains commands, source revisions, results and raw-log hashes. The full gate’s default lane passed 16,251 tests with 27 expected skips. Three documentation links failed the final build; their focused repair and strict rebuild pass. The coverage lane passed 548 interpreted-kernel tests and 16,251 compiled tests, reaching 100% of 42,760 statements and 12,942 branches. The Kokkos-selected slow lane passed all 148 tests. The Numba slow lane’s two historical-record failures were repaired and their four study cases pass; its other 145 tests had passed, with one expected skip. The separate statistics-cost test passes with both Numba and Kokkos.

Phase 1 review. The analytic and physical tests, source decomposition, private comparison identity, schema refusals and all admitted NumPy routes pass in the complete suite. The approved T2 correction and separate physical-polarisation oracle remain as recorded above. Historical Luna and TDSE witnesses are preserved. The primer and API descriptions now distinguish the two production current integrals from the physical polarisation’s third integral. No new physical constant or material coefficient was introduced.

Phase 2 review. The complete compiled and interpreted suites pass, including the current-column bound, embedded partial-chunk projection, masks, zero reuse, memory accounting, statistics and existing stepper paths. The approved logarithmic rate bound remains explicit about its libm assumptions. A timing-harness error called the Adams source seam on a Runge–Kutta-only driver; the harness now constructs and warms the declared source driver separately from the ten-step driver, and records both memory ledgers. This repair changes no production code or equation.

Phase 3 review. The native complete gate passes Python formatting, lint, strict typing, licence checks, C++ formatting and analysis, debug native tests, ASan/UBSan, 327 Python tests with six expected skips, and the distribution-wheel smoke test. The complete solprop CPU suite also passes with this backend selected. The CUDA compile record records the successful CUDA 12.8.93/GCC 11.5 build for sm_86 with four jobs. Compilation took 1,436 seconds; no CUDA device was allocated or executed. One pre-existing unused-function diagnostic remains visible in its evidence.

The final incoming SEP 23 phase-2 merge adds its completion report; its implementation was already present in this qualified source pair. The integrated schemas remain 33 and 22. The exclusive performance campaign, regenerated worked examples and the approved HIP execution campaign remain outstanding at this checkpoint. The SEP remains In progress and SEP 28 implementation remains deferred.

HIP verification and incoming radial admission — 2026-10-19#

The HIP execution record qualifies the schema-33 implementation on an AMD Instinct MI325X (gfx942), with HIP 7.14, AMD Clang 23, Kokkos 5.2.1, Kokkos-FFT 2.0.0, rocFFT and rocBLAS. The native test executable passed on the device. Solprop’s GPU lane passed 18 tests, the held-current backend ladders passed 34, and solprop-kokkos’s GPU lane passed 31. The latter includes the tiled band column, trapezoid diagnostics, repeatability and partial-chunk Raman cases. No numerical allowance was changed for this device.

Twelve fresh device processes cover mode averaging, multiple modes, radial, Cartesian, line and planar sources under both integration rules. Each records ten warmed source evaluations and three warmed ten-step windows, plus separate source and Runge–Kutta memory ledgers. Their source medians range from 0.227 to 5.351 ms; ten-step medians range from 9.016 to 200.226 ms. The complete resident inventories, including host and device storage, are 13.078–104.728 MiB for the Runge–Kutta driver and 22.382–230.138 MiB for the source driver. These are bounded execution checks, not a GPU baseline comparison or a cross-machine speedup claim.

The first environment setup selected Python 3.15, which the pinned Numba package refused. The device runner now selects Python 3.14 by default; the campaign used 3.14.7 with the unchanged lockfile. Both build attempts are retained. Upstream compiler deprecation and character-conversion diagnostics remain visible in the record; they did not prevent the successful build or tests.

During qualification, SEP 22 phase 3 became ready and was merged through ff862f2f. Its only conflict was adjacent imports: the merged code retains both the held-current certificate and the radial exponent bound. The latter is an admission check and does not change the admitted propagator or current assembly. All 149 affected operator, radial and held-current checks passed on the same HIP device after this merge. The CPU closing lanes are rerun on the integrated pair. SEP 23 phase 3 was still being edited in its own worktrees at this checkpoint and was not imported.

One rental was created. Evidence retrieval verified the archive and individual file hashes before deletion; a subsequent API inventory confirmed that the recorded resource was absent. Creation-to-deletion time was 932 seconds at USD 3.80/hour, an estimated USD 0.9838 against the approved USD 20 cap. The deletion watchdog was cancelled after confirmation. No campaign rental remains.

Incoming far-field statistics integration — 2026-10-19#

The completed SEP 23 phase-3 branches became available after the HIP rental was closed. Ordinary merges bf1d9b03 (solprop) and a48e6cab (solprop-kokkos) retain both implementations; only adjacent changelog additions conflicted. The current-source kernels, stream layouts and transform ordering are unchanged. The shared memory model now also counts the native far-field partials, and the registry checks the installed plugin’s declared statistics. The integrated schemas remain kernel 33 and output 22.

The final CPU lanes and regenerated worked examples use this source pair. The ongoing exclusive performance sweep retains its frozen pre-merge pair so that its alternating processes stay comparable. The merge adds statistics allocation and admission work, outside those timed source and step calls; the final memory inventory will distinguish that additional storage. The CUDA compile-only check is repeated for the combined native tree with the same four-job limit. The completed HIP evidence remains tied to its recorded revisions: it qualifies the held-current paths, but not the newly added SEP 23 device statistics. No second rental is started.

Integrated CUDA build follow-up — 2026-10-19#

The combined native tree passes the CUDA compile-only check. GCC 11.5 initially rejected two spectral-row count sums in the newly merged far-field pass under -Werror=sign-conversion. Commit a8ba4bb makes their 0/1 operands explicitly unsigned; formulas, loop extents and tolerances are unchanged, and no warning is suppressed. The repaired incremental build completed in 68 seconds with four jobs, linking the native library, Python extension, test executable and GEMM benchmark. The earlier failed attempt took 364 seconds and remains in the CUDA evidence. The original full build and its unchanged dependency objects are retained. No CUDA device was allocated or executed.

Incoming low-storage flow accounting — 2026-10-19#

Merged SEP 24’s committed closing work through a22996f8, including the e7826c83 reusable flow scratch and bb9034c9 z-table tests. The Numba modal low-storage march now forms its two step flows successively in one counted scratch volume, without a transient exponent array. The merge retained the held-current memory rows and the existing low-storage plasma and Raman tests. The new exact memory assertion accounts separately for the reference and compiled current layouts, using the constructed backend’s current ledger rows; their layouts need not occupy the same number of bytes.

Focused low-storage tests and the full integrated closing lanes remain due. The exclusive source and ten-step performance campaign retains its frozen source pair: its single-source Adams and Runge–Kutta paths do not execute this low-storage flow code. Final memory records and correctness checks use the integrated tree. No source equation or numerical tolerance changed in this merge.

Performance measurement scope — 2026-10-19#

The source-cost campaign measures a changed formulation on existing routes. Each workload has distinct legacy and held identities; the comparison is the cost of replacing the former production equation, not a same-equation regression. A separately configured Adams source evaluation and a ten-step Runge–Kutta window distinguish source cost from other backend work. Full adaptive work counts remain in the physics record.

The initial sweep expanded to 1,464 planned fresh processes and was interrupted when the maintainer questioned the elapsed time. Complete comparison groups are retained unchanged. Incomplete groups are preserved separately with hashes and rerun in full using five source samples, one warmup Runge–Kutta step and one measured ten-step window per process. Every remaining group retains three fresh rounds, alternating order, the unchanged baseline and two held runs for A/A spread. The report refuses mixed protocols within a group. Completed NumPy comparisons retain the gated and alternative-B prototypes; remaining groups concentrate on the production comparison. This reduces repeated work without changing the equations, propagation windows or the preselected slowdown rule.

The final performance review must report absolute Numba and Kokkos times as well as each backend’s ratio to its own old source. An improvement relative to one backend’s baseline does not imply that it is the fastest backend. The completed original modal/radial/Cartesian matrix already demonstrates this distinction: Kokkos’s one-thread radial trapezoid window drops from 4,867 to 2,077 ms, while Numba’s rises from 504 to 744 ms. These are short-window measurements, not predictions of full propagation time; profiling and the remaining matrix are still due.

The first cross-backend report incorrectly called the requested-18-thread runs an equal-thread comparison. Execution metadata showed that Kokkos reached 18 threads but Numba reached two. A lightweight runtime probe confirmed the cause: with OMP_PLACES=cores, OpenMP binds the main thread to a core’s two hardware threads, while the existing allocation helper only recognises a one-CPU binding. Its inferred allocation therefore shrinks to two CPUs. Explicitly listing one logical CPU per physical core restores the requested cap without changing the backend sources. The erroneous scaling interpretation was withdrawn. All 18-thread groups are repeated with matched explicit placement and assertions of actual pool sizes; superseded records are preserved separately with hashes. One-thread results remain valid. The general affinity-helper correction is a separate follow-up, not a change to the held-current implementation.

Closing integration and parity-fixture review — 2026-10-19#

Subsequent incoming changes are taken through main. Its revisions through 4ef7bf09 are merged, including SEP 22’s archive and SEP 28’s acceptance; SEP 28’s implementation still waits for this proposal. The earlier direct topic-branch integration of SEP 24 through a22996f8 remains in this tree and is identified separately from the changes already on main. The native main merge through 95b7502 changes ancestry without changing the already integrated executable code.

The expanded one/eight/ten-step, two-rule, two-pre-ionisation parity matrix initially built its constant pre-ionisation table only to the last save plane. The translated tables include an end margin, so validation correctly refused it. The fixture now uses the common linear.z_span. All 85 non-planar NumPy/Numba cases pass after that correction.

The planar fixture independently exposed an invalid coarse replay: its original 0.1 m problem explicitly requires 80 Adams steps, while the new matrix requested one, eight or ten. The pre-ionised reference overflowed before comparison. The maintainer approved a \(10^{-7}\) m window for this new planar matrix, retaining the grid, launch, responses, both pre-ionisation values, both integration rules, the requested step counts and all tolerances. The length was selected before testing its result. The fully ionised cold-plasma rate \(\omega_p^2/(2c\,\operatorname{Re}(n_{\rm eff})\,\omega)\), evaluated from the translated prefactor with its temporal crop scale removed, is at most \(2.76495\times10^5\ {\rm m}^{-1}\) over its active modes and frequencies. The proposed whole-window phase is therefore below 0.028 rad. This is a short-window design scale, not a proof of nonlinear stability or a substitute for the independent planar accuracy and convergence studies, whose lengths remain unchanged. All 12 corrected planar cases pass on NumPy/Numba (17.01 s), completing that matrix with the 85 non-planar cases (30.79 s) on macOS, Python 3.14.5. The integrated Kokkos matrix remains part of the closing CPU qualification. Lint, formatting, typing, public docstrings, spelling and licence checks pass on this source tree; SEP lifecycle tests pass (211, one expected skip).

The additional macOS build of the integrated native backend passes all 113 focused Kokkos checks in 70.07 s: the 96 one/eight/ten-step comparisons and 17 source, memory, scratch-reuse and benchmark-refusal checks. It uses Python 3.14.5, Clang 23.1.3, Kokkos 5.2.1, OpenMP, FFTW and Accelerate in the local Release profile, with one numerical thread per pytest worker. These are correctness checks on a concurrently used machine, not performance measurements.

Final test-budget correction — 2026-10-19#

The existing 0.1 m, 80-step uniform-planar field-plasma fixture passed its unchanged field, transverse-leakage and density assertions, but its peak rate differed by \(1.28\times10^{-7}\) relative, above the old empirical \(10^{-7}\) allowance. The maintainer approved propagating the existing spectral field allowance through the actual synthesis and rate table. This changes only the field-plasma total and species peak-rate assertions, retaining the original problem and all other checks.

For the permitted coefficient error \(B\), including the existing nonzero-\(k_y\) leakage allowance summed through the inverse transverse transform, the real inverse temporal transform and modal synthesis give \(\delta F = B(2N_{\rm active}/N_{\rm to})\max_n\sum_m|S_{nm}|\). Both reconstructions’ temporal and transverse FFT and synthesis rounding are added against their absolute operands. The maximum absolute nodal field is Lipschitz in this error. Every log-cubic interval’s derivative is checked at both endpoints and any internal quadratic vertex, establishing monotonicity for this table. Evaluating \(W(F_*\pm\delta F)\) then bounds the peak rate; the previously approved logarithm, polynomial and exponential evaluation bound is added. The allowance is computed from the reference before running the planar comparison. Here it is about \(4.1\times10^{-6}\) relative, against \(F_*\simeq2.0145\times10^{10}\ {\rm V/m}\). The unchanged field comparison remains the sensitive route-parity check.

The full planar module passes (10 tests, 31.90 s). Two independent bookkeeping fixtures were also repaired: the low-storage memory comparison now includes the reference’s actual embedded-current rows, and a synthetic timing record supplies the actual thread count required by the new identity guard. No production equation or implementation changes were needed. The full coverage run and affected-module repairs reach 100% of 42,836 statements and 12,970 branches. The NumPy/Numba slow functional lane passes 130 tests with 21 expected skips (705.13 s), and the Kokkos-selected slow functional lane passes 148 tests (789.24 s). Exclusive statistics-cost checks remain separate from these concurrently executed correctness checks.

Published SEP 24 closure — 2026-10-20#

The ordinary merges of solprop main through 7bfbab51 and native main through 496fbf5 bring in SEP 24’s completed archive and native validation guards. All previously integrated SEP 24 implementation is now also on main. The solprop executable implementation was already present; its incoming source changes only update documentation links. The native additions refuse an elided stage register beside an in-stage absorber, more than one separable flow volume, and reuse of a constant flow table before construction or after reinitialisation. They do not change the held-current pipeline or the fixed-step timing workloads.

The final CPU builds use these merged guards. The earlier CUDA and HIP records retain their actual revisions and scope; they are not relabelled as device tests of this later integration. The worked-example refresh uses the preceding native build, whose executed propagation paths are unchanged by these guards.

Worked-example refresh — 2026-10-20#

All fourteen worked examples admitting field-grid plasma pass the supported poe docs-examples workflow. Their generated plots, printed values, downloadable source archives and the README’s VUV figure are refreshed. The example record retains each source and raw-log hash, 1,974.676 seconds of script execution and 2,564.45 seconds for the complete rendering workflows. They ran with four numerical threads alongside correctness work, so these runtimes are provenance, not evidence of a performance change.

The refreshed figures retain the examples’ stated physical conclusions. This work checks the published end-to-end workflows; the analytic tests and dedicated campaign above supply the physics validation. The maintainer questioned the value of additional example runs, and no further example campaign is planned. Historical TDSE and Luna witnesses remain unchanged.

Closing scope amendment — 2026-10-20#

The maintainer requested prompt closure because other work depends on this SEP, explicitly allowing remaining performance issues to move to the roadmap and requiring them to be flagged here. The already running matched timing matrix is retained, including resolved slowdowns and A/A variation. The unstarted exhaustive compiled profiling pass is cancelled. Existing NumPy profiles remain evidence; complete compiled attribution and optimisation are not claimed and are no longer conditions for Final under this execution agreement.

The roadmap records the radial and modal cost questions and the Cartesian, line and planar handoff to SEP 28, together with the separately discovered OpenMP allocation-cap issue. The production definition remains \(A_J\); no alternate formulation or performance-driven numerical relaxation is introduced. Required correctness, schema, memory and statistics-cost checks remain closing requirements.

Phase 4 completion and performance review — 2026-10-20#

The complete CPU performance record contains 1,380 fresh processes in 144 comparison groups, with three alternating rounds, held/held A/A, per-case medians, variability, actual thread pools and memory ledgers. The benchmark ledger reports both the old/new ratios and absolute Numba/Kokkos costs. All corrected 18-thread compiled groups actually use 18 physical cores; the earlier two-thread Numba results are not used in this comparison.

Across fourteen plasma geometry/rule cases, the ten-step geometric-mean ratios are 0.976/0.983 for NumPy at one/eighteen threads, 1.145/1.341 for Numba, and 0.944/1.103 for Kokkos. None of the sixty non-plasma control groups has a resolved slowdown above 10%. At eighteen threads Kokkos is faster in absolute time in every measured plasma case, despite its own modal and spatial regressions. At one thread Numba remains faster for several spatial cases. These are ten-step windows, not adaptive full-run speedup claims.

The largest changes remain visible: Numba’s eighteen-thread radial source rises 67–72% and its window 53–67%, while Kokkos radial source falls 41–42% and its window 53–54%. Kokkos multimode and planar trapezoid windows rise about 60% and 59%; Numba’s corresponding increases are 64% and 61%. Some Numba A/A pairs are noisy; the flagged increases survive the full observed process ranges. The completed NumPy profiles explain added transforms and chunk passes. Compiled attribution is incomplete and explicitly deferred under the closing scope amendment. No formulation was substituted and no tolerance was relaxed to improve timing.

The twenty-four final memory inventories preserve every held-current buffer size from the frozen timing candidate. The incoming Cartesian and line far-field tables and host copies add 512 and 784 bytes respectively in these workloads; measured FFT-plan heap variation remains separately recorded. The private gated and alternative-B prototype comparisons retain their own identities and all fourteen one-thread NumPy geometry/rule groups. No benchmark output can resume into the production equation.

The full physics campaign, fixed-field replay, window investigation, source comparisons and actual work counts are complete as reported above. The raw intermediate scalar ladders near 1e-12 remain explicitly unbounded; their unnecessary tightening was stopped at the maintainer’s request. The spectral and endpoint observable residuals resolve the reported changes. No additional physical constant, material coefficient or literature claim was introduced by this closing work. Earlier source-verification limits remain unchanged.

Phase 5 completion and final review — 2026-10-20#

The closing CPU record records the integrated full gate: 16,451 tests passed, 27 expected skips, all static checks passed, and the strict documentation build passed. Coverage and its focused repairs reach 100% of 42,836 statements and 12,970 branches. The Numba slow functional lane passes 130 tests with 21 expected skips; the Kokkos-selected slow functional lane passes 148. The separate exclusive statistics-cost checks pass for both compiled backends. There is no remaining failed required check.

The native closing record adds the final main integration to the earlier complete gate. The incoming low-storage construction fixture initially omitted Cartesian transverse axes, so the existing extent validator fired before the intended new guard. Commit 5284a6e supplies axes consistent with the fixture’s existing unit transverse wavenumber; this changes only the construction test. After repair, Debug CTest, ASan/UBSan, all 336 Python tests (five expected skips), the distribution-wheel build and its isolated smoke pass. Static checks had already passed and the changed fixture’s formatting was rechecked. No gate stage was suppressed.

The final integrated executable pair is solprop def695fe and solprop-kokkos 5284a6e, incorporating main through 7bfbab51 and 496fbf5 respectively; subsequent solprop commits publish this evidence and archive the proposal. All earlier integrated SEP 24 work is now on main. The CUDA compile-only and MI325X execution records retain their actual earlier revisions and qualified scope; they are not relabelled as device tests of later statistics additions, CPU matrix extensions or construction guards. The single HIP rental was retrieved and destroyed, with absence confirmed and estimated cost USD 0.9838 of the USD 20 cap. No campaign rental or queued compute job remains.

All fourteen affected worked examples have been regenerated with provenance. Historical TDSE and Luna witnesses are preserved. Manual pages, docstrings, kernel/output contracts, current architecture and changelog describe the held source; kernel schema 33 and output schema 22 are the integrated contracts. The physical ungated PlasmaResponse.evaluate contract, general-modal refusal, envelope plasma, Kerr, Raman and TDSE behaviour remain as specified.

Handoff and remaining work. Phases 1–5 are complete under the recorded scope amendment. Mark this proposal Final and archive it. The roadmap retains (1) the OpenMP core-place allocation-cap defect and (2) compiled held-current cost attribution and optimisation. SEP 28 can use this completed implementation, the 144-group timing matrix and final memory inventory as its baseline; its implementation is not started here. The scalar precision limitation and device revision scope are recorded above, not silently claimed as resolved. Source changes remain locally committed on the topic branches; publishing or merging them into main remains a separate maintainer action.

Part III — History#

2026-10-20: Phases 1–5 completed and proposal marked Final. Remaining compiled performance attribution and optimisation are roadmap follow-ups under the maintainer’s closing amendment; required correctness checks pass.
2026-10-19: Drafted at the maintainer’s request, answering SEP 26 Q3 and Q10. The drafting probes of §4.4, §4.5 and §10 used FieldGrid, Collar and RunningIntegral at solprop 9eef5425, with no propagation.
2026-10-19: Revised before review at the maintainer’s instruction (relayed by the coordinating agent). There is no backward compatibility: the held source replaces the truncation outright. The public option, opt-in phases, record and resume rules and the promotion phase were removed. A private comparison hook serves validation only.
2026-10-19: Q1–Q6 resolved by the maintainer (the drafting agent’s recommendations).

AI review — 2026-10-19#

Reviewers and disposition. GPT-6 (AI); John Travers (human, recorded at his request). Review status is Reviewed; lifecycle status remains Draft. This records completion of the reviews, not acceptance. Part I has not been edited by the reviewer. The drafting agent should answer each numbered point in a separate Response to review subsection and make any resulting revisions to Part I there.

Assessment. The continuous held-current identity, the motivation for removing the polarisation edge, and retaining the existing non-plasma responses are well motivated. The committed SEP 26 record supports the quoted fixed-field differences. However, the proposed discrete spectral identity is conditional on a boundary extension that the admitted grids do not all satisfy; the two descriptions of the frequency-domain oracle are not the same discrete computation; and several mandatory tests assert incorrect or incomplete identities. Resolve the P1 points before acceptance. These findings do not establish that option B is unusable: they establish what its specification and qualification still need to show.

Evidence scope. Reviewed all of Part I against the architecture, conventions, SEP 0, the relevant derivations and reports in SEPs 18, 25 and 26, the current reference plasma passes, RunningIntegral, envelope source factors, output identity, and the Numba and Kokkos column structure. Numerical probes used solprop 7e7155c8, NumPy 2.5.2 and SciPy 1.18.1; the Kokkos source inspected was 391ca7d. These were synthetic fixed-current calculations and one independent quadrature check, with no propagation or backend qualification. The hashes and numbers in benchmarks/sep26/evidence/rate-plasma.json were inspected; the saved L6 fields were not reacquired or replayed.

R1 — P1: qualify the boundary extension in the periodic identity (§4.4–4.5, T1–T2, T6)#

The Fourier multiplier follows when the filter uses periodic samples. The existing RunningIntegral.filter instead uses point reflection. The draft recognises this condition but then asserts that all the new rows satisfy it, citing a \(4\times10^{-15}\) collar level from one SEP 18 grid. That is not an admission guarantee on the grids in §4.5.

Counterexample: construct FieldGrid.from_limits(1.0, 800e-9, (200e-9, 2e-6), 40e-15) and its band rule. It has \(N=512\), \(\Delta t=1.1118803173271735\times10^{-16}\) s and \(K=4\). The maximum collar value in the first or last \(K\) samples is \(1.5605\times10^{-8}\), while the proposed certificate is \(3.4286\times10^{-13}\): this grid is explicitly admitted. For \(J=[1+\operatorname{erf}((t+5\,\mathrm{fs})/(1\,\mathrm{fs}))]/2\), directly evaluating the two closed integrals in §4.4 differs from its claimed multiplier expression by \(2.57\times10^{-9}\) of the analytic active-band polarisation norm. The same comparison on run_short is \(1.21\times10^{-14}\). These are distinct from the rule’s interior multiplier error and the collar certificate.

Please either define periodic extension for the two new periodic integrations, or derive and budget the actual reflection residual at both integrations. In the latter case, let \(d_y=F_{\rm reflect}y-F_{\rm periodic}y\). Before rounding, the correct one-integral relation is

\[ \operatorname{DFT}[\mathcal J_Ry]_k =\frac{R(\theta_k)Y_k+\rho(\theta_k)\operatorname{DFT}[d_y]_k}{i\omega_k}. \]

Propagate this term through the nested definition. The certificate on \(g\) alone does not bound it for \(g'J\) and the outer integrand. Include the admitted 512-point grid, the supported smooth FFT sizes and currents reaching either taper in the tests. Specify what happens when the new boundary bound is not met; do not assume that small error in the run_short probe proves the identity for every grid.

R2 — P1: choose one discrete oracle and reconcile the comparison units (§4.4, §6 A, T6, §7.2)#

Option A as written subtracts from the old sampled polarisation. SEP 26’s rate_plasma.py instead constructs the held source directly from the sampled current. They agree continuously when \(P'=J\), but that is not an exact identity for \(P=I_R[J]\).

Write \(q=i\omega\), \(V=\operatorname{DFT}[gP]\), \(X=\operatorname{DFT}[gJ]\), \(Y=\operatorname{DFT}[g'P]\), \(Z=\operatorname{DFT}[g'J]\), and \(C=qV-X-Y\). Then

\[ A_P=V-Y/q-Z/q^2, \qquad A_J=X/q-Z/q^2, \qquad A_P-A_J=C/q. \]

Ignoring R1’s boundary residual, \(B=RX/q-R^2Z/q^2\). The proposed T6 bound accounts for \(B-A_J\), but it omits \(C/q\) if the oracle is §6’s \(A_P\). The existing fixed-field record explicitly measures a nonzero \(C\); it is not FFT rounding.

The review reproduced the draft’s run_short erf probe (\(t_c=-5\) fs, \(\sigma=1\) fs). Relative to the analytic active-band polarisation, the errors are:

rule	B	subtraction from P, \(A_P\)	direct current form, \(A_J\)
band	\(7.010\times10^{-8}\)	\(6.995\times10^{-8}\)	\(5.32\times10^{-16}\)
trapezoid	\(1.622\times10^{-4}\)	\(1.619\times10^{-4}\)	\(5.32\times10^{-16}\)

Thus the draft’s table is consistent with \(A_P\), while the SEP 26 replay uses \(A_J\). Please name them separately, use the direct-current form as the independent held-source reference, and retain \(A_P\) as a decomposition check with its measured closure residual if desired. Also distinguish raw nodal polarisation spectra from the projected, cropped, prefactor-weighted Q in the replay: carry those linear maps into every bound and divide the bound by the same denominator when reporting relative errors. As written, T6 and §7.2 mix a source norm with raw-polarisation operands and call the resulting estimate approximately \(5\times10^{-6}\) of the band.

R3 — P1: retain the additive edge error when the physical spectrum is small (§4.5, T1–T2)#

T2 bounds the rule error by \(\varepsilon_R(2+\varepsilon_R)\) times the analytic erf-rise spectrum. That omits the residual of the steady tail which §4.4 itself reports. Using R2’s direct oracle,

\[ B=R A_J+R(1-R)Z/q^2. \]

The second term does not vanish when the rise’s Gaussian spectrum becomes small. It needs an additive, operand-based bound; multiplying the oracle alone cannot bound it. Broader rises make the distinction particularly visible. Please carry that term, R1’s boundary terms, the appropriate product-alias terms and rounding separately in T2, including bins with negligible physical signal. A band floor may be used for relative reporting, with absolute bounds retained elsewhere.

Likewise, \(\kappa_R\le\varepsilon_R\) is not the general certificate bound. Even with a periodic filter,

\[ \kappa_R\le \varepsilon_R\frac{\|\operatorname{DFT}[g]\|_A}{\|\operatorname{DFT}[g]\|_2} +(1+\varepsilon_R)\kappa_{\rm ss}, \]

before rounding; reflection adds R1’s residual. The steady-source bound also carries the outer factor \(R\) on the collar residual. Define \(\varepsilon_R\) for the trapezoid on the actual active bins, since its certified_error is not the band-rule certificate. Explain explicitly that a steady-current certificate qualifies that particular collar identity, not aliasing of arbitrary unresolved \(gJ\) or \(g'J\) products.

R4 — P1: correct the analytic drift sign and the zero-drift tests (T4–T5)#

With the definitions in T4, \(E=-\partial_t A\) and \(A(\pm\infty)=0\) give

\[ \int fE\,dt=-[fA]_{-\infty}^{\infty}+\int f'A\,dt, \qquad J_\infty=+\frac{e^2}{m_e}\Delta f\,(A*\phi_{\sigma_i})(t_i). \]

The draft has a minus sign. An independent quadrature check, in dimensionless variables, with \(A_0=1\), \(\tau=3\), \(\omega_0=2.1\), \(\varphi=0.4\), \(t_i=0.7\), \(\sigma_i=0.8\), \(f_0=0.2\) and \(\Delta f=0.3\), gives \(\int fE\,dt=+0.1396845403804866\) and the same positive smoothed-potential value. Please correct the analytic reference rather than changing the implementation’s sign to satisfy the proposed test. State whether this test isolates the free current or also constructs an ionisation-loss current consistent with the prescribed fraction.

T5’s second assertion is also false: \(J\) vanishing before the taper does not make held and gated sources agree. A compact current with \(\int J\,dt\ne0\) leaves \(P_\infty\ne0\), exactly the counterexample in §4.1 and T4(b). Require both the current and its residual polarisation to vanish for that equivalence test, and keep a separate test that removes the residual-polarisation edge. T4(c) should not make the total removed term depend only on \(J_\infty\): it also depends on the residual polarisation, which T4(b) deliberately varies independently.

R5 — P2: state the two-edge continuation when the leading taper is occupied (§4.1, §4.4, T9)#

The plateau equality and trailing-only formula for \(H\) in §4.1 assume \(g'J=0\) throughout the leading taper. T9 deliberately violates that assumption. For the stated zero pre-window current, integration by parts gives

\[ J_h(t)=g(t)J(t)-\int_{t_a}^{t}g'(s)J(s)\,ds. \]

On the plateau this is \(J(t)-\int_{\rm lead}g'J\,dt\), and its final held value is \(H=-\int_{\rm lead+trail}g'J\,dt\). This is consistent with the discrete definition subtracting both edges, but not with an unconditional statement that it equals \(J\) on the plateau or that \(H\) depends on the trailing taper alone. Ionisation in the leading taper can also populate it without pre-ionisation; admission does not constrain the field there.

Please give both formulas, label the empty-leading-taper limit, and make T9 test the plateau offset and final value against them. Also correct the stated area of \(g'P\): in general \(\int g'P\,dt=-\int gJ\,dt\), not minus the polarisation at an unspecified end point. Distinguish the time endpoint from the propagation coordinate \(z\) in that statement. These are clarification of the proposed continuation, not a request to add another source policy.

R6 — P1: derive the work test for the held source, including window terms (T8, §7.5)#

The held source is deliberately different from the windowed transform of \(J\). Even with exact integration, \(qA_J=X-Z/q\). Relative to the raw periodic transform \(\widehat J\), its current-equivalent difference is

\[ qA_J-\widehat J=\operatorname{DFT}[(g-1)J]-Z/q. \]

There is no general bound on its work against \(E\) using only \(\varepsilon_R\) and the field’s out-of-band fraction. A field overlapping a taper can have a finite contribution from either term while occupying only active bins. With nonzero final drift, comparing a periodic transform of \(J\) with the continued current also requires the specified endpoint convention.

A concrete counterexample uses R1’s 512-point grid and a pre-ionised Drude current, with its constant coefficient scaled out: set \(E_n=\cos(2\pi 9n/N+\pi/4)\) and \(J_n=[\sin(2\pi 9n/N+\pi/4)-\sin(\pi/4)]/\omega_9\). This is a single in-band field mode with \(J'=E\) and \(J(t_a)=0\). Normalising the work sum by \(\|\widehat E\|_A\|\widehat J\|_A\), the raw-current work is \(-9.8\times10^{-17}\), whereas the direct held-current work is \(-1.3707\times10^{-3}\). The field’s out-of-active-band norm fraction is only \(1.47\times10^{-15}\). The finite window term is therefore present even before introducing the running-integral error.

Please derive the discrete work identity for the proposed continuation, including the filter multiplier, both window terms, endpoint/reflection effects and the projection/crop. Specify the real-field half-spectrum weights and raw-DFT normalisation. Where the field has negligible overlap with these contributions, bound the remaining work defect in terms of those actual operands. Test that limit and a deliberately occupied taper. The old gated source’s work difference from \(J\) also includes \((g-1)J\); \(\max_{\rm taper}|E|\,\|g'P\|_1\) alone bounds only its polarisation-edge part. Preserve SEP 18’s population/loss-work defect as a separate check, since changing the final source integral does not remove it.

R7 — P2: make the rounding budget implementable through the whole new operator (§4.6)#

The new terms have different units until propagated through the outer integral. For example, a prefix bound for \(a=g'J\) uses trapezoid increments \(\Delta t(\tilde a_n+\tilde a_{n-1})/2\), not the unscaled sum of \(|\tilde a|\). The latter is not a current error. The FIR also needs its own operand bound, including absolute tap weights and point-reflected endpoint samples; cancellation in \(\tilde a\) cannot be used to bound the error in forming it.

Please express the budget as error propagation through named operators: the existing error in \(J\) enters both \(gJ\) and \(g'J\); the derivative and products feed the first filter and scan; uncertainty in \(S_a\) feeds the inner closure and outer integral; and uncertainty in the outer total feeds the final closure. Then apply absolute projection weights, FFT, crop and source factors. The \(4\varepsilon(|S_a|T+|S|)\) expression can cover arithmetic on already exact totals, but does not by itself cover the computed totals’ errors. Retain the existing protected arithmetic and qualify the FFT constant on supported smooth sizes, as SEP 25 requires. This can be a concise recurrence, not a catalogue of every kernel operation.

R8 — P1: distinguish comparison outputs and reject incompatible continuation (§5, §6, §9, §11)#

The private gated hook changes the differential equation while §5 explicitly excludes it from the run identity. Phase 4’s external provenance flag helps a reader, but does not stop two HDF5 outputs or resumable checkpoints from having the same configuration hash under different sources. It also makes the new plasma_source metadata ambiguous on a gated comparison run.

Please specify a private benchmark output identity/label for the source form, or make gated comparison output explicitly non-resumable and segregated from production output. This does not require a public option or a compiled legacy path. For old production files, state the concrete rejection boundary and the identity/schema field that changes. The kernel schema bump alone is not a stated saved-output identity rule: RunMetadata.configuration_hash has its own schema and records. No backward compatibility is required, but silently continuing an old equation with the new source must not be an accidental consequence of leaving the version decision to the implementer. Test the new rejection/identity contract.

R9 — P2: align the route matrix and public response contract with current admission (§2, §5–6, §7.4)#

The scope lists general modal plasma, but the current general-modal manual’s admission section permits the regions’ cubic and silica tensor Raman responses; it does not expose a field-plasma material route. The generic gas/plasma keywords are explicitly refused beside a provider. Its local fields also have three components, whereas T6 and §7.4 only exercise one and two. Please identify the existing admitted general-modal plasma path if one is intended, or say that the change applies to every route that already admits field plasma and preserves general-modal refusals. Adding that physics route would be additional scope. List the line free-space case explicitly as either covered by its shared Cartesian path or outside the admitted matrix. Tests should cover the actual matrix, including the mixed Kerr/Raman paths whose collar placement changes.

Separately, PlasmaResponse.evaluate currently returns the physical ungated \(P=I_R[J]\) and calls itself the backend oracle. It receives neither \(g\) nor \(g'\), so it cannot produce §4.4’s periodic representative with its unchanged signature. Specify its intended continuing meaning and how its physical-current/response check is separated from the new assembled-source oracle. A docstring saying only that every PlasmaResponse is now held would misdescribe this public call. Also qualify the envelope comparison in §4.2: shock=False freezes its inverse frequency factors at \(\omega_0\), so the exact \(1/\omega^2\) identity describes the frequency-dependent envelope policy.

R10 — P2: recompute live support after closure, and budget its cost (§6, Phase 3)#

The statement that \(g'J\) is zero until the trailing taper excludes T9’s supported leading-taper case. More fundamentally, closing its integral subtracts \(nS_a/N\), which is nonzero across the window even when the unclosed edge integral is confined to the trailing taper. The outer integrand is therefore not supported only where the original current or edge was live.

Please preserve band_support only where its exact-zero precondition holds. Either determine support from each actual new row, as the current Kokkos code does, or carry the analytic ramp separately and derive the corresponding scan and closure totals. Write the final correction over the whole output row, including samples skipped by the scans. Add empty-row, leading-edge, trailing-edge and scratch-reuse cases, so a previous node cannot leak through an unwritten region. Measure the revised support and extra full-row work in the hot-path benchmark; the existing quiet-window saving cannot simply be assumed unchanged.

R11 — P2: specify independent convergence and consistent spectral scaling (§7.3, §9)#

The convergence ladder is a good requirement, but the tightest run has zero difference from itself. It needs a bound on its remaining error from the last three resolved levels, with further tightening if that evidence is inconclusive; the five listed tolerances must not become a fixed endpoint that automatically certifies the reference. Record atol, maximum-step and save-plane policy, and use the resulting residual bound for each form when deciding whether their difference is resolved. No particular convergence order should be assumed from the rtol ratios alone.

For the window control, common frequency bins are not sufficient to compare raw DFT coefficients. Convert each spectrum to the same continuous-transform normalisation (sample spacing included), Fourier origin and amplitude convention, and state how the identical physical launch is rebuilt on each grid. Retain band energy comparisons when bins do not coincide.

Finally, §6’s assertion that the derivative changes only in the near infrared and §9’s expectation that propagated differences are attenuated are stronger than the evidence. The committed source record has a nonzero UV difference, and nonlinear propagation can amplify a perturbation. Keep the numerical expectations labelled as hypotheses and let the matched-input controls, integration envelopes and window control determine the result; no expected sign or attenuation should be a pass condition.

R12 — P2: record the approved full-run exception and make the handoff public (§11–12)#

§7.3’s full 0–4.05 m comparisons conflict with SEP 0 §7 and the standing qualification limit in AGENTS.md. This was clarified with the maintainer during review.

Maintainer decision — John Travers, 2026-10-19: allow the full-length §7.3 validation campaign as an explicit exception because these runs should be fast. The drafting agent should record that exception in the proposal. The ordinary backend qualification and hot-path timings remain bounded as specified in §7.4 and the standing rules. This review did not execute or schedule the campaign.

Also replace the private machine name in Phase 4 with its public compute requirement (CPU execution); put actual hardware in the eventual benchmark ledger. Replace coordination through an unnamed “master” with a concrete cross-SEP integration responsibility. Q3’s resolved statement says the TDSE certificate fix is already done, while its recommendation still asks for a future roadmap item: mark that recommendation as superseded by the recorded resolution and cite the public change. No additional maintainer decision is needed on the full-run question.

Sources checked and remaining limits#

The bibliographic details of Geissler et al. were checked against the APS record, and Brunel against the Optica record. Their full articles were not available in the accessed publisher views, so their equations were not independently verified from those articles. The held-source and drift identities above were re-derived from the equations stated in the SEP. The Planck taper and equation (7) were checked in the McKechan, Robinson and Sathyaprakash paper. The chapter title and radix-two/twiddle assumptions were checked in the SIAM chapter 24 excerpt; this does not independently certify \(c_F=8\) for every FFT implementation. No new physical constant or material coefficient was transcribed. The CPU propagation campaign, production implementation, device builds and empirical cost estimate remain work for the implementation phases.

2026-10-19: GPT-6 completed the AI review, R1–R12; John Travers’s human review was recorded at his request. Set review status to Reviewed, retain lifecycle status Draft, and leave Part I unchanged. The maintainer approved the full §7.3 validation runs as a qualification-scope exception (R12). The drafting agent’s responses remain due.

Response to review — 2026-10-19#

The drafting agent (Opus 5.5) revised Part I; the review text above is unchanged. The coordinating agent decided each point under the maintainer’s delegation. One decision is returned to the maintainer: Q2, the change of implementation (R1–R3).

The implementation choice (R1, R2, R3, R7, R10). Before answering the points one by one, the agent reconsidered B against the direct-current form \(A_J=X/q-Z/q^2\), which R2 showed is exact to \(5\times10^{-16}\) under both rules.

What \(A_J\) costs. Two extra real temporal transforms per node and component, against one filter-and-scan saved, because \(P\) is no longer formed. The per-route counts are in §6.
Indications. In numpy_ref one band running integral costs about four real FFTs on run_short’s lattice, so \(A_J\) is a net saving there. In compiled code the estimate runs from neutral (single-mode modal) to about \(+20\%\) of a nonlinear evaluation (Cartesian), against B’s estimated 10–25% of the plasma column.
Radial. Projecting three arrays would triple the dominant GEMM, so the radial route transforms per node and projects once in frequency.

The cost is acceptable, so \(A_J\) is now the definition and the recommended implementation (§1, §4.4, §6). B is kept only as a later optimisation, admitted against \(A_J\) within a derived budget and with periodic filters. Q2 is reopened as a changed recommendation for the maintainer.

R1, resolved by the change. \(A_J\) adds no running integral, so the reflection residual does not arise. The column’s two remaining integrals keep SEP 18’s boundary budget unchanged. If B is pursued later, it must use periodic extension for its new integrals (§6, “Kept as a later optimisation”). The 512-point grid remains in T7, and T2 runs on it.
R2, accepted.
- \(A_P\) and \(A_J\) are named separately (§4.4). \(A_J\) is the definition and the independent reference. \(A_P\) is T6’s decomposition check, asserting \(A_P-A_J=C/q\) within the budget and reporting \(\lVert C/q\rVert\).
- §4.4 defines the assembled source \(Q=pW\,\mathrm{crop}[\sum\Pi(\cdot)]\), and every bound and relative figure in §7 is carried through these maps.
- §7.2 (i) now compares two implementations of the same \(A_J\), so its bound is the rounding budget, not \(5\times10^{-6}\).
- The probe table is the review’s.
R3, accepted.
- T2’s tolerance is additive and absolute per bin: the collar residual of the steady tail, the product aliases, the tail mass and rounding. It covers bins with negligible signal, and a band floor is used only for relative reporting.
- \(\kappa_R\) is removed, since \(A_J\) has no rule multiplier. B’s additive term \(R(1-R)Z/q^2\) is named in §6 for any later use.
- §4.5 now states that \(\kappa_{\rm ss}\) certifies the collar identity only, not aliasing of arbitrary sampled products.
R4, accepted.
- T4’s reference is \(J_\infty=+(e^2/m_e)\Delta f\,(A*\phi_{\sigma_i})(t_i)\); the implementation is not touched. T4 now isolates the free current, with the loss current zero; the loss current is covered by T1, T6 and the column runs.
- T4(c) treats the removed term as the sum of a drift part and a residual-polarisation part, each linear in its own amplitude.
- T5 requires both \(J\) and \(\int J\,dt\) to vanish for the equivalence, and keeps a separate residual-polarisation case.
R5, accepted.
- §4.1 gives the general continuation (plateau \(J-c_-\), final \(H=-\int_{\rm lead+trail}g'J\)) and the labelled empty-leading-taper limit. It notes that ionisation can occupy the leading taper without pre-ionisation.
- §4.1 replaces the area statement by \(\int g'P\,dt=-\int gJ\,dt\), uses \(t_\pm\) for the tapers, and reserves \(z\) for propagation.
- T9 tests \(c_-\) and \(H\).
R6, accepted in a simplified form.
- §4.7 derives the discrete work identity for \(qA_J\): the half-spectrum weights, the raw-DFT Parseval form, both taper terms, the periodic window term \(HM_1\) and the out-of-band term.
- T8 gates the field-off-the-tapers limit and reports one occupied-taper case without gating it.
- The claim that \(\varepsilon_R\) and the out-of-band fraction bound the work is withdrawn. SEP 18’s population and loss-work check stays separate.
- Projection and crop are not carried into the work test, which is nodal; the assembled-source parity covers them.
R7, accepted. §4.6 is now a short recurrence through named operators: the current’s column bound, the products, projection, FFT (with \(c_F\) qualified on the supported smooth sizes), the two divisions and assembly. With \(A_J\) it is SEP 26 §4.5’s budget plus a second division. The closure and FIR terms of B are no longer needed.
R8, accepted.
- §5: gated comparison runs carry a private benchmark label in metadata and in the hashed configuration. Their output refuses to resume and is written only to the benchmark scripts’ own directories. There is no public option.
- §6: the rejection boundary for old production files is the output SCHEMA_VERSION bump. RunMetadata.configuration_hash digests it, so every old resume key differs. The kernel schema also moves.
- Both refusals are tested (§7.1, “Seam and output”). The concurrent cleanup’s output schema 21 and TDSE descriptor v5 are noted; numbers are taken at implementation.
R9, accepted.
- Scope is every route that already admits the field plasma: modal and radial (_over_nodes), Cartesian (_nonlinear_free), line free space (the same path with one \(y\) sample, covered) and planar (_nonlinear_slab). General-modal refusals are preserved (§2, §3).
- §7.4 tests that matrix, with plasma alone, with Kerr, with Raman, and with both.
- PlasmaResponse.evaluate keeps its meaning as the physical ungated \(P\), separate from the private assembled-source oracle (§5).
- The envelope comparison is qualified for shock=False (§4.2).
R10, resolved by the change. \(A_J\) adds no closure ramp. The two remaining scans keep their existing support logic, and the new pointwise rows \(gJ\) and \(g'J\) are written in full. The hot-path benchmark measures the column with its third scan removed. If B is pursued later, its support must be recomputed per actual row (§6).
R11, accepted.
- §7.3 replaces the fixed five-level ladder. It tightens until the tightest level’s residual is bounded from the last three levels by \(d_{\rm last}r/(1-r)\), assumes no order, and records atol, the maximum step and the save-plane policy.
- The window control uses continuous-transform normalisation, rebuilds the launch from the same pulse definition, and compares band energies.
- The near-infrared-only and attenuation statements are now labelled hypotheses, not pass conditions (§6, §7.3, §9).
R12, accepted.
- §7.3 records the maintainer’s full-length exception.
- Phase 4 says “CPU execution”, and hardware goes in the benchmark ledger; the execution handoff stays in benchmarks/sep27/README.md.
- The unnamed coordinator is replaced by the implementer’s integration responsibility at merge (§11).
- Q3’s roadmap recommendation is marked superseded by the TDSE certificate bug fix, cited as pending on refactor/remove-compat-paths (f6b76751).

Drafting probes for this revision, at solprop 2f30fb0c, with no propagation:

The work identity of §4.7 closed to \(3\times10^{-15}\) of the work for a plateau field and a stepped current on run_short’s lattice.
33 rows of \(N=2048\) took 104 µs per numpy.fft.rfft and 441 µs per band running integral.

Checks. git diff --check, codespell, uv run pytest tests/test_seps.py and uv run poe docs (strict, 8 workers).

2026-10-19: Part I revised in response to R1–R12. \(A_J\) replaces the in-column form as the definition, pending the maintainer’s confirmation of the reopened Q2. Status stays Draft.
2026-10-19: Q2 resolved by the maintainer: \(A_J\) is the definition and implementation; the integration rule’s narrower meaning accepted.