SEP 26 — An edge-free gated TDSE source: hold the current instead of switching it off#

SEP

26

Title

An edge-free gated TDSE source: hold the current instead of switching it off

Author

John Travers

Drafting agent

Opus 5.5 (2026-10-18)

Status

Final

Review status

Reviewed

Reviewers

John Travers (human); GPT-6 (AI)

Type

Change

Created

2026-10-18

Resolution

Accepted 2026-10-18 by the maintainer, after the human and AI reviews (R1–R13) and the drafting agent’s response (branch docs/sep-0026-accept)

Requires

SEP 12 (the tdse_native source and its time_collar gate); SEP 20 (the haCC response, the atomic window, L5, L6 and Q30)

Supersedes

—

Superseded-by

—

Implementation

phase 1: Opus 5.5 (AI) for John Travers, branch feature/sep-0026-phase-1 (solprop and solprop-kokkos); phase 2: Opus 5.5 (AI) for John Travers, branch feature/sep-0026-validation; phase 3 (promotion and the final gate): Opus 5.5 (AI) for John Travers, branch feature/sep-0026-phase-3

Manual pages

edge-free sections of the haCC helium response, TDSE in capillaries and TDSE configuration; SEP 26 paragraphs of resuming and differences from Luna

Part I — Proposal#

1. Abstract#

A gated TDSE source today is the transform of \(g(t)S(t)\). Here \(S=J-j_1\) is the atom’s nonlinear remainder current, and \(g\) is the Planck gate: the atomic window’s, or the grid’s temporal collar. When \(S\) is still nonzero in the trailing taper, the gate switches it off over a few femtoseconds. That switch-off is not an electron’s response, but it puts near-infrared content into the source.

For haCC at the run_short peak-power plane, removing that switch-off from both gates of the same 100 fs current reduces their 60-against-100 fs band difference by about 73% (§2). This SEP adds the source filter "gated_derivative". On the active band it replaces \(\mathcal F[gS]\) by \(\mathcal F[g\,\partial_tS]/(i\omega)\), evaluated as \(\mathcal F[gS]-\mathcal F[g'S]/(i\omega)\) with the analytic gate derivative \(g'\). The result is the transform of a current held at its taper-weighted value after the window instead of being switched off. A steady current then contributes nothing at \(\omega\neq0\), all time variation inside the window is kept, and the form has no free parameter.

The discrete form is admitted only on lattices that resolve the tapers, and each configuration carries a computed steady-current certificate (§4.4). The filter is opt-in until its validation passes and the maintainer has decided the remaining Q30 exception (§11).

Success means:

  • the analytic and fake-provider tests close within their derived budgets;

  • the archived L5 currents reproduce the supporting study’s same-current split;

  • the actual L5 comparison, an L6 segment, the SAE controls and the rate-plasma measurement are reported.

2. Motivation and scope#

The problem. SEP 20 L5 failed its 1% band tolerance. On-axis haCC(1) rows on the 60 fs window and on a separately driven 100 fs window differ by 4.61% (peak-power plane) and 1.43% (maximum-UV plane) over 90 nm–1.6 µm, almost all of it above 900 nm. The maintainer kept the 60 fs default with a recorded exception (SEP 20 Q30).

The supporting provider study is summarised, with its identities, in benchmarks/sep26/evidence/0133-summary.md (reference 5). It found three things:

  • The late current is box-converged. Changing the box from 80 to 240 bohr changes the 0.9–1.6 µm source by 0.13%. That is a box-to-box sensitivity: no box is an independently valid late-time reference after its own norm loss.

  • The escaped-electron term dominates what the gate removes. It accounts for 99.6% of the nir norm that the 60 fs gate removes from the 100 fs current: a near-steady drift of about \(3.3\times10^{-3}\) au at +50 fs.

  • The exact split (§4.1), applied to that same 100 fs current under both gates:

\[ \mathcal F[gS]=\frac{\mathcal F[gS']}{i\omega}+\frac{\mathcal F[g'S]}{i\omega}. \]

Same-current split (both gates applied to the 100 fs current; the 60 fs-driven current is not used):

case

60/100 fs gate difference, band: full → edge-free

nir (0.9–1.6 µm)

uv (90–200 nm)

edge term over own nir source, 60 / 100 fs

haCC(1) peak power

4.60% → 1.22%

12.5% → 3.33%

0.387% → 0.387%

10.6% / 11.0%

haCC(5) peak power

4.57% → 1.22%

12.4% → 3.33%

0.384% → 0.384%

10.6% / 10.9%

haCC(1) maximum UV

1.44% → 1.45%

4.36% → 4.42%

0.349% → 0.349%

1.7% / 2.2%

What this shows:

  • Peak power. Removing the edge term reduces the band difference by about 73%. Both gates carry an edge term of about 11% of their own nir source, so a longer window alone does not remove it.

  • Maximum UV. The drift is ten times smaller, and the difference does not shrink.

  • What remains is the window sensitivity left after the edge is removed. It is not a measured physical error, because there is no converged late-time reference.

  • The 90–200 nm difference does not change when the edge term is removed.

The edge term is a property of the gate, not of the model: any gated source whose current is nonzero in the trailing taper carries it, SAE included.

Who needs it. Every near-infrared result of a windowed or collared TDSE run: haCC’s run_short campaign, and SAE runs with source_filter="time_collar".

In scope.

  • The new source form on numpy_ref, with its discrete definition, admission condition, certificate and rounding budget.

  • The record, descriptor and schema changes.

  • Last-evaluation diagnostics.

  • Validation: the L5 replay, an L6 segment, the SAE controls and an offline rate-plasma measurement.

  • A separate, later default promotion.

Out of scope.

  • The window default (Q4).

  • An explicit drift-current model, collisions and recombination (§10).

  • The unfiltered source_filter="none" route (§4.5).

  • Any change to the rate model’s plasma route; only its offline measurement is in scope (§4.6, Q3).

  • Compiled backends: TDSE stays NumPy-only (SEP 12’s recorded exception).

3. Context#

The source assembly is solprop.backends.numpy_ref.tdse.

  • atomic_sources forms the unscaled remainder \(S=J-j_1\) for tdse_native. For source_filter="time_collar" it multiplies \(S\) by _source_gate(problem): the atomic window’s gate (AtomicWindowRecord.gate_values) when a window is set, else grids.t_window_fine. It then applies density and remainder_scale.

  • project_atomic_sources:

    • projects the nodal current with the modal weights and taper scale;

    • transforms it on the fine lattice and crops it to the coarse bins (passes.to_frequency);

    • multiplies by current_numerator over \(\operatorname{Re}n_{\rm eff}\). The numerator is \(-\text{crop\_scale}/(2\varepsilon_0c)\) on active_mask bins (\(\omega>\omega_{\min}/2\)) and zero elsewhere;

    • forms the embedded same-row diagnostic with an absolute rounding floor (_roundoff_floor).

    The caller applies the spectral collar grids.omega_window once. The last evaluation’s diagnostic is exposed as metadata()["tdse_last_evaluation"].

  • Gates are solprop.grid.planck_taper profiles: an atomic window \((t_-,t_-+w,t_+-w,t_+)\) with \(w=5\) fs by default (SEP 20 §4.6), or the grid’s temporal Collar. TDSEAtomicWindow checks only that \(0<w<T_a/2\), and resolve_atomic_window that at least four samples are retained. Neither checks that the tapers are resolved.

  • TDSEGasResponse.source_filter resolves to "time_collar" for haCC and for any windowed response, and to "none" for an unwindowed SAE response. Descriptors solprop.tdse-descriptor.v1–v3 record it (kernel contract, versions 18 and 27). An unwindowed SAE "time_collar" response uses v2, and v3 requires haCC or a window.

  • admit_tdse refuses a duplicate Kerr, plasma or ionisation response beside a TDSE response.

Manual pages: TDSE capillary, TDSE configuration and the haCC response. Roadmap: the haCC item after SEP 20, and the SAE width-sensitivity limit of the native qualification (recorded in the development archive).

4. Theory#

The transforms use solprop’s internal FFT convention, \(\tilde f(\omega)=\int f(t)e^{-i\omega t}\,dt\) (conventions.md §2), in which \(\mathcal F[f']=i\omega\mathcal F[f]\). In the optics convention, replace \(i\omega\) by \(-i\omega\) throughout. Four things are kept apart below: the continuous identity (§4.1), the discrete definition (§4.4), its alias error, which is qualified by admission and a certificate (§4.4), and floating-point error, which is budgeted (§4.5).

4.1 The continuous identity and the new source#

Let \(g\) be a Planck gate with breakpoints \(z_-<o_-\le o_+<z_+\), so that \(g=0\) outside \([z_-,z_+]\), and let \(S\) be the remainder current. Since \(gS\) vanishes at both ends of its support, the product rule gives the exact identity

\[ \mathcal F[g\,S'](\omega)=i\omega\,\mathcal F[gS](\omega)-\mathcal F[g'S](\omega). \]

Continuous definition. For \(\omega>0\), the edge-free source is

\[ \boxed{\;\hat S_{\rm ef}(\omega)=\frac{\mathcal F[g\,S'](\omega)}{i\omega} =\mathcal F[gS](\omega)-\frac{\mathcal F[g'S](\omega)}{i\omega}\;} \]

That is today’s source minus an edge term. Here \(g'\) is the analytic derivative of the gate, nonzero only strictly inside the tapers; \(S\) itself is never differentiated.

Hold interpretation. Let \(S_{\rm h}(t)=\int_{-\infty}^{t}g\,S'\,dt'\) and \(H=\int gS'\,dt\). Then \(\mathcal F[S_{\rm h}]=\mathcal F[gS']/(i\omega)+\pi H\delta(\omega)\), so \(\hat S_{\rm ef}=\mathcal F[S_{\rm h}]\) at every \(\omega\neq0\). The held current is:

  • zero before the window;

  • \(S(t)-c_-\) on the plateau, with \(c_-=\int g'_-S\,dt\) the leading-taper term (§4.3);

  • held at \(H\) after \(z_+\).

The form keeps every time variation of \(S\) inside the window. That includes the turn-on of the drift current during the pulse, its Brunel-type emission (reference 3). Only the switch-off is replaced by a hold. Three cases show what this means:

  • A current constant over the whole window has \(S'=0\), so \(\hat S_{\rm ef}=0\).

  • A current that rises inside the plateau and is then steady, for example \(S=H\Theta(t-t_c)\) with \(g(t_c)=1\), keeps its rise: \(\hat S_{\rm ef}=He^{-i\omega t_c}/(i\omega)\). Only the trailing switch-off is lost. For constant \(S\) on the trailing taper, \(\mathcal F[g'_+S]=S\,\mathcal F[g'_+]\) (with \(g'_+\le0\)), and that is exactly the term subtracted.

  • A current that varies inside the trailing taper keeps \(gS'\) there. Its variation after the window is dropped, which is the window’s finite-memory error; this SEP does not remove it.

4.2 The gate derivative and its evaluation#

On an edge with outer breakpoint \(a\) and inner \(b\), let \(\Delta=b-a\) (negative on a falling edge) and \(s=(x-a)/\Delta\in(0,1)\). The gate is \(g=1/(1+e^{\zeta})\) with \(\zeta=1/s-1/(1-s)\) (the _taper_edge exponent), and

\[ g'(x)=\frac{1}{\Delta}\,h(s),\qquad h(s)=\frac{1}{4\cosh^2(\zeta/2)}\left(\frac{1}{s^2}+\frac{1}{(1-s)^2}\right)>0 . \]

As an analytic function, \(g'\) is \(C^\infty\) and integrates to \(\pm1\) over each edge; it is zero outside the tapers and tends to zero at the breakpoints.

Evaluation.

  • \(s\) is formed first. If \(\min(s,1-s)\le1/740\), the value returned is the asymptotic zero, and no reciprocal is formed. In that region \(|h|<10^{-300}\), since \(\cosh^{-2}(\zeta/2)\le4e^{-738}\) and the rational factor is at most \(2\cdot740^2\).

  • Otherwise \(h\) is evaluated as written. All terms are finite: \(|\zeta|<740\) and \(s^{-2}\le740^2\).

  • Points with \(s\le0\) or \(s\ge1\), including the exact breakpoints, return zero.

The evaluated function therefore differs from the analytic \(g'\) by at most \(10^{-300}/|\Delta|\) in the cut-off region, and by a few rounding errors elsewhere (§4.5). It never forms an infinite intermediate. The existing gate’s clipping is not reused for the derivative. planck_taper_derivative sits beside planck_taper in solprop.grid, and both AtomicWindowRecord and Collar expose it.

4.3 The leading-taper term#

Let \(W_-=\Delta t\sum_n|g'_-(t_n)|\) be the sampled weight of the leading edge (1 up to sampling; T1 checks it). In raw DFT units (§4.4), on each active bin,

\[ \left|\frac{E_{-,k}}{i\omega_k}\right|\le\frac{W_-}{\Delta t\,\omega_k}\max_{t_n\in(z_-,o_-)}|S_n|, \qquad \left\|\frac{E_-}{i\omega}\right\|_{B}\le\frac{W_-}{\Delta t}\max_{\rm lead}|S_n|\Big(\sum_{k\in B}\omega_k^{-2}\Big)^{1/2} \]

over a band \(B\).

Why it is small.

  • haCC and windowed SAE start at \(t_s\ge z_-\) in the field-free ground state with \(A(t_s)=0\), so \(S(t_s)=0\).

  • The drive in the leading taper is below 1% of its peak at the run_short planes (SEP 20 §4.6), and \(S\) is at least cubic in it.

  • An unwindowed collared SAE run has its leading collar at the lattice start.

Subtracting both edges is the parameter-free form, and it holds the pre-window value, which is zero. The last-evaluation diagnostic records the ratio \(\rho_-\) and this bound (§5).

4.4 The discrete definition, admission and certificate#

Lattice and normalisation. The fine response lattice has \(N\) samples \(t_n=(n-N/2)\Delta t\), period \(T=N\Delta t\), \(\omega_k=2\pi k/T\), \(\omega_s=2\pi/\Delta t\) and \(\omega_N=\omega_s/2\). The raw DFT is \(x_k=\sum_n x_ne^{-2\pi ikn/N}\), with no \(\Delta t\) weight and its phase origin at \(n=0\). For a smooth \(x\) supported in one period, Poisson summation gives

\[ \Delta t\,x_k=(-1)^k\sum_m\mathcal F[x](\omega_k+m\omega_s), \qquad (-1)^k=e^{-i\omega_kT/2}. \]

Discrete definition. Let \(X_k=\mathrm{DFT}[g(t_n)S_n]_k\), today’s source, and \(E_k=\mathrm{DFT}[g'(t_n)S_n]_k\). The source is

\[ \hat S_{{\rm ef},k}=X_k-\frac{E_k}{i\omega_k}\quad\text{on active bins},\qquad 0\ \text{elsewhere}. \]

The common factor \((-1)^k\) cancels between the two terms. Bin \(k=0\) and every bin below \(\omega_{\min}/2\) lie outside active_mask, where current_numerator is already zero, so no further cutoff is needed. The largest weight is \(1/\omega_{\rm lo}\le2/\omega_{\min}\) at the lowest active bin. For 90 nm–1.6 µm, \(\omega_{\min}=1.18\times10^{15}\) rad/s (187 THz) and the active edge is at 94 THz (3.2 µm).

Alias error. If the samples are those of a smooth \(S\), the \(m=0\) terms reproduce the continuous identity. The \(m\ne0\) terms of \(X\) are today’s alias, unchanged. Those of \(E/(i\omega)\) are new, and they depend on how well the sampled taper resolves \(g'\). They are not small on every admitted lattice. With a taper shorter than a sample, no sample of \(g'\) need be nonzero, and the steady-current cancellation fails completely; a review probe gave ratio 1.0 on such a lattice.

The asymptotic decay of a Planck taper’s spectrum (super-algebraic, from its essential singularity) motivates resolving the taper well. It is not used as a bound. Instead:

  • Admission. gated_derivative requires each taper to span at least 128 lattice samples (\(w\ge128\,\Delta t\)), for the atomic window’s tapers or the collar’s. Translation refuses a narrower taper and names the minimum.

  • Steady-current certificate. At translation, for the actual lattice, gate and active band \(A\), compute

\[ \kappa_{\rm ss}=\frac{\big\|\mathrm{DFT}[g]-\mathrm{DFT}[g']/(i\omega)\big\|_A}{\|\mathrm{DFT}[g]\|_A}. \]

This is exactly the discrete source of \(S\equiv1\) relative to today’s. Translation refuses with a clear error if \(\kappa_{\rm ss}>10^{-10}\). The value is recorded in the run metadata and the last-evaluation diagnostic, not in the descriptor, so the identity never depends on a floating-point result. The certificate is one transform of a fixed gate per configuration.

What the certificate shows. A drafting-time probe of the window gate on the haCC lattice (\(N=16384\), \(\Delta t=6.254\times10^{-18}\) s, active band up to 80 nm) gave:

taper samples

10

20

40

56

64

96

128

800 (5 fs, the default)

\(\kappa_{\rm ss}\)

7e-5

4e-6

6e-9

9e-11

2e-11

3e-14

7e-16

6e-16

The SAE parent’s collar (about 2000 samples per taper) gives 2e-16. The 128-sample admission leaves margin below the threshold, and the certificate, not this table, is the qualification. It certifies the steady part exactly. For a general current, T3 checks the identity on resolved lattices against an independent spectral-derivative oracle, with refinement.

4.5 Rounding budget#

The budget is absolute and operand-based, in the raw FFT units of the existing diagnostics. It is carried through multiplication, projection, transform, crop, division and subtraction. Let \(c_F\varepsilon\log_2N\) be the normwise relative error bound of the forward transform (reference 2). It is assumed for the 2-, 3- and 5-smooth sizes solprop’s lattices use. T3 checks the constant at a power-of-two size and at a \(2^a3^b5^c\) size; any other size is flagged in the diagnostic as having an unqualified constant. By Cauchy–Schwarz, a perturbation \(\delta x\) moves any single bin by at most \(\sqrt N\|\delta x\|_2\).

The source error per active bin is bounded by the sum of:

  • Today’s budget. The existing _roundoff_floor, from the raw operands \(|g|(|J|+|j_1|)\), unchanged, so that shadow cancellation is accounted for.

  • The edge transform. \(c_F\varepsilon\log_2N\,\sqrt N\,\|g'(|J|+|j_1|)\|_2/\omega_{\rm lo}\), using the raw operands for the same reason.

  • Gate-derivative evaluation. A relative error of at most \((8+|\zeta_n|)\varepsilon\) at sample \(n\). The 8 is §4.2’s operation count; \(|\zeta|\) amplifies the exponent’s own rounding inside \(\cosh\). This gives \(\varepsilon\sqrt N\,\|(8+|\zeta|)\,g'(|J|+|j_1|)\|_2/\omega_{\rm lo}\).

  • The division and subtraction. \(2\varepsilon(|X_k|+|E_k|/\omega_k)\).

Each term is multiplied by the absolute projection weights, the density, the scale and \(|\text{current\_numerator}|\), as _roundoff_floor does today. Aliasing is not part of this budget; it is governed by §4.4.

The spectral-derivative oracle (T3) has its own budget. Its intermediate \(i\omega\,\mathrm{DFT}[S]\) reaches \(\omega_N\), so its terms are \(c_F\varepsilon\log_2N\sqrt N(\omega_N\|S\|_2+\|g\,\mathrm{IDFT}(i\omega\,\mathrm{DFT}S)\|_2)/\omega_{\rm lo}\). T3’s tolerance is the sum of the two budgets.

Numerical differentiation, for comparison. The supporting study’s centred differences have relative error about \((\omega\Delta t)^2/6\), which is \(3\times10^{-3}\) at 90 nm on the haCC lattice. Its split closed to 3.3–5.3×10⁻⁴ of the band norm. The identity form needs no derivative of \(S\).

4.6 Consistency with the existing pipeline#

  • Remainder and shadows. \(S\) is the unscaled \(J-j_1\), formed first as today, and the edge term uses the same \(S\). Density, remainder_scale, projection, taper scale, crop, current_numerator, \(1/\operatorname{Re}n_{\rm eff}\) and the spectral collar are linear, and act identically on both terms.

  • Window and padding. Outside the atomic window, \(S\), \(g\) and \(g'\) are zero on the lattice. The gate’s breakpoints are the record’s, never moved to samples, and they may be off-grid and asymmetric. The drive taper \(gE\) is unchanged.

  • The grid collar. For an unwindowed gated response, \(g\) is the collar, whose zeros lie inside or on the lattice, so §4.1 holds. The same admission and certificate apply.

  • source_filter="none". There is no gate, so this SEP does not touch it.

4.7 Every electron once, and the rate-plasma measurement#

  • The atomic current. \(S\) contains the atom’s whole nonlinear current: the bound part, the ionisation current and, for haCC, the escaped term. The form keeps all of its in-window variation. It removes only the gate’s switch-off, which is not an electron’s response, and it adds no current.

  • The linear current. \(j_1\) stays subtracted. It is not gated or held, and it is carried by the linear index (κ₁, SEP 20 §4.8).

  • Kerr and plasma. tdse_native contains its cubic. optmat_cubic and tdse_shadow admit only "none". admit_tdse still refuses a rate plasma beside a TDSE response.

The rate model. In continuous time its plasma current \(J_{\rm pl}=J_{\rm free}+J_{\rm loss}\) (Drude current plus the ionisation-loss current of passes.plasma_current) is constant after the pulse (reference 4). solprop’s discrete rate route, however, gates the polarisation \(P=\rho\int J_{\rm pl}\,dt\) with t_window_fine before its transform. Writing \(J=P'\), its source contains

\[ i\omega\,\mathcal F[gP]=\mathcal F[gJ]+\mathcal F[g'P], \]

whereas the held-current form of the same current is \(\mathcal F[gJ]-\mathcal F[g'J]/(i\omega)\). Holding \(P\) instead would hold a different quantity.

The offline measurement (Phase 2, reported only; the production rate route is unchanged). Recompute the rate model’s nodal \(J_{\rm pl}\) and \(P\) with passes.plasma_scan and passes.plasma_current on fixed saved fields: the saved planes of SEP 20’s L6 rate run, projected to its nodes. Use the run’s own running-integral rule (responses.plasma_integral) and density. Then form three things on the fine lattice, with the route’s crop and polarisation prefactor:

  • \(Q_{\rm route}=\mathrm{DFT}[gP]\);

  • \(Q_{\rm held}=(\mathrm{DFT}[gJ]-\mathrm{DFT}[g'J]/(i\omega))/(i\omega)\);

  • the closure residual \(\|i\omega\,\mathrm{DFT}[gP]-\mathrm{DFT}[gJ]-\mathrm{DFT}[g'P]\|\).

The closure residual measures how far the discrete running integral departs from \(P'=J\). Report \(\|Q_{\rm route}-Q_{\rm held}\|/\|Q_{\rm route}\|\) by band, the closure residual and the certificate of \(g\). This is a fixed-field source comparison, not a propagated trajectory.

5. Proposed interface#

  • TDSEGasResponse.source_filter gains "gated_derivative", the gate of "time_collar" with the edge term subtracted (§4.1).

    • It is opt-in through Phases 1 and 2. The omitted value resolves exactly as today: "time_collar" for haCC and for any windowed response, and "none" for an unwindowed SAE response.

    • Promotion to the haCC and windowed default is Phase 3, after the conditions of §11.

    • It is admitted for tdse_native with any gate: windowed SAE, windowed haCC, and unwindowed SAE or haCC on the grid collar.

    • It is refused for a taper below the §4.4 admission or a certificate above \(10^{-10}\).

  • "time_collar" keeps its formula and code path. Its descriptor bytes, identities and resume keys are unchanged exactly. Recomputing its sources is required to agree with recorded values within the §4.5 budget; a same-build bitwise repeat may be recorded as evidence, but is not a requirement (SEP 25).

  • Last-evaluation diagnostic. tdse_last_evaluation["projection"] gains an edge entry for a gated response, with the same projection, spectral collar and raw FFT modal units as the existing entries. It covers two bands, the grid’s active_mask and inband_mask, and records:

    • the norms of the truncating source \(c\,X\), the edge term \(c\,E/(i\omega)\) and the corrected source \(c\,\hat S_{\rm ef}\), where \(c\) is the current factor times the collar;

    • the ratios edge/truncating and edge/corrected;

    • the §4.5 floor of each norm;

    • \(\rho_-\), which is \(\max_{\text{rows, lead samples}}|S|\) over \(\max_{\text{rows, window}}|S|\);

    • the band bound of §4.3 and \(\kappa_{\rm ss}\) (also in the run metadata).

    A ratio whose denominator is at or below its floor is None, with a reason string (“denominator below rounding floor”); \(\rho_-\) is None with “zero remainder” when \(\max|S|=0\). The admission excludes an empty sampled taper. For "time_collar" the same entry is computed, so that the two forms can be compared on one evaluation. These are diagnostics, never gates. Per-call history is not stored: per-call edge analysis is an offline replay of raw archives (§7.3, §7.5).

response = replace(hacc_preset(1), source_filter="gated_derivative")  # opt-in (Phases 1-2)

6. Implementation strategies#

Where. In solprop.grid, planck_taper_derivative (§4.2). In numpy_ref, atomic_sources also returns the gated edge array \(g'S\), scaled the same way, and project_atomic_sources:

  • projects, transforms and crops the edge array as it does the current;

  • divides it by \(i\omega\) on active_mask bins;

  • subtracts it before current_numerator;

  • extends the embedded coarse projection, _roundoff_floor and the edge diagnostic.

The admission and certificate live in translation, beside resolve_atomic_window and the collar resolution.

Options.

  • (a) Frequency-domain subtraction, as above (recommended). It costs one extra fine FFT of one modal column per call (two with the embedded map), and \(S\) is never differentiated.

  • (b) A time-domain held current. This needs a cumulative quadrature and a periodic ramp. It is less accurate and not simpler.

  • (c) Differentiate \(S\). Centred differences have an \(\mathcal O((\omega\Delta t)^2)\) error, and a spectral derivative amplifies rounding by \(\omega_N/\omega\). Both serve only as test oracles.

Records and seam.

  • Descriptor v4. "gated_derivative" is recorded in a new solprop.tdse-descriptor.v4. It has v3’s layout, with model and atomic_window (which may be null), plus the new value, so it covers unwindowed SAE on the collar as well as windowed SAE and haCC.

  • v1–v3 stay unchanged byte for byte, and each refuses "gated_derivative". _validate_filter_schema admits v4 only with that value.

  • SCHEMA_VERSION goes from 29 to 30 (kernel-contract row 30). No field moves, and a version-29 consumer refuses the problem. solprop-kokkos accepts version 30 and keeps refusing TDSE by name.

  • Output schema. No bump: source_filter is already in the resume key whenever it is not "none", and the last-evaluation metadata is not part of the archive schema. The κ₁ cache is keyed by basis identity and is unaffected.

Backends. NumPy only, as for every TDSE route; numba_cpu and Kokkos refuse TDSE by name. There is no GPU work.

Cost. Each call adds one fine FFT, one nodal array (\(n_{\rm nodes}\times N\)) and the extra diagnostic work. Phase 1 measures this with a bounded hot-path benchmark and memory ledger (§7.2) before calling it negligible.

Dependencies. None.

7. Verification and validation#

Equality is exact only for descriptor bytes, identities, resume keys, losslessly copied arrays, masks and call accounting. Matched source recomputations use the §4.5 budgets. Propagation and resume comparisons use accuracy envelopes (SEP 25). Each tolerance is derived before evaluation and stated beside its assertion. “Raw units” means the unweighted DFT of §4.4. Per-bin bounds hold on every active bin; band norms are 2-norms over the stated bins.

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

  • T1 The gate derivative.

    • Against the independent form \(-g(1-g)\,d\zeta/dx\), built from planck_taper’s own \(g\) where \(0.01<g<0.99\) (so \(g(1-g)\) is not cancellation-limited), the tolerance per sample is the §4.5 evaluation budget plus the gate’s own \((4+|\zeta|)\varepsilon\).

    • Against a fourth-order centred difference of planck_taper at two spacings, the error must fall by \(16\pm2\times\) between them. That is a convergence check, not a bound from a sampled derivative.

    • The sampled weight \(\Delta t\sum|g'|\) of each edge is reported against 1 at 64, 128 and 800 samples per taper. The 128- and 800-sample cases agree within the certificate threshold. No exponential-convergence claim is made.

    • Values are finite and non-negative in \(h\) on both sides of every breakpoint, at the exact breakpoints, at nextafter neighbours and at the §4.2 cut-off. Widths run from \(10^{-30}\) s to 1 s, and include SI femtosecond scales. Warnings are errors.

  • T2 Steady and non-steady currents. The tolerance is the §4.5 budget (absolute, raw units) plus \(\kappa_{\rm ss}\|X\|_A\).

    • (a) \(S\equiv H\) on the whole lattice: \(\hat S_{\rm ef}=0\), while the truncating form is nonzero.

    • (b) \(S=0\): both forms give exactly zero.

    • (c) An interior smooth rise followed by a steady tail keeps the analytic rise spectrum (T4’s oracle) and differs from (a)’s truncating source.

    • (d) A current with a slope through the trailing taper keeps \(gS'\) there: the result equals the spectral-derivative oracle (T3).

  • T3 Identity against a spectral derivative, with refinement. Use a band-limited synthetic \(S\): a smooth rise, a steady drift, a Gaussian-envelope oscillation and a slope through the trailing taper. Compare \(\mathrm{DFT}[g\cdot\mathrm{IDFT}(i\omega\,\mathrm{DFT}S)]/(i\omega)\) with §4.4’s form on active bins. The tolerance is the sum of the two §4.5 budgets. Cases:

    • asymmetric and off-grid breakpoints;

    • tapers at the admission limit and at 2×, 4× and 8× it, where agreement must not worsen as the taper resolution grows;

    • the highest retained frequency inside the band;

    • a power-of-two \(N\) and a \(2^a3^b5^c\) \(N\).

    The same synthetic case also records the centred-difference split’s \(\mathcal O((\omega\Delta t)^2)\) miss.

  • T4 The analytic held step. Take \(S=\tfrac H2[1+\operatorname{erf}((t-t_c)/\sigma)]\), with the rise on the plateau at distance \(d\) from both plateau edges. In raw units, the oracle on active bins is

\[ \frac{(-1)^k}{\Delta t}\,\frac{He^{-\omega_k^2\sigma^2/4}e^{-i\omega_kt_c}}{i\omega_k}. \]

The tolerance, per bin, is the sum of three terms:

  • the rise’s mass outside the plateau, \(H\operatorname{erfc}(d/\sigma)/(\Delta t\,\omega_k)\);

  • the alias \(He^{-(\omega_N\sigma)^2/4}/(\Delta t\,\omega_k)\);

  • the §4.5 budget.

  • T5 A known oscillation is unchanged. Take \(S=A\cos(\omega_0t)e^{-(t/\tau)^2}\) inside the plateau, with \(t_p\) the distance from its centre to the nearest taper. Per bin in raw units, the two forms differ by at most \(A e^{-(t_p/\tau)^2}W/(\Delta t\,\omega_k)\) plus the budget. Here \(W=\Delta t\sum|g'|\).

  • T6 The leading taper. An \(S\) with known values in the leading taper gives the recorded \(\rho_-\) and satisfies both §4.3 bounds.

  • T7 Admission and certificate.

    • The review’s counterexample is refused: \(N=16384\), \(\Delta t=6.254\times10^{-18}\) s, an \(80\Delta t\) window and a \(\Delta t/4\) taper.

    • So are a 127-sample taper and an under-resolved collar.

    • A 128-sample taper is admitted, with \(\kappa_{\rm ss}\le10^{-10}\) recorded in the metadata.

    • The recorded certificate agrees with an independent recomputation in the test within \(10\varepsilon\log_2N\) relative. It does not enter the descriptor or the identity.

7.2 Fake-provider tests and the benchmark (default suite)#

These use the existing SAE and haCC fake providers.

  • A fake post-pulse drift. A fake current that rises inside the plateau and then stays steady through the trailing taper. Under "gated_derivative" the source keeps the rise spectrum and has no switch-off: it matches T2(c) within the budget, and the edge diagnostic’s edge/truncating ratio equals T2(c)’s. Under "time_collar" it matches today’s formula within the budget.

  • Unchanged routes. For "time_collar" and "none", descriptors, identities and resume keys are exactly unchanged, and their sources match stored witnesses (with provenance) within the §4.5 budget.

  • Resolution and refusals.

    • The omitted filter resolves as today (Phases 1–2).

    • "gated_derivative" is refused with optmat_cubic or tdse_shadow, and with an under-resolved taper.

    • A window with "none" is refused.

    • v1–v3 refuse the new value, v4 refuses the old values, and a version-29 consumer refuses a version-30 problem.

  • Identity and resume. Write a v4 descriptor and round-trip and resume it for each of: unwindowed SAE on the collar, windowed SAE, windowed haCC and unwindowed haCC. A "time_collar" run refuses to resume under the new filter, and the reverse. Every historical v1–v3 case still validates.

  • Every electron once. The source equals \(\mathrm{DFT}[gS]-\mathrm{DFT}[g'S]/(i\omega)\) with the same \(S\), density and scale, within the budget; no term enters twice, and \(j_1\) never enters.

  • Diagnostics and calls.

    • The edge entry holds the norms, floors and None-with-reason cases of §5.

    • Its values change with the last evaluation and keep no history.

    • The atomic ledger is exactly the same as under "time_collar": no extra atomic calls.

  • Hot-path benchmark (benchmark lane, bounded). Time atomic_sources plus project_atomic_sources under both filters on a fake-provider problem at the haCC lattice size, with warmup and interleaved repeats, reporting the distribution. Record the extra arrays in the memory ledger.

7.3 The L5 replay (offline, no new atomic calls)#

benchmarks/sep20/window.py analyse gains a filter option. It reads the archived Phase 3 currents, hashed in benchmarks/sep20/evidence/manifest.json:

  • the L4 candidate rows, which are the separately driven 60 fs current;

  • the 100 fs reference.

It writes a derived analysis record. That record keeps each raw current’s descriptor and identity, and names the replay filter and analysis policy; the raw atomic evaluation is not relabelled. Three comparisons are reported separately.

  • (i) Recorded figures. "time_collar" reproduces the recorded L5 figures within the §4.5 budget, propagated through the ratio.

  • (ii) Same-current split, the anchor. Both gates are applied to the 100 fs current. The metric is the dimensionless ratio \(\delta=\|F_{100}-F_{60}\|_B/\|F_{100}\|_B\), with \(F\) the edge-free source and \(B\) the band.

    • The expected values are those of §2: 1.22% band and 3.33% nir at peak power, and 1.45% and 4.42% at maximum UV.

    • The supporting figures used centred differences. Their split residual is \(r\le5.4\times10^{-4}\) of the band norm (3.3–5.3×10⁻⁴ measured). Let \(e_g\) be the replay’s measured difference between the analytic and the centred-difference \(g'\) edge terms, relative to the band norm.

    • The requirement is \(|\delta-\delta_{\rm ref}|\le(2+\delta)(r+e_g)\) in the band. That is about \(1.1\times10^{-3}\) (0.11 percentage points) for \(e_g\le10^{-5}\). nir and uv are reported.

  • (iii) Actual L5 with the new filter. The separately driven 60 fs current is compared with the 100 fs current, both edge-free. This is reported, with no tolerance borrowed from (ii).

    • Expectation: close to (ii), plus the startup and drive difference that L5’s common-window comparison measured (0.036–0.046% band). The actual peak-power UV figure is about 0.405%, against 0.387% for the same-current split.

    • It remains a recorded failure of L5’s 1% band tolerance if it exceeds 1%.

7.4 L6 segment (CPU nodes; no GPU rental needed)#

From SEP 20’s L6 launch and pump, with haCC(1) fast at rtol=1e-2, 32-thread runs, the provider pin of the existing CPU segment record and the full TDSE archive. Execution details are in benchmarks/sep26/README.md.

Runs.

  • (a) "gated_derivative" at 60 fs, about 1 h.

  • (b) "gated_derivative" at 100 fs, about 1.7 h.

  • (c) "time_collar" at 100 fs, about 1.7 h.

  • The existing "time_collar" 60 fs CPU segment is the current form. If the pin has moved since it was recorded, rerun it.

These total 140–175 core-hours.

Metric. At the common endpoint, compare complex spectra by band (full, 90–200 nm, 0.9–1.2 µm and 1.2–1.6 µm) as \(\|a-b\|_B/\|b\|_B\), with \(b\) the current-form endpoint (as in SEP 20 L6). The floor is the two runs’ interior-restart agreement on the same segment, the 0.073% full-band fresh-history figure of SEP 20 L6. A difference below that floor is reported as unresolved.

Gates. L6’s mechanical and sensible-optics gates.

Investigation threshold (maintainer, R6). A 90–200 nm difference between (a) and the current form above 0.5% is investigated, not failed. The cause is named by three checks:

  • matched-input sources at a few saved planes, which separate the direct source change from the propagated one;

  • pinned scientific inputs, so that only the filter differs;

  • a tighter-rtol control of the pair (rtol=3e-3), which separates integration error from propagated physical sensitivity.

rtol=1e-2 is a local tolerance and does not certify a 0.5% global comparison.

Reported.

  • Every pair’s band differences: (a) against the current form, (a) against (b), and (c) against the current form.

  • Energy, peak power, calls and wall time.

Expectation, not a result. The (a)–(b) difference above 900 nm should be well below the current form’s 60-against-100 fs difference.

7.5 SAE controls#

  • L6 per-call archives (the sae-full and sae-window segment archives, hashed in the manifest).

    • Recompute each call’s source under both forms offline, with the certificate and the T3 oracle on the real currents.

    • Report the edge fractions by band.

    • The first calls share the launch field, which gives a single-field collar-against-window comparison.

  • SEP 12’s saved-current width study. Replay benchmarks/sep12/evidence/native-qualification/source-filter-study.py with the new form, on its hashed external archive (retrieval in benchmarks/sep26/README.md). Whether the 0.75/1.25 width sensitivities fall is a hypothesis to test.

  • One SAE propagation. Rerun the L6 sae-window segment (60 fs, 75–93 core-hours) with the new form, and report it as in §7.4.

Pass for the SAE controls, which the SAE research record needs before it moves:

  • the replays reproduce their recorded "time_collar" figures within budget;

  • every certificate is at most \(10^{-10}\);

  • the SAE rerun passes L6’s mechanical and sensible-optics gates;

  • the edge-free width sensitivity is no larger than the truncating one in any band.

If any of these fails, the SAE research record stays on "time_collar" and the finding is reported. Recorded SAE results keep their identities and are not relabelled; unwindowed SAE keeps "none" as its default.

8. Documentation#

  • haCC response. The identity, the hold interpretation, admission and certificate, the opt-in, and, after Phase 2, Q30 restated from §7’s measured values.

  • TDSE capillary. The derivation and a worked comparison of the two forms on a synthetic drift, as a tested guide block.

  • TDSE configuration, resuming, kernel contract (row 30, descriptor v4) and the Luna differences source-filter paragraph.

  • Docstrings, the changelog citing SEP 26, the roadmap, and benchmarks/sep26/.

9. Compatibility and impact#

  • Phases 1–2 change no default. Nothing changes unless a response names "gated_derivative". Opting in changes the near-infrared source by about 11% of the 0.9–1.6 µm source at the peak-power plane and about 2% at maximum UV (same-current edge fractions, §2).

  • Phase 3 makes it the haCC and windowed default, if §11’s conditions hold. New runs then get v4 descriptors; old files resume with their recorded filter.

  • Seam. SCHEMA_VERSION 30 and descriptor v4. No output-schema bump (§6). "time_collar" keeps its formula, path, identities and resume keys exactly, and its numbers within the rounding budgets (SEP 25).

  • Q30 (expectation, not a result). The exception should reduce to the window sensitivity that remains without the edge term: about 1.2% of the band and 3.3% of the nir at peak power, and 1.4–1.5% and 4.4% at maximum UV, in the same-current split. The actual L5 figures (§7.3 iii) decide it.

  • Luna has no TDSE source; only the existing paragraph is updated.

10. Alternatives considered#

  • Option 2, an explicit drift current. The provider would deliver the escaped term’s steady part, and solprop would propagate it ungated. It is rejected for now:

    • it needs work in both projects;

    • the steady part is an arbitrary choice with its own turn-on edge;

    • for a collisionless model it gives nothing that the hold does not already give at \(\omega\neq0\).

    Reconsider it for collisions or recombination, where the drift decays and its decay radiates.

  • A longer default window. About 1.67× the steps, and it still carries an edge term of the same size.

  • Fitting and subtracting a steady part. Arbitrary, and it has its own edge.

  • Differentiating \(S\) numerically. Less accurate or rounding-amplified (§4.5); used only as oracles.

  • Band-first filtering. A spectral mask cannot remove a step’s in-band content.

  • Redefining "time_collar" in place. That would invalidate recorded identities.

11. Phases and milestones#

  1. Phase 1, implementation, opt-in (feature/sep-0026-phase-1, plus a schema-acceptance commit in solprop-kokkos).

    • §4.2’s derivative, the edge-free assembly, the admission and certificate, and the last-evaluation diagnostic.

    • Descriptor v4 and SCHEMA_VERSION 30.

    • T1–T7, §7.2’s tests and the bounded hot-path benchmark with its memory ledger.

    • Coverage of the touched code, the manual pages and the changelog.

    • Gate: the per-phase checks plus the full gate, because the schema changes.

  2. Phase 2, validation (feature/sep-0026-validation).

    • §7.3 offline (minutes) and §7.5’s offline SAE controls.

    • §7.4’s L6 segments and the SAE rerun on CPU nodes, with no GPU rental (about 215–270 core-hours in all).

    • §4.7’s offline rate-plasma measurement.

    • Each result is reported in Part II.

  3. Phase 3, promotion and final gate.

    • If §11’s promotion conditions hold, make "gated_derivative" the haCC and windowed default, and move the SAE record if §7.5 passed.

    • Restate Q30 in the manual and the roadmap (SEP 20 is Final and takes only an erratum note).

    • Run the end-of-SEP full gate, the slow lane, the coverage lane and the Kokkos CPU lane (schema acceptance and refusal).

Blocking conditions.

  • Acceptance: this revision’s response to R1–R13 is reviewed, and Q4 is recorded as deferred to Phase 3.

  • Default promotion: Phase 2 has passed, meaning T1–T7, §7.3 (i)–(ii), §7.4’s gates with every investigation closed, and a certificate of at most \(10^{-10}\) on every replayed configuration. The maintainer has also decided Q4 and Q30’s remaining exception from §7.3 (iii) and §7.4.

  • Final: every phase is merged and reported, the end-of-SEP lanes pass, and the manual is updated.

12. Open questions#

Resolved by the maintainer on 2026-10-18:

  • Q1 Naming and the old form. The new value is "gated_derivative". "time_collar" stays as the truncating comparison form, with its formula and code path. Its descriptor bytes, identities and resume keys are exactly unchanged, and its numerical recomputation agrees within rounding budgets (SEP 25).

  • Q2 SAE. The SAE research record moves to the new form only if §7.5’s pass conditions hold; otherwise it stays on "time_collar". One SAE L6 rerun is funded. Unwindowed SAE keeps "none".

  • Q3 The rate plasma. This SEP does not change it. Phase 2 measures it offline (§4.7), and the measured number decides whether a follow-up SEP is needed.

  • Q5 L6 scope. All three segments.

  • Q6 The SEP 12 archive. Located. It is identified by hash, and its retrieval is described in benchmarks/sep26/README.md.

Deferred to Phase 3:

  • Q4 Q30’s exception and the default. Whether to promote "gated_derivative" (R11), whether the 60 fs default window stays, and how Q30’s remaining exception is restated. The maintainer decides from §7.3 (iii) and §7.4.

13. References#

  1. 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

  2. N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM (2002), ch. 24 (the FFT’s normwise rounding error). doi:10.1137/1.9780898718027

  3. 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

  4. 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

  5. SEP 26 evidence summary of the provider’s late-current study, benchmarks/sep26/evidence/0133-summary.md in the solprop repository: its equations, metric, tables, limitations and the SHA-256 identities of its records. For provenance, it is Pyroraptor report 0133 and its taper-edge addendum (2026-10-18).

Part II — Implementation record#

Phase 1 report — implementation, opt-in (2026-10-18)#

Scope. Phase 1 of §11: §4.2’s derivative, the edge-free assembly, the admission and certificate, the last-evaluation diagnostic, descriptor v4 and kernel schema 30, T1–T7, §7.2’s tests, the bounded hot-path benchmark and the manual. Implementer: Opus 5.5 (AI) for John Travers. Branches feature/sep-0026-phase-1 in solprop (from e1a8274; main at b20697f, SEP 22 phase 1, merged in at d24ca70) and in solprop-kokkos (from 2c8b9ca: 3462b62, de2a805). Merge commits are the coordinating agent’s to record on integration.

Delivered.

  • src/solprop/grid.py: planck_taper_derivative, planck_taper_derivative_error, Collar.window_derivative.

  • src/solprop/nonlinear/tdse.py: source_filter="gated_derivative", opt-in; an omitted filter resolves as before; a window refuses only "none".

  • src/solprop/translate/tdse.py: descriptor v4 (version 3’s layout, recording only gated_derivative; versions 1–3 refuse it and version 4 refuses every other filter); the 128-sample admission (GATED_DERIVATIVE_TAPER_SAMPLES), checked when a descriptor is formed and on every reconstructed record; AtomicWindowRecord.gate_derivative, TDSEDescriptor.source_breakpoints, source_gate_derivative and source_gate_derivative_error.

  • src/solprop/translate/tdse_source.py: steady_current_certificate, cached per configuration and band, and STEADY_CURRENT_CERTIFICATE_LIMIT.

  • src/solprop/translate/problem.py: SCHEMA_VERSION 30; KernelProblem validation refuses a certificate above 1e-10 and checks the recorded collar for both gated filters.

  • src/solprop/backends/numpy_ref/tdse.py: atomic_edge_source (the scaled g'S), the edge-free projection project_atomic_sources(..., edge=), the §4.5 budget (FFT_ERROR_CONSTANT) and the edge diagnostic. backend.py passes the edge array and records metadata()["tdse_source"]; backends/memory.py adds the tdse_edge_sources ledger row.

  • Tests: tests/test_planck_taper_derivative.py (T1); tests/backends/test_tdse_edge_free.py (T2–T7, every electron once, the seam’s refusals); tests/backends/test_tdse_unchanged_routes.py with the witness tests/data/sep26_unchanged_sources.{npz,json}, written from e1a8274 by tests/data/generate_sep26_unchanged_sources.py; tests/translate/test_tdse_descriptor_v4.py; tests/driver/test_tdse_gated_derivative_run.py (the manufactured SAE binding and the fake haCC provider); tests/edge_source_cases.py; the schema literals of five existing modules.

  • Benchmark: benchmarks/sep26/hot_path.py, its record benchmarks/sep26/evidence/hot-path.json, the ledger entry in docs/dev/benchmarks.md, and how to run it in benchmarks/sep26/README.md.

  • Manual: docs/guide/media/tdse-hacc.md (identity, hold, admission, certificates, diagnostic, opt-in), tdse-capillary.md (the derivation and a tested worked comparison on a synthetic drift), tdse-configuration.md (the filter, its refusals, a tested example), docs/guide/results/resuming.md, docs/guide/workflows/luna.md, docs/dev/kernel-contract.md (row 30, descriptor v4; its stale “saved-output schema: 19” header, left by SEP 22, now reads 20). Each page produced or extended ends with a link to this SEP. CHANGELOG.md.

  • solprop-kokkos: 3462b62 accepts schema 30 and keeps refusing TDSE by name (its refusal test now includes gated_derivative); de2a805 updates its last schema literal.

Deviations from Part I, and why.

  1. Evaluating \(g'\) (§4.2). \(1/(4\cosh^2(\zeta/2))\) is evaluated as \(e/(1+e)^2\) with \(e=e^{-|\zeta|}\): written as \(\cosh^2\) it overflows for \(|\zeta|>709.8\), inside the admitted \(|\zeta|<740\). \(s\) and \(r=1-s\) are both formed from \(x\), as \((x-a)/\Delta\) and \((b-x)/\Delta\), so that neither comes from a cancellation. The analytic function, the cut-off at \(\min(s,1-s)\le1/740\) and the zeros at and beyond the breakpoints are as specified.

  2. The derivative’s rounding bound (§4.5). A first-order count of that evaluation gives \((17+2.5|\zeta|)\varepsilon\), used as \((18+3|\zeta|)\varepsilon\); §4.5’s \((8+|\zeta|)\varepsilon\) came from a shorter count. Near a breakpoint \(\zeta\approx1/s\) carries \(s\)’s own relative rounding times \(|\zeta|\), which no evaluation avoids.

  3. T1’s oracle tolerance. The closed form \(-g(1-g)\,d\zeta/dx\) takes \(1-g\) from a second gate evaluation at the mirrored point \(a+b-x\) (exact on the test’s dyadic lattice): forming \(1-g\) from \(g\) amplifies its rounding up to 100-fold where \(0.01<g<0.99\). The oracle’s budget is derived as \((20+4|\zeta|)\varepsilon\) (two gate evaluations at \((8+2|\zeta|)\varepsilon\) each by the same count, \(d\zeta/dx\) and the products), in place of “the gate’s own \((4+|\zeta|)\varepsilon\)”.

  4. The edge-transform budget term (§4.5) also carries the rounding before the transform, \((n_{\rm nodes}+6)u\) of the projected raw operands (the subtraction \(J-j_1\), the products by \(g'\), density and scale, the taper division and the node reduction), which §4.5 does not list. \(c_F=8\) (FFT_ERROR_CONSTANT) starts from Higham’s radix-2 constant, about 3.4, with room for radix-3 and radix-5 levels; T3 measured 0.092 and 0.106.

  5. T4’s tolerance adds a fourth term, the per-bin steady-current residual \(H|r_k|\) (the discrete edge-free source of \(S\equiv H\)), for the trailing taper’s alias, which §4.4 governs and the three printed terms omit. It is about \(10^{-16}\) of the source.

  6. T7’s agreement. At admitted resolutions \(\kappa_{\rm ss}\) is rounding (\(3.9\times10^{-16}\), against \(5.2\times10^{-16}\) from an independent numpy.fft recomputation), so two transforms cannot agree relatively. The test reads “within \(10\varepsilon\log_2N\) relative” as absolute agreement in \(\kappa_{\rm ss}\), which is itself relative to \(\lVert\mathrm{DFT}[g]\rVert_A\): \(1.3\times10^{-16}\) against \(2.7\times10^{-14}\). On an alias-dominated time_collar gate (20 samples, \(\kappa_{\rm ss}=8.3\times10^{-6}\)) the two agree to \(4.8\times10^{-17}\) absolutely but only \(5.8\times10^{-12}\) relatively: the numerator cancels terms the size of the denominator, so relative agreement is about \(\varepsilon/\kappa_{\rm ss}\) at best.

  7. Interface (§6). atomic_sources keeps its two-tuple; the edge array comes from the new atomic_edge_source, with the same \(S\), density and scale, instead of a third return value. project_atomic_sources takes edge= keyword-only: required for gated_derivative, refused for "none" and optional for "time_collar", which then has no edge entry. The backend always passes it.

  8. The certificate is also computed for "time_collar" and recorded with certificate_enforced: false, so that the two forms compare on one evaluation, as §5 asks of the diagnostic. It is never enforced there.

Decisions left open by Part I. The run-metadata record is metadata()["tdse_source"]: the filter, the certificate (None if the band has no truncating content), its limit, whether it is enforced and the gate’s breakpoints. The edge entry holds source_filter, units, steady_current_certificate, fft_constant_qualified (a 2-, 3- and 5-smooth size); per band (active, inband) the bin count, the three norms, their floors and both ratios with their reasons (the corrected floor is the truncating floor plus the edge floor, which includes the division and subtraction); and leading_taper (ratio, reason, lead_samples, sampled_weight and the per-band bound). The band bound uses the absolute projection of \(|\rho sS|\), an upper bound on the modal current. The §4.5 terms use \(1/\omega_{\rm lo}\) on every bin, as printed. No ADR: nothing here outlasts the SEP’s own record.

Verification. On an 18-core Apple M5 Max (macOS 27, arm64), Python 3.14.5, NumPy 2.5.2, SciPy 1.18.1; xdist bounded to 8 workers, one thread per worker (SOLPROP_KOKKOS_THREADS=1, OMP_PROC_BIND=false); solprop-kokkos built from de2a805 against d24ca70 (CPU, OpenMP, local profile, Release) and installed in the worktree environment. The machine was shared (load average 5–8) throughout.

Gate, on cd16e74 (the merge d24ca70 and five test, figure and docstring commits; the SEP text was uncommitted):

  • uv run poe check: green (ruff, basedpyright, numpydoc, codespell, reuse; the default lane 15830 passed, 29 skipped; strict Sphinx). Its first run found the one_run structure figure stale at schema 29, which poe figures redrew.

  • uv run poe cov: 100% of statements and branches (41097 and 12438). Its first run missed one branch, the empty leading taper, which a test now covers.

  • uv run poe test-slow: the functional slow tests 147 passed, 1 skipped (on d24ca70 plus the figure). The serial statistics-cost check then measured 27.2% against its 25% gate at a load average near 8; rerun alone at load 9–11 it passed. Nothing in this phase touches the statistics channel, so this was machine load.

  • Kokkos-selected default lane (SOLPROP_TEST_BACKEND=kokkos uv run poe test): 15829 passed, 29 skipped.

  • solprop-kokkos: tests/test_api_contract.py and tests/test_backend.py, then its whole Python suite, 319 passed and 5 skipped, against the editable build.

Results, each tolerance derived before the comparison and stated beside its assertion:

test

claim

measured

T1 closed form

within the derivative’s and the oracle’s bounds

at most \(2.0\varepsilon\) relative, against at least \(38\varepsilon\)

T1 fourth-order difference

the error falls \(16\pm2\)-fold

15.96 (\(2.46\times10^{-6}\to1.54\times10^{-7}\))

T1 edge weight

128 and 800 samples agree within \(10^{-10}\)

64 samples \(1+2.9\times10^{-11}\); 128 and 800 samples 1 to rounding

T1 finiteness

finite, \(h\ge0\), exact zeros, no floating-point exception

widths \(10^{-30}\) s to 1 s; every breakpoint, nextafter and cut-off neighbour

T2 (a) steady

\(\lVert\hat S_{\rm ef}\rVert_A\le\) budget \(+\kappa_{\rm ss}\lVert X\rVert_A\)

\(6\times10^{-21}\) and \(7\times10^{-21}\) against \(2.6\times10^{-17}\) and \(4.3\times10^{-17}\) (window, collar); truncating \(1.4\times10^{-5}\) and \(9.3\times10^{-7}\)

T2 (b) zero

exactly zero under both forms

exact; ratios None with their reasons

T2 (c), T4 held step

per bin within the four terms

at most \(10^{-3}\) of the tolerance; \(3.7\times10^{-16}\) relative

T3 identity

per bin within both budgets; no worse with resolution

at most \(4.5\times10^{-6}\) of the tolerance; \(3.5\)–\(8.3\times10^{-16}\) (4096) and \(4.2\)–\(5.1\times10^{-16}\) (3840) relative, tapers of 128–1024 samples

T3 FFT constant

observed \(\le c_F=8\)

0.092 (4096) and 0.106 (3840 \(=2^8\cdot3\cdot5\))

T3 centred difference

its miss is recorded

\(1.7\times10^{-3}\) against \((\omega_{\max}\Delta t)^2/6=3.4\times10^{-3}\)

T5 oscillation

the forms differ by at most \(Ae^{-(t_p/\tau)^2}W/(\Delta t\,\omega_k)\) plus the budget

at most \(1.2\times10^{-3}\) of it

T6 leading taper

the recorded \(\rho_-\); both §4.3 bounds

\(\rho_-=0.0200\) to one rounding; 0.58 of the per-bin bound, 0.47 and 0.33 of the band bounds

T7 admission

the counterexample, 127 samples and a 121-sample collar refused; 128 admitted

as stated; \(\kappa_{\rm ss}=3.9\times10^{-16}\) at 128 samples, in tdse_source

T7 certificate

an admitted taper with a failing certificate is refused

20 fs (400-sample) taper, 90–200 nm, \(q=1\): \(1.8\times10^{-10}\), refused

§7.2 unchanged routes

identities exact; sources within twice the recorded floor

exact; within it, for none, the collar and the window from e1a8274

§7.2 runs

v4 round trip and resume for unwindowed and windowed SAE and haCC; cross-filter resume refused; equal ledgers

as stated; atomic and logical ledgers exactly equal

§7.2 fake drift

the run’s edge ratio equals the direct T2 (c) one within its floors

under both forms, on each run’s own problem

The unchanged-route witness found a real regression during development: with the edge array omitted, time_collar’s operand floor had lost its gate. It was fixed before the first commit of the assembly.

Certificates of the default configurations (run_short: 60 fs over 90 nm–1.6 µm, the six-node Gauss field-scale rule, 366 active bins from \(6.1\times10^{14}\) to \(2.3\times10^{16}\) rad/s):

configuration

lattice

samples per taper

\(\kappa_{\rm ss}\)

haCC(1), haCC(5) and haCC(1) fast, 60 fs window with 5 fs tapers

16384 × 6.25 as

799

\(5.06\times10^{-16}\)

haCC(1), full lattice, grid collar

16384 × 6.25 as

3395

\(5.69\times10^{-15}\)

SAE research record (SEP 12, \(q=32\)), full lattice, grid collar

65536 × 1.56 as

13581

\(4.37\times10^{-15}\)

SAE research record, 60 fs window

65536 × 1.56 as

3198

\(4.84\times10^{-16}\)

The haCC presets were represented by the contract copies of their records in tests/hacc_cases.py, since the provider is not installed here; the certificate depends only on the lattice, the gate and the band.

Hot path (benchmarks/sep26/evidence/hot-path.json, at 988add7, load 6.4): one source assembly and projection on the run_short lattice took 0.657 ms (MAD 0.013 ms) on the pre-SEP-26 path, 1.131 ms (0.016 ms) for time_collar with its edge diagnostic and 1.129 ms (0.018 ms) for gated_derivative: 1.72 times, with 3.1 MB more peak traced allocation against a 4.0 MB tdse_edge_sources ledger row. Against minutes per atomic call it is negligible; against the source assembly alone it is not.

Not run. None of Phase 2 (the L5 replay, the L6 segments, the SAE controls, the rate-plasma measurement). No GPU, cluster or Linux run; CI’s Linux, Windows and lowest-version lanes run on the pull request. solprop-kokkos’s scripts/check.sh (native GoogleTest, sanitizers, clang-tidy, the wheel smoke test) was not run; its only native change is the schema literal of the build record. Its Python suite passed (319 passed, 5 skipped) against the editable build.

Sources checked. The derivative of McKechan et al.’s eq. (7) (reference 1) was re-derived and checked numerically against the closed form and a fourth-order difference (T1). The FFT constant starts from Higham’s Theorem 24.2 (reference 2) as the implementer knows it; the chapter was not re-read for this phase, and the constant is qualified by T3’s measurement, not by the citation. No physical constant or coefficient was transcribed.

Limitations.

  • numpy_ref only; numba_cpu and solprop-kokkos refuse every TDSE problem by name (SEP 12’s recorded exception).

  • The certificate is relative to \(\lVert\mathrm{DFT}[g]\rVert_A\). A long, well-resolved taper on a band far above its frequencies can be refused although its absolute edge error is at rounding (T7: \(1.8\times10^{-10}\)). No default or campaign configuration is near this.

  • time_collar runs now pay the edge diagnostic: one more fine transform and about 3 MB per evaluation.

  • The §4.5 budget is a conservative first-order indicator (T2 (a) sits four orders below it), not a certificate of the atomic integration.

  • The leading-taper band bound is an absolute-projection upper bound, loose by 2–3 times on the T6 case.

Open questions.

  • Q7 (maintainer, Phase 3): should the certificate stay relative to the truncating source, or be compared with the band’s own rounding budget, so that a resolved taper on a UV-only band is not refused? Phase 2 should record it on every replayed configuration either way.

  • Q8 (maintainer, Phase 3): keep computing the edge diagnostic under time_collar, or compute it only on request once Phase 2’s comparisons are made?

  • Q9 (Phase 2): the SEP 20 scripts (run_short_hacc.py, segment.py, window.py) have no source-filter option yet.

Handoff to Phase 2.

  • Opt in with replace(response, source_filter="gated_derivative"). The run then writes a v4 descriptor and records execution["tdse_source"]. A time_collar file refuses to resume under it and the reverse, so §7.4’s runs (a)–(c) are fresh runs.

  • For the offline replays (§7.3, §7.5), build a TDSEResult from the archived currents and call atomic_sources, atomic_edge_source and project_atomic_sources; the edge diagnostic gives the band norms and ratios with their floors directly. steady_current_certificate(descriptor, omega, active) gives any gated descriptor’s certificate without an evaluation.

  • Add the source-filter option to the SEP 20 scripts first (Q9); the derived analysis record must keep each raw current’s descriptor and identity (§7.3).

  • Rebuild solprop-kokkos from its feature/sep-0026-phase-1 (schema 30) before any Kokkos lane on a Phase 2 tree.

Phase 2 report — validation (2026-10-19)#

Scope. Phase 2 of §11: §7.3’s L5 replay, §7.5’s offline SAE controls and SAE rerun, §7.4’s three L6 segments, §4.7’s offline rate-plasma measurement and the certificate of every replayed configuration. Implementer: Opus 5.5 (AI) for John Travers, on branch feature/sep-0026-validation from 392c2305. Part I is unchanged. Phase 1’s open questions were decided by the coordinating agent on 2026-10-18:

  • Q7: the certificate stays relative to the truncating source. The refusal of a long taper on a UV-only band remains a known limitation (below).

  • Q8: the edge diagnostic stays under time_collar.

  • Q9: in scope here.

Delivered.

  • Q9. benchmarks/sep20/run_short_hacc.py and segment.py take --source-filter {default,time_collar,gated_derivative} and --window-duration-fs (with --atomic-window window). Both are recorded in the provenance (requested and resolved filter) and in run.json. The run summary also keeps tdse_source and the last evaluation’s edge entry. benchmarks/hacc_perf/slurm/segment.sbatch passes SEGMENT_OPTIONS through.

  • benchmarks/sep20/window.py. analyse --source-filter replays the raw L5 currents as a derived record, schema solprop.sep20-window.v2. The record keeps each raw descriptor identity, the replay filter, the replay descriptors and their certificates. The replay descriptor must differ from the raw one only in its filter and version. The analysis adds a same-current comparison, the 0.9–1.6 µm band and the edge fractions. run takes --source-filter. benchmarks/sep20/sources.py forms the edge-free source and its §4.5 floor by band.

  • Replays, benchmarks/sep26/.

    • l5.py (§7.3 (i)–(iii));

    • sae_calls.py (the L6 per-call SAE archives);

    • sae_width.py (SEP 12’s width study);

    • rate_plasma.py (§4.7);

    • publish.py, which copies a record into the tree without host names or local paths.

  • Tests. tests/test_sep20_window.py (new), tests/test_sep20_segment.py (extended) and tests/test_sep26_publish.py (new).

  • Evidence, benchmarks/sep26/evidence/.

    • l5-*.json and l5-recorded-descriptors.json;

    • sae-calls.json, sae-width.json and rate-plasma.json;

    • l6-*.json;

    • manifest.json, which holds the hashes of the raw segment outputs, the original records, the private pump, the jobs and the core-hours.

  • Documentation. benchmarks/sep26/README.md and benchmarks/sep20/README.md.

Inputs. Every raw input was hashed against its manifest before use, and everything matched:

  • the L5 currents (benchmarks/sep20/evidence/manifest.json, raw_files_sha256.session);

  • the L6 runs (phase4.l6_runs);

  • the SEP 12 width study’s eight cases (source-filter-provenance.json, raw_case_hashes, read only).

The private pump was used only where it was already staged, with mode 600. Only its hash d9a512d3… is recorded. The private run_short.py was not used.

Deviations from Part I, and why.

  1. L5 descriptor identity. The L5 currents were recorded at the provider pin ea2bfcaa. At the current pin 876b72d1, the preset’s provenance text differs: its source strings, one added limitations line and the calibration’s basis_identity. That changes the descriptor identity, although no number and no other field changes. The recorded configurations were regenerated at the Phase 3 solprop revision db4ca407 with the provider at ea2bfcaa. Each hashes exactly to its recorded identity (l5-recorded-descriptors.json). window.py admits a harness only if its configuration differs from the recorded one in those four fields alone.

  2. (i)’s budget propagates the combined floor of both sources into both the numerator and the denominator, \((2F+2\delta F)/(\lVert r\rVert-2F)\). This is conservative.

  3. (ii)’s \(e_g\) is the sum, over both gates, of \(\lVert c(\mathrm{DFT}[g'_{\rm cd}S]-\mathrm{DFT}[g'S])/(i\omega)\rVert_B\), relative to \(\lVert F_{100}\rVert_B\). The record also carries half a unit in the last printed digit of the expected figures. It was not needed: every deviation is within the derived tolerance alone.

  4. SAE “recorded figures”. The L6 archives keep no per-call source figures. Their recorded figure is each run’s tdse_last_evaluation complete-source norm, reproduced within twice its recorded floor.

  5. The current-form 60 fs segment was not rerun. The provider pin has not moved since benchmarks/hacc_perf/evidence/perf.json (job 483804, solprop d482cdf0): time_collar keeps its formula and path. The SAE rerun is compared with SEP 20 Phase 4’s sae-window run at ea2bfcaa. No native file differs between ea2bfcaa and 876b72d1 (only pyroraptor/hacc/*.py changed), so the comparison differs only in the filter and the solprop revision.

  6. Rate plasma. The rate problem is rebuilt from the case’s settings with a placeholder pulse, because the pump is private and off this machine. The pulse enters no grid, map or coefficient. The coarse grid was checked equal to the run’s saved grid. The L6 rate run saves two planes, 3.969 m and 4.05 m, and both were measured.

Verification and results.

Machines. The offline replays ran on an 18-core Apple M5 Max (macOS 27, arm64). The software was Python 3.14.5 and NumPy 2.5.2, with Pyroraptor 876b72d1 (Clang 23, OpenMP, Release) installed in the worktree environment. Each replay used 1–4 threads, one job at a time. The load average was 5.8–6.0 before and after the final pass at 7ad9f0dc, and each replay took 3–6 s.

The L6 segments ran on cluster CPU nodes:

  • 2 × EPYC 7763 nodes for haCC, and a 96-core node for SAE, as in Phase 4;

  • GCC 16.2, Release, OpenMP, x86-64-v3, solprop 594b0c68;

  • each in a 32-CPU block with --hint=nomultithread --distribution=block:block.

Placement was verified with taskset and numactl:

run

job

node

CPUs

NUMA domains

elapsed

core-hours

(a)

483988

node19

0–31

0–1

1 h 15

39.8

(b)

483989

node19

32–63

2–3

1 h 46

56.4

(c)

483990

node20

0–31

0–1

1 h 35

50.7

SAE

483991

node14

0–31

0–2

1 h 21

43.2

With the build and the analysis, the total is 191.5 core-hours, against 215–270 estimated. Timing was not the aim; (a) and (b) shared a node.

§7.3 L5 (l5-replay.json):

  • (i) Recorded figures. All eight recorded time_collar ratios are reproduced: full lattice and common window, band and uv, both planes. The largest change is \(3.2\times10^{-17}\), against budgets of \(3.0\times10^{-12}\) to \(1.1\times10^{-11}\). Passed.

  • (ii) Same-current anchor. Edge-free, against 0133’s figures; the tolerance is \((2+\delta)(r+e_g)\) with \(r=5.4\times10^{-4}\). Passed in the band at both planes (the requirement), and nir and uv within the same tolerance:

    plane

    band δ (expected)

    nir

    uv

    \(e_g\)

    tolerance

    peak power

    1.2224% (1.22%)

    3.3260% (3.33%)

    0.3877% (0.387%)

    \(1.3\times10^{-6}\)

    \(1.09\times10^{-3}\)

    maximum UV

    1.4521% (1.45%)

    4.4238% (4.42%)

    0.3498% (0.349%)

    \(2.2\times10^{-7}\)

    \(1.09\times10^{-3}\)

    The study’s own centred-difference split, replayed here, gives 1.2226% and 1.4524% in the band.

  • (iii) Actual L5, the separately driven 60 fs current against the 100 fs one. This is reported, and it is still a recorded failure of L5’s 1% band tolerance at both planes:

    plane

    filter

    band

    nir

    uv

    common window (band)

    peak power

    time_collar

    4.607%

    12.49%

    0.405%

    0.036%

    peak power

    gated_derivative

    1.230%

    3.342%

    0.405%

    0.047%

    maximum UV

    time_collar

    1.428%

    4.315%

    0.365%

    0.046%

    maximum UV

    gated_derivative

    1.439%

    4.379%

    0.365%

    0.052%

    As §7.3 expected, (iii) is close to (ii) plus the startup and drive difference. The edge term is 10.7% (60 fs) and 11.0% (100 fs) of each source’s own nir at peak power, and 1.7% and 2.2% at maximum UV.

§7.5 SAE controls:

  • L6 per-call archives (sae-calls.json, 9 evaluations per run).

    • Both runs’ recorded last-evaluation norms are reproduced: \(1.1453056830\times10^{7}\) and \(1.1495518654\times10^{7}\), each a change of \(5.6\times10^{-7}\) against budgets of \(8.7\times10^{-5}\) and \(8.0\times10^{-5}\).

    • The T3 spectral-derivative oracle on the real currents agrees with the identity form to \(2.7\)–\(5.8\times10^{-16}\) (band) and at most \(3.3\times10^{-15}\) (1.2–1.6 µm) relative, on every call of both runs.

    • Edge over truncating source, band / nir / 1.2–1.6 µm:

      • collar (sae-full): 0.07% / 0.26–0.32% / 0.65–0.82%;

      • 60 fs window: 2.5–2.9% / 9.5–10.4% / 21.5–23.3%.

      In 90–200 nm it is at most \(2.6\times10^{-5}\) for both.

    • The shared launch field, window against collar, changes from 2.17% to 0.88% (band), from 8.14% to 3.25% (nir) and from 0.119% to 0.119% (uv), going from time_collar to gated_derivative.

  • SEP 12 width study (sae-width.json).

    • All 66 recorded direct-ordering figures are reproduced (48 width sensitivities and 18 window comparisons). The largest change is \(2.7\times10^{-7}\) of its budget.

    • The edge-free width sensitivity is no larger than the truncating one in every one of 48 case × gate × band comparisons; none is unresolved. It falls by 3–1000 times. For example, the 60 fs launch window, narrow against matched, over 120–1600 nm goes from 3.48% to 0.82%, and the 120 fs and 240 fs windows fall to below \(10^{-4}\).

    • The 60-against-120 and 240 fs window comparisons do not fall: 8.71% → 8.83% full band and 7.12% → 6.71% over 120–1600 nm (launch). That is the windows’ history difference, as for haCC.

  • SAE rerun (l6-gates-sae.json, l6-sae-pairs.json).

    • L6’s mechanical and sensible-optics gates pass: energy over launch 0.9856, over rate 0.9949, and UV over rate 1.27.

    • Against Phase 4’s time_collar sae-window: 0.318% full, 0.0087% in 90–200 nm (below the floor), 0.145% at 0.9–1.2 µm and 1.94% at 1.2–1.6 µm.

    • Against sae-full (collar), the window’s endpoint sensitivity falls from 0.323% to 0.082% (full band) and from 1.35% to 0.72% (1.2–1.6 µm).

  • Pass for the SAE controls (§7.5): yes, all four conditions. The replays reproduce their recorded figures, every certificate is at most \(10^{-10}\), the rerun passes L6’s gates, and the edge-free width sensitivity is no larger than the truncating one in any band. The SAE research record may therefore move in Phase 3.

§7.4 L6 segments. haCC(1) fast, rtol=1e-2, 32 threads and the full archive (l6-hacc-pairs.json, l6-gates-*.json). Every run completed with 9 atomic calls and 8 accepted steps. Every run passes L6’s mechanical and sensible-optics gates:

  • energy over launch 0.9852;

  • energy over rate 0.9945;

  • UV over rate 1.240;

  • UV over sae-full and over sae-window, 0.975.

Endpoint spectra are compared as \(\lVert a-b\rVert_B/\lVert b\rVert_B\). The floor is 0.073% (SEP 20 L6’s restart sensitivity), and a figure below it is unresolved.

pair (relative to)

full

90–200 nm

0.9–1.2 µm

1.2–1.6 µm

above 0.9 µm

(a) gated 60 fs vs current form (time_collar 60 fs)

0.334%

0.0088% (unresolved)

0.152%

2.03%

0.345%

(a) vs (b) gated 100 fs

0.091%

0.087%

0.058%

0.77%

0.131%

(c) time_collar 100 fs vs current form

0.357%

0.087%

0.163%

1.30%

0.256%

(b) vs (c), both 100 fs

0.091%

0.0022% (unresolved)

0.035%

0.47%

0.079%

  • Endpoint energy is \(1.34001\times10^{-3}\) J for (a)–(c) against \(1.34003\times10^{-3}\) J for the current form. Peak power is 590.64, 590.64, 590.65 and 590.69 GW, and the 90–200 nm energy is 4.8800 µJ for all four.

  • The certificates recorded in each run’s metadata are \(5.14\times10^{-16}\) (a), \(5.06\times10^{-16}\) (b and c) and \(4.46\times10^{-16}\) (SAE). The last evaluation’s \(\rho_-\) is at most \(2.3\times10^{-11}\).

  • Investigation threshold. The 90–200 nm difference between (a) and the current form is 0.0088%. That is 57 times below 0.5%, and below the floor. No investigation was triggered, so the matched-input and rtol=3e-3 controls were not run.

  • Expectation. The (a)–(b) window sensitivity above 900 nm is 0.131%, against the current form’s 0.256% (c against the current form). It is about half, not “well below”; at 1.2–1.6 µm it is 0.77% against 1.30%. The 90–200 nm window sensitivity, 0.087%, is the same under both filters, as in L5.

§4.7 rate plasma (rate-plasma.json). This is a fixed-field source comparison on the 33-node rule, with the band running-integral rule. The closure residual is \(3\)–\(10\times10^{-7}\) of \(i\omega\,\mathrm{DFT}[gP]\) in every band, within that rule’s certified error of \(4.9\times10^{-6}\):

plane

\(\lVert Q_{\rm route}-Q_{\rm held}\rVert/\lVert Q_{\rm route}\rVert\): band

uv

nir

0.9–1.2 µm

1.2–1.6 µm

3.969 m

3.37%

\(1.8\times10^{-5}\)

8.40%

3.24%

18.7%

4.05 m

7.36%

\(1.4\times10^{-5}\)

18.3%

7.15%

38.2%

\(P\) is held at its peak at the lattice end; its final-to-peak ratio is 1.0. The collar’s edge term \(\mathrm{DFT}[g'P]\) is of the same size as this difference. The collar’s certificate is \(1.8\times10^{-14}\). The production rate route is unchanged; Q3 decides whether a follow-up SEP is needed.

Certificates of every replayed configuration, each at most \(10^{-10}\):

configuration

\(\kappa_{\rm ss}\)

haCC lattice, 60 fs window

\(5.06\times10^{-16}\) (offline); \(5.14\times10^{-16}\) in run (a)

haCC lattice, 100 fs window

\(5.53\times10^{-16}\) (offline); \(5.06\times10^{-16}\) in runs (b) and (c)

SAE lattice, grid collar (sae-full)

\(4.37\times10^{-15}\)

SAE lattice, 60 fs window

\(4.84\times10^{-16}\) (offline); \(4.46\times10^{-16}\) in the SAE rerun

SEP 12 cases, narrow, matched and wide gates (3000–5000 samples per taper)

\(3.1\)–\(8.1\times10^{-16}\)

rate route, fine collar (§4.7)

\(1.8\times10^{-14}\)

The offline and in-run values differ at rounding level, because they were computed by different FFT builds and machines.

Checks.

  • ruff, ruff format and basedpyright on benchmarks/sep20, benchmarks/sep26 and the touched tests: clean.

  • uv run pytest tests/test_sep20_window.py tests/test_sep20_segment.py tests/test_sep20_benchmarks.py tests/test_sep26_publish.py tests/test_seps.py: 245 passed, 1 skipped (the process-SEP skip).

  • codespell, git diff --check and uv run poe docs for this text.

§11 promotion conditions, item by item.

  • T1–T7: passed in Phase 1 and not rerun here.

  • §7.3 (i) and (ii): passed.

  • §7.4’s gates: passed on all three runs and on the SAE rerun. No investigation was opened, so none is left open.

  • Certificates: at most \(10^{-10}\) on every replayed configuration; the largest is \(1.8\times10^{-14}\).

  • The maintainer’s decision on Q4 and Q30: below.

Maintainer’s decision on Q4 and Q30 (2026-10-18, relayed by the coordinating agent; conditional). The 60 fs default window stays. gated_derivative becomes the haCC and windowed default in Phase 3. Q30’s exception is restated at the measured actual-L5 size under the new form: 1.23% band, 3.34% nir and 0.405% uv at peak power, and 1.44%, 4.38% and 0.365% at maximum UV. The condition was that the L6 segments show nothing unexpected: the 90–200 nm change stays below the 0.5% investigation threshold, and every investigation is closed. The condition holds. The 90–200 nm change is 0.0088% ((a) against the current form), and no investigation was opened.

Sources checked. No physical constant or coefficient was transcribed. Two derivations were rechecked numerically, as described above:

  • the held-current identity, against the spectral-derivative oracle on the real SAE currents;

  • the discrete closure \(P'=J\), against the running-integral rule’s certified error.

Limitations.

  • The certificate stays relative to the truncating source (Q7). A long, well-resolved taper on a band far above its frequencies can be refused although its absolute edge error is at rounding level. No configuration measured here comes near this: the largest certificate is \(1.8\times10^{-14}\).

  • The window sensitivity that remains under gated_derivative (L5: 1.23% and 1.44% band) is still a sensitivity, not an error against a converged late-time reference. L5 still fails its 1% band tolerance under either filter.

  • The L6 comparisons are single trajectories at rtol=1e-2. Figures below the 0.073% floor are reported as unresolved, and no tighter-tolerance control was run because none was triggered.

  • The rate-plasma figure is a fixed-field source comparison on two saved planes, not a propagated trajectory.

Open questions.

  • Q10 (maintainer, with Q3): the rate route’s gated polarisation differs from a held current by 3–7% of the band source and 8–18% in the near infrared at the L6 planes, and by \(10^{-5}\) in the UV. Does this justify a follow-up SEP?

Not run.

  • §7.4’s matched-input, pinned-input and rtol=3e-3 investigation controls, because no investigation was triggered.

  • Any GPU or Kokkos lane, the full gate, the slow lane and the coverage lane; these are Phase 3’s.

  • A rerun of the current-form 60 fs segment, because the provider pin has not moved.

Handoff to Phase 3.

  • Promote gated_derivative to the haCC and windowed default. The maintainer’s condition holds.

  • Move the SAE research record, since §7.5 passed.

  • Restate Q30 at the actual-L5 figures above.

  • Run the end-of-SEP lanes.

  • Rebuild solprop-kokkos for schema 30 before the Kokkos lane.

Phase 3 report — promotion and the final gate (2026-10-19)#

Scope. Phase 3 of §11: the default promotion, the move of the SAE research record, Q30 restated, the Q3/Q10 rate-plasma record, and the end-of-SEP lanes. Implementer: Opus 5.5 (AI) for John Travers. Branch feature/sep-0026-phase-3 from 4efb87bd, the merge of Phase 2. Its commits are 37853788 (the promotion, the record, the manual and the changelog) and 2e3fb485 (the roadmap and SEP 20’s erratum), followed by this report. Merge commits are the coordinating agent’s to record. solprop-kokkos is unchanged: its main at f40728d already accepts schema 30.

Delivered.

  • Promotion. src/solprop/nonlinear/tdse.py: an omitted source_filter resolves to "gated_derivative" for haCC (windowed or on the grid collar) and for any windowed response. An unwindowed SAE response keeps "none". "time_collar" stays selectable, and its formula, path, descriptor bytes, identities and resume keys are unchanged. The resolution happens in the response record before translation, so no descriptor version, kernel schema or output schema moves. Recorded files keep their explicit "time_collar" and resume under it only. The docstring states the promotion. src/solprop/translate/tdse.py: the admission refusal now also names source_filter='time_collar' as a remedy, because an omitted filter on an unresolving lattice is now refused instead of truncated.

  • Tests.

    • tests/hacc_cases.py’s haCC contract fixture now names "time_collar". Its lattices give 1–5 fs tapers of 4–62 samples, below the 128-sample admission. They test the binding, records and archive, not the source form; without the pin, 157 tests were refused at translation.

    • tests/nonlinear/test_tdse_spec.py, tests/translate/test_tdse_descriptor_v4.py and tests/translate/test_tdse_contract.py assert the new resolution (haCC windowed and unwindowed, windowed SAE, unwindowed SAE) and that the named truncating form is kept.

    • tests/translate/test_tdse_descriptor_v4.py gains a test that an omitted filter on a 127-sample taper is refused, not silently truncated, while the named "time_collar" builds a v3 descriptor.

    • tests/test_sep20_window.py follows the harness’s new haCC default.

  • The SAE research record (§7.5, Q2): benchmarks/sep26/evidence/selected-gated-derivative.response.json. It holds SEP 12’s record with source_filter="gated_derivative" under the new calibration identity sep26-native1537-cut400-q32-gated-derivative-research-v1. The calibration’s source text and its two filter-related limitations are rewritten; every other field is identical. SEP 12’s selected-filtered.response.json is unchanged; its SHA-256 de50ad2a… still matches SEP 20’s manifest. tests/test_sep26_research_record.py checks the old hash and identity, that the two records differ only in the filter and the calibration’s identity, source and limitations, and that the new record builds a v4 descriptor with a distinct identity on SEP 12’s parent lattice at \(q=32\), with a certificate within the limit.

  • Manual.

    • docs/guide/media/tdse-hacc.md: the default; the restated window limitation (Q30); a separate SAE window limitation; the edge-free L6 table; the corrected L5 wording. Its example now asserts the default and builds the truncating form.

    • tdse-configuration.md: the resolution table; “The default, and what it changed”; the tested example; the reconstruction section, which now downloads the moved record and keeps SEP 12’s.

    • tdse-capillary.md: the source-filter row; the default note; “The research record after SEP 26”.

    • docs/guide/results/resuming.md: resuming files written under the old default.

    • docs/guide/workflows/luna.md: the default.

    • docs/dev/kernel-contract.md: row 30 and the version-4 paragraph; no schema change.

  • Benchmarks. The docstring of benchmarks/sep20/run_short_hacc.py and benchmarks/sep20/README.md: --source-filter default is now gated_derivative for haCC, and --source-filter time_collar repeats a SEP 20 haCC run.

  • Roadmap (docs/dev/roadmap.md): the SEP 26 listing updated; Q30 restated in the haCC item; the SAE window sensitivity in the TDSE qualification item; a new item, The field-grid rate plasma’s collar truncation (SEP 26 Q3, Q10).

  • SEP 20. One erratum note under Q30, pointing to this SEP.

  • Changelog. A Changed entry marking the breaking default; the Phase 1 Added entry no longer says “no default changes”.

Deviations from Part I, and why.

  1. The contract fixtures keep "time_collar" rather than moving to the default. Their coarse lattices are not admitted under "gated_derivative". Refining them would change the cost and meaning of 157 binding, record and archive tests, which are about neither source form. The new default’s resolution, admission and runs are tested separately; tests/driver/test_tdse_gated_derivative_run.py covers the runs.

  2. The SEP 20 harness keeps SEP 12’s time_collar record as its default --sae-record, so that SEP 20’s SAE runs and Phase 2’s comparisons repeat unchanged. The moved record is selected with --sae-record. haCC’s harness default follows the response’s own default and so moves.

Decisions.

  • An omitted filter on an unresolving lattice is refused at translation. It never falls back to "time_collar": that would be a silent fallback, and the refusal names the remedies.

  • The moved record’s limitations state that the new form was propagated only on the 60 fs window segment. On the full-lattice collar it was evaluated offline per call, with an edge term of 0.07% of the band.

  • No ADR: the decision is recorded here and in the changelog.

Q30 restated (manual and roadmap), from Phase 2’s figures under the new form:

  • Actual L5, against 100 fs. At peak power, 1.23% of the band, 3.34% above 900 nm and 0.405% at 90–200 nm. At maximum UV, 1.44%, 4.38% and 0.365%. Both still fail L5’s 1% band tolerance, and the 60 fs default is kept with this exception (the maintainer’s decision of 2026-10-18).

  • L6 propagated window sensitivity, (a) against (b). 0.091% full, 0.087% at 90–200 nm and 0.131% above 0.9 µm.

  • Advice. Use a 100 fs window where the near infrared matters.

  • The SAE limitation, stated separately. The 60-against-120/240 fs window sensitivity is about 8.7% (time_collar) and 8.8% (edge-free) of the full band. The new form does not reduce it. It does reduce the filter-width sensitivity 3–1000 times and the window-against-collar differences (single field 2.17% → 0.88% band; L6 endpoint 0.323% → 0.082% full).

Q3/Q10, the rate plasma. Reported only; the rate route is unchanged. The route’s gated \(P\) differs from the held current by 3.4% of the band and 8.4% in the nir at 3.969 m, and by 7.4% and 18% at 4.05 m, against about \(1.5\times10^{-5}\) in the UV. The roadmap records this as an open item for the maintainer: a possible follow-up SEP for the field-grid rate plasma’s collar truncation.

Verification. The machine was an 18-core Apple M5 Max (macOS 27, arm64), with Python 3.14.5, NumPy 2.5.2 and SciPy 1.18.1. xdist was bounded to 8 workers with one thread per worker (SOLPROP_KOKKOS_THREADS=1, OMP_PROC_BIND=false). solprop-kokkos main f40728d (schema 30) was built from a clean clone (CPU, OpenMP, local profile, Homebrew Clang 23) and installed in the worktree’s environment. The lanes ran sequentially on 2e3fb485, with this report not yet written. The machine was shared, with other agents’ work running, and the load average was 4.1–10.4 (1-minute 3.7–9.1):

lane

result

wall

uv run poe check

green: ruff, format, basedpyright, numpydoc, codespell, reuse; the default lane 15920 passed, 29 skipped; strict Sphinx

15.4 min

uv run poe test-slow

147 passed, 1 skipped; the statistics-cost check passed

6.8 min

uv run poe cov

100% of statements and branches (41542 and 12580). The lane exited 1 on one test (below)

11.2 min

SOLPROP_TEST_BACKEND=kokkos uv run poe test

15920 passed, 29 skipped

10.3 min

SOLPROP_TEST_BACKEND=kokkos uv run poe test-slow

148 passed; the statistics-cost check passed

6.0 min

The coverage lane’s one failure was tests/backends/test_numpy_ref_backend.py::test_a_gradient_capillary_propagates_on_the_field_grid[numba_cpu]. The test asserts no log record, and caught an INFO record from solprop.translate.linop (a raised tolerance floor). It concerns neither TDSE nor anything this SEP changed, and it passed in the default and Kokkos lanes on the same tree. Rerun alone, the module passed three times under the lane’s coverage configuration, once without coverage, and serially with the failing run’s random seed (130 passed, 1 skipped each time). It depends on which tests share a worker, so it is recorded as an intermittent failure that predates this branch, not fixed here. The coverage total was measured on the full run, which reached 100%.

Before the lanes, the TDSE-related modules (tests/nonlinear, tests/translate, the TDSE backend, driver and output modules, the SEP 12, 20 and 26 scripts) passed. So did the TDSE guide pages run with their slow snippets (8 passed). The lanes ran these modules again.

Not run. No GPU, cluster or Linux run; CI’s Linux, Windows and lowest-version lanes run on the pull request. solprop-kokkos’s own scripts/check.sh was not run: nothing changed there.

Backend parity. TDSE runs on numpy_ref only. numba_cpu and solprop-kokkos refuse every TDSE problem by name, gated_derivative included (tested in Phase 1 and in the Kokkos lanes here). This is SEP 12’s recorded exception, which SEP 26 §2 carries (“Compiled backends: TDSE stays NumPy-only”). It satisfies SEP 0’s backend condition for Final through that exception, whose roadmap item SEP 12 holds.

Sources checked. No constant, coefficient or citation was transcribed. Every figure restated in the manual, the roadmap, the changelog and the moved record was checked against Phase 2’s report and evidence (l5-replay.json, l6-hacc-pairs.json, sae-width.json, sae-calls.json, l6-sae-pairs.json, rate-plasma.json).

Limitations.

  • The moved SAE record has not been propagated on its own full-lattice collar under the new form. Its propagated check is the 60 fs window segment; the collar was evaluated offline per call.

  • The window sensitivities restated under Q30 are sensitivities, not errors against a converged late-time reference. L5 still fails its 1% band tolerance.

  • The coverage lane carries the intermittent numba_cpu log-record failure above.

  • Phase 1’s report says that each page it produced or extended ends with a link to this SEP. That held for the three TDSE pages, not for resuming.md or luna.md; their SEP 26 paragraphs now link it inline.

Open questions.

  • Q10 (maintainer, with Q3; moved to the roadmap): does the rate plasma’s collar truncation justify a follow-up SEP?

  • Q1–Q9 are resolved. Q4 and Q30 were decided by the maintainer on 2026-10-18 (Phase 2 report) and applied here.

Handoff. No phase follows. Final status, the archive move and the link fixes are in a separate commit on this branch. The roadmap keeps two informal items: the rate-plasma item (Q10), and the window and propagation questions in the haCC and TDSE qualification items.

Part III — History#

  • 2026-10-18: Drafted at the maintainer’s choice of the held-current remedy (option 1) after Pyroraptor report 0133’s taper-edge addendum.

  • 2026-10-18: Q1–Q6 resolved by the maintainer (the coordinating agent’s recommendations); §7.5 and §11 updated: one SAE L6 rerun funded, the SEP 12 archive located, and an offline rate-plasma measurement added to Phase 2. Merged into main as a Draft for review.

AI review — 2026-10-18#

Reviewers and scope. GPT-6 (AI) reviewed all of Part I against SEP 0, the conventions, the current TDSE source, descriptor, output and diagnostic code, and the qualification records of SEPs 12, 19, 20 and 25. John Travers requested that his human review also be recorded. The review status is Reviewed; the proposal remains Draft. This records completion of review, not acceptance or resolution of the findings below. Part I has deliberately not been edited. The drafting agent should respond to R1–R13 individually, with the changed sections or an explicit reason for leaving a recommendation open.

Assessment. The continuous product-rule identity and the held-current interpretation are sound away from zero frequency. Keeping shadow subtraction first, a distinct filter identity, the existing active mask and the separate field absorber is a coherent design. The proposal is not yet ready for acceptance: the discrete validity claim currently includes admitted configurations where it fails, some test oracles are incorrect, and several validation claims exceed their evidence.

R1 — P1: qualify the discrete identity on the admitted lattices (§4.4, §6, T1–T3)#

The continuous identity does not make the sampled formula exact to rounding for arbitrary admitted gates. TDSEAtomicWindow only checks a positive taper shorter than half the duration; resolve_atomic_window checks the window length and at least four retained samples, without resolving either taper. An otherwise valid window with a taper shorter than a sample can have no nonzero derivative samples. Then DFT[g'S] is zero although DFT[gS] is not, and the proposed steady-current cancellation fails completely.

A review probe used \(N=16384\), \(\Delta t=6.254\times10^{-18}\) s, an atomic duration \(80\Delta t\) and taper \(\Delta t/4\). The existing resolver admitted it. For \(S=1\), the norm of the proposed source divided by that of the truncating source was 1.0 on \(5.9\times10^{14}<\omega<2.1\times10^{16}\) rad/s. With the 60 fs window and 5 fs tapers on the same lattice, that ratio was \(5.9\times10^{-16}\) instead.

Please state a lattice/taper/retained-band validity condition and how translation handles a configuration outside it. Either bound and test the alias contribution for the admitted domain, or explicitly limit the rounding-level claim to qualified configurations. A four-sample taper alone is not a rounding-level certificate. Include asymmetric/off-grid breakpoints, widths near the admission limit and the highest retained frequency, with refinement checks. The two campaign estimates and one low-band synthetic signal do not cover the public interface.

Also separate the exact continuous identity, the discrete definition, aliasing error and floating-point error in the prose. A saddle-point asymptotic with unspecified algebraic factors is not a numerical upper bound. A trigonometric interpolant has a line spectrum; formulate its Fourier bound using coefficient sums or the total variation of its spectral measure, rather than an ordinary \(L^1\) density. For T1, \(C^\infty\) regularity alone does not imply an exponential trapezoidal error bound, and a sampled maximum of \(g^{(5)}\) is not by itself an upper bound on that derivative.

R2 — P1: preserve the turn-on source in the steady-tail tests (T2, §7.2)#

T2’s last bullet incorrectly extends the zero-source result for a current constant over the whole window to a current that turns on inside the plateau. For \(S=H\Theta(t-t_c)\) with \(g(t_c)=1\), the retained source is \(H e^{-i\omega t_c}/(i\omega)\), not zero. Its smooth counterpart is precisely T4. Deleting that term would delete the in-window emission that §4.1 intends to keep. The fake-provider test’s wording about a steady post-pulse tail has the same ambiguity.

Please split the assertions: a globally constant current cancels to the qualified sampling/rounding floor; an interior rise followed by a steady tail retains the analytic rise spectrum while losing only the artificial trailing switch-off. Include a current varying through the taper to verify that \(gS'\) is retained there, and a zero-current case. In the review’s smooth-rise probe, the source was nonzero and agreed with the correctly normalised erf oracle to \(4.3\times10^{-15}\) relative.

R3 — P1: use operand-based, conditioned rounding budgets (§4.4, T2–T3)#

An FFT normwise bound does not directly give the same relative bound after masking and division by frequency. The active bins can be arbitrarily small compared with the full transform, and \(1/\omega\) is not a uniform scaling. Nor can T3’s error budget be proportional only to \(\|\hat S_{\rm ef}\|\): that is the result of a subtraction and may vanish while its operands and their rounding errors do not. The measured 11% edge fraction in one near-infrared example is not a general conditioning bound.

Please derive an absolute budget through multiplication, projection, FFT, mask/crop, division and final subtraction. For example, the FFT contribution from \(g'S\) can conservatively use a constant times \(\varepsilon\log N\,\sqrt{N}\|g'S\|_2/\omega_{\rm lo}\) for a raw, full DFT, before including the actual projection/scaling bounds; a sharper bound needs its own justification. Retain the raw \(|J|+|j_1|\) operands for shadow cancellation, add gate and derivative evaluation errors, and keep aliasing separate. Apply the same logic to the spectral-derivative oracle, whose intermediate spectra can be much larger than the final source. Specify the FFT assumptions behind the constant, including any supported non-power-of-two sizes. Higham’s FFT chapter is a suitable starting point, but does not certify the stated weighted/cropped bound without this additional conditioning analysis.

R4 — P2: correct the DFT normalisation, phase origin and signed-edge example (§4.1, §4.3–4.4, T4–T6)#

The raw DFT has neither \(\Delta t\) weighting nor its phase origin at zero (conventions §2). For the lattice in §4.4, Poisson summation needs the factor \(e^{-i\omega_kT/2}=(-1)^k\) when relating a raw coefficient to the centred continuous transform. The factor cancels in a consistently written identity, but not when an analytic spectrum is used as a complex-valued oracle. T4 currently labels a continuous transform as being in DFT units. Its raw oracle should be

\[ \frac{(-1)^k}{\Delta t}\, \frac{H e^{-\omega_k^2\sigma^2/4}e^{-i\omega_kt_c}}{i\omega_k}, \]

up to the stated tail and alias errors. With \(t_c=2\) fs and \(\sigma=2\) fs on R1’s resolved lattice, this matched at \(4.3\times10^{-15}\) relative; the expression as printed in T4 gave a relative discrepancy of 1.0.

Please give all T4–T6 bounds in the same units and specify whether they are per-bin or band norms. T5’s \(A e^{-(t_p/\tau)^2}\) alone is a time-domain amplitude, not a raw spectral-error bound. The leading-taper pointwise bound in §4.3 does not directly bound a band 2-norm; summing it introduces \(\sqrt{\sum_{k\in B}\omega_k^{-2}}\), with the appropriate DFT scale and sampling error. Finally, §4.1’s steady-tail example has an extra minus sign: \(\mathcal F[g'_+S]=S\mathcal F[g'_+]\) for constant \(S\); \(g'_+\) is already negative.

R5 — P2: make derivative evaluation safe near the breakpoints (§4.2, T1)#

Clipping \(\zeta\) does not prevent overflow in \(\Delta/u^2\) or \(\Delta/v^2\). The printed expression evaluated at nextafter(0, 1) for the valid rising edge \((a,b)=(0,1)\) returned a nonfinite value, with overflow/division-by-zero warnings, although the mathematical derivative tends to zero. Multiplying a tiny clipped factor by infinity cannot repair that. The clipped numerical gate also is not literally the analytic \(C^\infty\) gate at every endpoint.

Please require a scaled or logarithmic evaluation, or a justified branch that returns the asymptotic zero before forming dangerous reciprocals. Distinguish the analytic function from its finite-precision evaluation, and test both sides of all breakpoints, exact endpoints, very small valid widths and representative SI scales with warnings treated as errors. The existing gate’s clipping is not a sufficient derivative algorithm.

R6 — P1: separate physical propagation changes from numerical failure (§7.4)#

A negligible direct UV component of the removed source does not bound the UV endpoint change in a nonlinear march. Changing the near-infrared field changes later atomic driving and therefore later UV generation. The schematic perturbation equation is \(\delta y'=D\delta y+N'[y]\delta y+\delta N[y]\); the response term can transfer an initially near-infrared perturbation into the UV. SEP 20’s L4 discussion explicitly calls SEP 19’s transfer figures study targets, not transferable error bounds.

Please remove the inference that a change above 0.5% necessarily means an implementation error. The reviewer recommends treating it as a threshold for investigation: check matched-input sources, pin all scientific inputs, and use controlled/tighter-step comparisons to distinguish numerical error from propagated physical sensitivity. A hard accuracy gate instead requires an independently justified envelope. Define the endpoint metric and denominator, its floor, and how integration error is resolved at rtol=1e-2; that local tolerance alone does not certify a 0.5% global comparison.

Maintainer disposition (2026-10-18). John Travers agreed: use the 0.5% value as an investigation threshold, with the cause determined by convergence and controlled comparisons. The drafting agent should incorporate this decision into §7.4.

R7 — P2: distinguish the same-current decomposition from actual L5 (§2, §7.3, §9)#

The supporting taper addendum’s 4.60% → 1.22% and 1.44% → 1.45% figures apply both gates to the 100 fs current. Actual L5 compares a separately driven 60 fs current with the 100 fs current. The supporting report distinguishes these, and benchmarks/sep20/window.py deliberately reports an additional common-window comparison to expose the different startup and drive histories. They are not the same experiment. For example, its actual peak-power UV comparison is about 0.405%, whereas the same-current gate comparison is about 0.387%.

Please reproduce the same-current split as an anchor, then report the actual new-filter L5 comparison separately, preserving each raw current’s origin and identity. Do not require the latter to match the former using a tolerance derived only from the differentiation residual. That residual concerns closure of a sum; it does not separately bound both differentiated terms’ errors. A ratio metric also needs the reference norm and its perturbation in the bound. State whether \(10^{-3}\) means a dimensionless ratio (0.1 percentage point) and pin the denominator. Replaying raw currents under a new filter should create an explicit derived analysis record, with the original descriptor and analysis policy retained, not relabel the raw atomic evaluation.

The reported norm reduction can be described as a reduction by about 73%; it is not an additive or orthogonal allocation of 73% of the error to one mechanism. Keep box-to-box sensitivity, finite-window sensitivity and error against a converged physical reference distinct. The supporting study itself states that no tested box is an independently valid late-time reference after norm loss. Thus the remaining difference should not automatically be called a measured physical error or evidence that every remaining contribution is genuine emission.

R8 — P2: align compatibility promises with SEP 25 (§5, §7.2–7.3, §9, Q1)#

The repeated promises of bitwise-identical sources, diagnostics and L5/L6 replay conflict with SEP 25 and the current testing contract. Preserving the old filter’s formula and execution path is appropriate, but numerical recomputation can differ across builds, FFTs, thread counts and adaptive histories without being a regression.

Please retain exact equality for descriptor bytes, identities, resume keys, stored arrays copied losslessly, masks and accounting. Use justified rounding budgets for matched source calculations and accuracy envelopes for independent propagation or resume. Stored numerical witnesses need provenance and a comparison budget. A same-build bitwise observation may be recorded as evidence, not made a portable scientific requirement. Clarify Q1’s intended guarantee in these terms rather than weakening the numerical checks or changing the old filter.

R9 — P2: define diagnostics, their storage and the v4 cases (§5–6, §7.2)#

“Relative to the source” needs an explicit denominator: the truncating source, the corrected source, or both. These differ markedly for the steady-current control, where the corrected source is zero. Define each band, projection/weighting, collar placement, units and floor. Define \(\rho_-\) for a zero remainder, an empty sampled taper and multiple nodal rows. A ratio below its floor should be unavailable with an explanation, not NaN, infinity or a misleading zero.

The existing backend keeps only _tdse_last_projection and exposes it through tdse_last_evaluation; adding an edge dictionary there does not persist per-call history. Please decide whether the promise is a last-evaluation diagnostic or an actual per-evaluation record. For the latter, name the storage/callback, call identity and predictor/rejected-step labelling, bounded retention policy, and any archive schema implications. Extend the tests to distinguish these semantics and verify that no extra atomic calls occur.

For descriptor v4, explicitly include unwindowed SAE with gated_derivative and the grid collar: today that response follows the v2 layout, while v3’s validator requires haCC or an atomic window. Test round trips and resume for that case as well as windowed SAE and haCC, all historical v1–v3 cases, and invalid schema/filter combinations. An unchanged optical-output schema is plausible, but should follow this complete record/storage analysis.

R10 — P2: define the deferred rate-plasma measurement in current variables (§4.6, §11, Q3)#

The rate route gates a polarisation, whereas this proposal holds a current. If \(J=P'\), the existing continuous source contains \(i\omega\mathcal F[gP]=\mathcal F[gJ]+\mathcal F[g'P]\) before the common propagation factor. Simply applying the proposed formula to \(P\) would hold a different quantity; it would not implement the same current comparison.

Please give the offline experiment’s exact observable, derivative/integral ordering, Drude and ionisation-loss treatment, and discrete factors. Identify which saved fields or currents suffice to reconstruct it. Distinguish an offline comparison on fixed saved fields from an alternative propagated trajectory. This can remain a reported measurement and leave the production rate route unchanged, as Q3 intends.

R11 — P2: make default promotion and qualification milestones explicit (§7.5, §11–12)#

Phase 1 changes an omitted filter to the new form before Phase 2 has established the real-current comparisons, SAE controls and propagated consequences. Q2 defers promotion of the SAE research record, but ordinary new windowed SAE runs would already receive the changed default. Q4 also defers a maintainer decision until the new measurements, despite the closing statement that no question remains.

The reviewer recommends implementing the option first and promoting the default only after Phase 2 and the maintainer’s Q30 disposition. Specify what happens if the controls fail, what “pass” means for the SAE controls, and which decisions block acceptance, default promotion and Final status.

Maintainer disposition (2026-10-18). John Travers agreed: keep gated_derivative opt-in until Phase 2 has passed and he has decided the remaining Q30 accuracy exception. The drafting agent should align §5, §9, §11 and the deferred decisions in §12 with this ordering.

The final milestone should explicitly include the end-of-SEP full gate, slow, coverage and applicable Kokkos CPU qualification required by the current phased-SEP policy. A schema update in Phase 1 does not replace those final lanes. Add the required bounded hot-path benchmark/memory ledger, including projection, the embedded map and diagnostic work, before describing the overhead as measured negligible. No full-length propagation is requested by this review.

R12 — P2: make the public proposal independently reproducible (§1–3, §7, §12–13)#

The central numerical claims currently rely on an unpublished provider development report and addendum without a public bibliographic record. The proposal also embeds machine-specific execution plans and an ignored archive location in the maintainer’s checkout. These do not satisfy SEP 0’s requirement that the proposal stand alone; passing the current local-reference regular expression would not establish compliance.

Please publish a self-contained solprop evidence summary with the relevant equations, metrics, tables, limitations and immutable input/analysis identities, or cite a stable public report. Give artifact retrieval instructions that do not depend on a particular checkout. Keep private scheduling and storage details in the execution handoff; hardware belongs in measured benchmark evidence. No new physical claim should depend on access to a private development register. The reviewer could inspect the supporting report, but did not independently regenerate its atomic data or its addendum tables.

R13 — P3: correct and sharpen the primary-source citations (§13)#

Reference 3’s author is F. Brunel, not P. Brunel, and the cited 1987 paper concerns absorption at an overdense plasma surface. For the gas ionisation-current mechanism invoked here, add or use 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. The publisher records verify the author and the distinction in scope (1987 paper, 1990 paper).

The Planck profile was checked against equation (7) of McKechan et al., and its derivative was independently re-derived; that paper does not establish the new discrete error bound. Higham’s chapter and FFT assumptions were checked as described in R3. The bibliographic record of Geissler et al. was checked; its full derivation and the Brunel full texts were not independently re-audited. No new physical coefficient or constant was transcribed.

Review evidence and limits. The small numerical probes above used NumPy/SciPy and the existing gate/window resolver, with no native provider or propagation calls. They test specific claims and counterexamples, not the proposed implementation’s qualification. The production source, output and archive schemas are unchanged by this review.

Verification. git diff --check and codespell on this SEP passed; uv run pytest tests/test_seps.py passed (197 tests, one expected process-SEP skip); uv run poe docs passed with -W -n --keep-going, including the required tutorial executions. The checks used this review branch’s own locked environment. A separate comparison confirmed that Part I is unchanged. Full package, slow and coverage lanes were not run because this change contains review documentation only.

  • 2026-10-18: GPT-6 completed the AI review (R1–R13); John Travers’s human review was recorded at his request. Review status set to Reviewed, lifecycle status kept at Draft. John confirmed the investigation threshold in R6 and validation before default promotion in R11. The drafting agent owns the remaining responses and proposal revisions.

Response to review — 2026-10-18#

The drafting agent (Opus 5.5) revised Part I. The coordinating agent decided each item under the maintainer’s delegation, and the maintainer decided R6 and R11. The review text above is unchanged. Section numbers refer to the revised Part I: the former §4.5 and §4.6 are now §4.6 and §4.7, and §4.5 is the new rounding budget.

  • R1, accepted, via a computed certificate.

    • §4 now separates the continuous identity (§4.1), the discrete definition, its alias error (§4.4) and floating-point error (§4.5).

    • §4.4 adds an admission rule: each taper must span at least 128 samples.

    • §4.4 also adds the steady-current certificate \(\kappa_{\rm ss}=\|\mathrm{DFT}[g]-\mathrm{DFT}[g']/(i\omega)\|_A/\|\mathrm{DFT}[g]\|_A\), computed at translation for the actual lattice, gate and active band. Translation refuses a value above \(10^{-10}\), and the value is recorded in the run metadata. It is kept out of the descriptor so that identities never depend on a floating-point result.

    • A drafting-time table of \(\kappa_{\rm ss}\) against taper resolution is given; the default 5 fs taper has 800 samples and \(6\times10^{-16}\).

    • The saddle-point estimate is kept only as motivation, labelled as such, and the \(L^1\) “bound” is removed.

    • The new T7 refuses the review’s counterexample (ratio 1.0).

    • T3 covers asymmetric and off-grid breakpoints, the admission limit and 2–8× it with a refinement check, the highest retained frequency, and two FFT sizes.

    • T1 no longer claims an exponential trapezoid bound or treats a sampled \(g^{(5)}\) as a bound. It uses an independent closed form, a convergence-rate check and a reported edge weight.

  • R2, accepted. T2 is split into four cases: (a) a globally constant current gives zero to the budget plus the certificate floor; (b) zero current; (c) an interior rise plus a steady tail keeps the analytic rise; (d) a slope through the taper keeps \(gS'\). §4.1 states the step case \(He^{-i\omega t_c}/(i\omega)\). The §7.2 fake-provider drift now rises inside the plateau and is required to keep that rise.

  • R3, accepted. The new §4.5 gives an absolute, operand-based budget in raw FFT units. It sums today’s floor (raw \(|J|+|j_1|\)), the edge transform \(c_F\varepsilon\log_2N\sqrt N\|g'(|J|+|j_1|)\|_2/\omega_{\rm lo}\), the derivative-evaluation error, and the division and subtraction. Each term is scaled by the absolute projection and the factors. Aliasing is kept separate (§4.4). The FFT assumption covers 2-, 3- and 5-smooth sizes; T3 checks two sizes, and other sizes are flagged. The spectral-derivative oracle has its own budget, and T3 uses the sum.

  • R4, accepted.

    • §4.4 states the raw DFT normalisation and the \((-1)^k\) origin factor.

    • T4 uses the reviewer’s corrected oracle.

    • T4–T6 are in raw units, per bin unless a band norm is stated. T5’s bound is now spectral: \(Ae^{-(t_p/\tau)^2}W/(\Delta t\,\omega_k)\).

    • §4.3 gives both the per-bin and the band-norm bound, the latter with \(\sqrt{\sum\omega_k^{-2}}\) and the sampled edge weight.

    • The extra minus sign in §4.1 is removed.

  • R5, accepted. §4.2 evaluates \(g'\) through \(s=(x-a)/\Delta\). It returns the asymptotic zero, with no reciprocal formed, when \(\min(s,1-s)\le1/740\) (where \(|h|<10^{-300}\)), and it distinguishes the analytic function from its evaluation. T1 tests both sides of every breakpoint, the exact endpoints, nextafter neighbours, widths from \(10^{-30}\) s to 1 s and SI scales, with warnings as errors.

  • R6, accepted (maintainer). In §7.4, 0.5% in 90–200 nm is an investigation threshold, not an implementation-error inference. The cause is named by matched-input sources, pinned inputs and an rtol=3e-3 control. §7.4 defines the endpoint metric, its denominator (the current-form endpoint) and its floor (SEP 20 L6’s 0.073% restart sensitivity). It also states that rtol=1e-2 certifies no 0.5% global comparison.

  • R7, accepted.

    • §2 labels the table as the same-current split and says “reduces the norm difference by about 73%”. It calls the remainder the window sensitivity that remains, with no converged late-time reference.

    • §7.3 reports (i) the recorded figures, (ii) the same-current anchor and (iii) actual L5, each with its own expectation. (iii) borrows no tolerance.

    • (ii)’s tolerance is the dimensionless ratio \(\delta=\|F_{100}-F_{60}\|_B/\|F_{100}\|_B\), within \((2+\delta)(r+e_g)\), about 0.11 percentage points.

    • The replay writes a derived analysis record that keeps the raw descriptor.

    • §9 states Q30’s expectation in the same terms.

  • R8, accepted. §5, §7 (preamble, §7.2, §7.3) and §9 restrict exact equality to descriptor bytes, identities, resume keys, lossless copies, masks and accounting. Matched source recomputations use the §4.5 budgets, and propagation uses envelopes. The old filter keeps its formula and code path; a same-build bitwise repeat is evidence, not a requirement. Q1 is restated in these terms.

  • R9, accepted, with the smaller scope.

    • §5 makes the edge entry a last-evaluation diagnostic in tdse_last_evaluation, with no per-call store and no archive change. Per-call analysis is an offline replay (§7.3, §7.5).

    • The bands are active_mask and inband_mask, in the existing raw modal units with the collar. Both the truncating and the corrected norms are reported, with each ratio’s denominator stated and a floor for each.

    • A ratio below its floor is None with a reason. \(\rho_-\) is defined over rows, with the zero-remainder case covered; the admission excludes an empty taper.

    • §6 and §7.2 add descriptor v4 for unwindowed SAE on the collar, windowed SAE, windowed and unwindowed haCC, with round trips and resume for each. All v1–v3 cases and the invalid combinations are covered, and a test confirms no extra atomic calls.

  • R10, accepted. §4.7 defines the offline rate-plasma measurement in current variables.

    • The route’s source is \(i\omega\mathcal F[gP]=\mathcal F[gJ]+\mathcal F[g'P]\), and the held form is \(\mathcal F[gJ]-\mathcal F[g'J]/(i\omega)\).

    • \(J=J_{\rm free}+J_{\rm loss}\) is recomputed on fixed saved L6 rate fields, with the run’s own running-integral rule, crop and prefactor.

    • A closure residual measures the discrete \(P'=J\).

    • It is reported only; the production rate route is unchanged.

  • R11, accepted (maintainer). gated_derivative is opt-in through Phases 1 and 2 (§5, §9). Phase 3 promotes it only after Phase 2 passes and the maintainer decides Q4 and Q30. §7.5 defines “pass” for the SAE controls, and says the record stays on "time_collar" if they fail. §11 lists what blocks acceptance, promotion and Final, and adds the end-of-SEP full, slow, coverage and Kokkos CPU lanes. §7.2 adds a bounded hot-path benchmark with a memory ledger before any “negligible” claim. §12 reopens Q4 as deferred, and the “no question remains” line is removed.

  • R12, accepted in part.

    • Part I no longer names machines, checkout paths or scheduling. Those are in benchmarks/sep26/README.md.

    • The new benchmarks/sep26/evidence/0133-summary.md holds the equations, metric, tables, limitations and the SHA-256 identities of the provider study’s records, together with its revisions and retrieval by hash and archive layout. It is cited as reference 5, with the provider report as provenance.

    • Not done: the raw atomic arrays stay in the evidence archive and are not yet public, as for SEPs 12 and 20.

  • R13, accepted. Reference 3 is now F. Brunel, J. Opt. Soc. Am. B 7, 521 (1990), for the gas ionisation current. The 1987 paper concerns absorption at an overdense plasma surface, so it is dropped. Reference 1 cites eq. (7) and reference 2 cites the chapter.

Checks. git diff --check, codespell on the changed files, uv run pytest tests/test_seps.py and uv run poe docs (strict).

  • 2026-10-18: Part I revised in response to R1–R13 (above). Status stays Draft; gated_derivative is opt-in until Phase 3.

  • 2026-10-18: Accepted by the maintainer after the review and response; Part I frozen.

  • 2026-10-18: Status set to In progress; phase 1 started on feature/sep-0026-phase-1 (solprop and solprop-kokkos).

  • 2026-10-18: Phase 1 implemented on feature/sep-0026-phase-1 (solprop and solprop-kokkos) and reported in Part II; awaiting the coordinating agent’s review.

  • 2026-10-19: Phase 2 (validation) run on feature/sep-0026-validation and reported in Part II; the maintainer’s conditional Q4/Q30 decision is recorded there and its condition holds. Awaiting the coordinating agent’s review.

  • 2026-10-19: Phase 3 on feature/sep-0026-phase-3: "gated_derivative" became the haCC and windowed default, the SAE research record moved to it under a new identity, Q30 was restated and the rate-plasma question (Q10) moved to the roadmap. The end-of-SEP lanes passed, with one intermittent coverage-lane failure unrelated to this SEP. Reported in Part II.

  • 2026-10-19: Final, prepared on feature/sep-0026-phase-3 for the maintainer’s approval at merge. Every phase is reported and the end-of-SEP lanes passed (Phase 3 report). TDSE runs on numpy_ref only, under SEP 12’s recorded exception; numba_cpu and solprop-kokkos refuse it by name. Moved to the roadmap: the rate plasma’s collar truncation (Q3, Q10), the remaining window sensitivity (Q30, haCC and SAE) and the propagation of the moved SAE record on its own collar. Moved to archive/.

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