line-window-seeding#

Ledger row: line-window-seeding. Code under review: src/pyrite/montecarlo/spectrum/cxr_seeds.py (kinematic_cxr_seeds, absorption_edge_seeds, characteristic_line_seeds) and src/pyrite/_cxr_windows.py::build_window_plan. Acceptance evidence the row cites: line-spectrum error budget.

Initial fresh-context verification, 2026-09-17. The derivation below was written from the ledger row, the docstrings, and the cited upstream rows before the function bodies were read. No implementation file was modified.

The initial discrepancy verdict records the pre-fix implementation. Section 6 records the independent re-verification after commits 16b9d38c and 971648f1; the current verdict is anchored, with human sign-off pending.

Initial verdict summary (pre-fix)#

  • Claim: line-window-seeding. Code: montecarlo/spectrum/cxr_seeds.py::kinematic_cxr_seeds, ::absorption_edge_seeds, ::characteristic_line_seeds, and _cxr_windows.py::build_window_plan. Source: the vacuum resonance of line-energy-dispersion (Zhai SI Eq. 10), the finite-time width of finite-time-lineshape, \(\mu\) from absorption-length, and xraydb lines from characteristic-radiation.

  • Filters: units pass; limiting cases pass (both); signs and conventions pass.

  • Re-derivation: differs on two stated properties. The kernel conventions match exactly.

    1. The per-segment offset between the vacuum and in-medium resonance is \(\Delta E\simeq -E\,\delta\,\beta\cos\Theta/(1-\beta\cos\Theta)\). The docstring calls this “\(O(\delta)\sim10^{-5}\) relative, inside the tail_widths margin”. Its ratio to the margin \(\tau\,w_\varepsilon\) has no \(O(\delta)\) bound. It is measured at up to \(5.4\,w_\varepsilon\) against a margin of \(2\,w_\varepsilon\), above 300 eV, on hopg 300 keV tilt 85.

    2. The edge search picks the steepest bracket anywhere in \(\pm5\%\) of the xraydb energy, and a bracket already seeded is then skipped. Together these leave out every secondary shell that lies within 5% of a stronger jump. It is neither seeded nor reported as skipped. WSe\(_2\) loses Se L2 (\(\mu\) +35%), W M4 (+36%), and W L1 (+12.5%).

  • Verdict: discrepancy. Both findings are grid-coverage policy and not radiation physics. Neither one shows up in the coverage measured here. It also does not show up in the twelve-case window campaign as reported.

  • Suggested ledger change: set Status to discrepancy, and record the two diverging terms and this write-up in Checks and Notes.

1. Independent derivation#

Units are \(c=1\): lengths and times in Å, and \(\omega\) in Å\(^{-1}\), so \(E=\hbar c\,\omega\). A straight segment has velocity \(\mathbf v=\beta\hat{\mathbf v}\), length \(L\), and flight time \(t_L=L/\beta\). The observation direction is \(\hat{\mathbf n}\), and \(\cos\Theta=\hat{\mathbf v}\cdot\hat{\mathbf n}\).

1.1 Resonance, vacuum and in medium#

The emitted field of a segment is proportional to \(\int_0^{t_L}e^{i\Phi(t)}\,dt\), with the phase \(\Phi(t)=\omega t-(\mathbf k+\mathbf g)\cdot\mathbf v\,t\) and \(\mathbf k=n(\omega)\,\omega\,\hat{\mathbf n}\). Stationarity, \(d\Phi/dt=0\), gives

\[ \omega\,\bigl(1-n\,\beta\cos\Theta\bigr)=\mathbf v\cdot\mathbf g . \]

In vacuum (\(n=1\)) this is the ledgered resonance

\[ \omega_0=\frac{\mathbf v\cdot\mathbf g}{1-\mathbf v\cdot\hat{\mathbf n}}, \qquad E_0=\hbar c\,\omega_0 . \]

A segment radiates only when \(\omega_0>0\). The vacuum denominator satisfies \(1-\beta\cos\Theta\ge1-\beta>0\), so the sign test falls on \(\mathbf v\cdot\mathbf g\) alone.

With \(n=1-\delta\),

\[ \omega_1=\frac{\mathbf v\cdot\mathbf g}{1-\beta\cos\Theta+\delta\,\beta\cos\Theta}, \qquad \frac{\omega_1-\omega_0}{\omega_0} =-\frac{\delta\,\beta\cos\Theta}{1-\beta\cos\Theta}+O(\delta^2). \]

The relative shift is \(O(\delta)\) times a geometric factor. That factor is order unity for side emission, but it is enhanced for forward-going segments. When \(\beta\to1\) and \(\Theta\to0\) it tends to \(2\gamma^2/(1+\gamma^2\Theta^2)\).

1.2 Feature width and the shift in feature widths#

Near the vacuum root, the detuning rate is \(\Delta=(1-\beta\cos\Theta)(\omega-\omega_0)\). The segment integral is

\[ \Bigl|\int_0^{t_L}e^{i\Delta t}dt\Bigr|^2 =t_L^2\,\operatorname{sinc}^2\!\Bigl(\frac{\Delta t_L}{2}\Bigr), \qquad \operatorname{sinc}x=\frac{\sin x}{x}. \]

Its first zero is at \(\Delta t_L/2=\pi\). The zero lies at \(|\omega-\omega_0|=\pi/a_w\) with \(a_w=(1-\beta\cos\Theta)\,t_L/2\). In energy, the segment’s own feature width is

\[ w_{\rm seg}=\frac{2\pi\hbar c\,\beta}{(1-\beta\cos\Theta)\,L}. \]

Dividing the first-order shift by this width removes the geometric denominator:

\[ \frac{|E_1-E_0|}{w_{\rm seg}} =\delta\,|\cos\Theta|\,\frac{E\,L}{2\pi\hbar c} =\delta\,|\cos\Theta|\,\frac{L}{\lambda}. \]

This ratio is the number of refractive phase cycles accumulated over the segment. It is not \(O(\delta)\): it is \(\delta\) multiplied by \(L/\lambda\), which is about \(10^2\)–\(10^5\) for these segments. The measured maximum of the ratio is 0.13 at hopg 30 keV and 1.28 at hopg 300 keV.

The planner’s margin is not \(w_{\rm seg}\). It is \(\tau\,w_\varepsilon\), where \(\tau\) is tail_widths \(=2\) and \(w_\varepsilon\) is the \(\varepsilon\)-weighted lower-quantile width, which is near the narrowest. The per-segment condition the docstring asserts is therefore

\[ \boxed{\; \frac{E\,\delta\,\beta\,|\cos\Theta|}{1-\beta\cos\Theta}\;\le\;\tau\,w_\varepsilon \;} \]

No property of the construction guarantees this condition. With the free-electron estimate \(\delta=r_e\lambda^2n_e/(2\pi)\), graphite has \(\delta\approx1.9\times10^{-5}\) at 5 keV. That grows as \(E^{-2}\) to a few \(\times10^{-3}\) near 300 eV, where the C K edge adds its own anomalous structure. Forward segments (\(1-\beta\cos\Theta\ll1\)) raise the left-hand side further. The kernels’ 300 keV feature widths are 0.2–0.3 eV. Violation is therefore expected for forward-going segments with \(E\lesssim1\) keV at high beam energy.

The margin is not the only coverage mechanism. A window spans the \([\varepsilon/2,1-\varepsilon/2]\) quantile band of the whole population, and a shifted segment in the band interior stays covered. The quantity that matters for coverage is the in-medium weight that falls outside the window. Section 3 measures it.

1.3 Below ~300 eV#

The tabulated Re \(n\) can exceed 1 below roughly 300 eV (carbon: 6.24–285 eV, per _kernels.py::_in_medium_kinematics). In that range \(\delta<0\) and the shift changes sign, so the root moves upward. The magnitude of \(\delta\) is no longer small, and \(1-n\beta\cos\Theta\) can approach zero, which is a Cherenkov-like pole. A vacuum root there can be many feature widths from the kernel’s root. It can also belong to a segment that the kernel rejects, because the fixed-point iteration fails its convergence check and returns a NaN denominator. Vacuum seeding therefore gives no margin guarantee below about 300 eV. Only the population band, or the backbone, covers that region. The bandwidths of the hopg and wse2 cases start at 10–50 eV, so this region is inside the band.

1.4 Dropped-reflection bound#

Let \(W_r\) be the radiating proxy weight of reflection \(r\), and \(W=\sum_rW_r\). Reflections are sorted by ascending \(W_r\), and they are dropped while the cumulative dropped weight \(D\) satisfies \(D\le\varepsilon W\). Each kept reflection is covered from its lower weighted quantile \(q_-\) to its upper weighted quantile \(q_+\). The quantile is defined as the smallest ordered value whose cumulative weight reaches the target. That definition leaves strictly less than \(\varepsilon W_r/2\) below \(q_-\) and at most \(\varepsilon W_r/2\) above \(q_+\). The uncovered weight is therefore

\[ U\le D+\varepsilon\,(W-D)\le(2\varepsilon-\varepsilon d)\,W<2\varepsilon W, \qquad d=D/W\le\varepsilon . \]

The tail margin only shrinks \(U\). The bound is relative to the total radiating proxy weight. That total includes weight whose resonance lies outside \([\texttt{start},\texttt{stop}]\), so relative to in-band weight alone the bound can be looser by the in-band fraction.

1.5 Edges#

The attenuation that the kernels apply is \(\mu=2r_e\lambda\sum_in_if_{2,i}\), with \(f_2\) taken from the Chantler/FFAST table (absorption-length). The kernel evaluates \(\mu\) on a tabulation grid that contains every native Chantler node, and it interpolates \(\log\mu_i\) against \(\log E\). A step in \(\mu\) therefore lies inside one or more adjacent native Chantler brackets. That position is fixed by the Chantler node energies, not by the xraydb edge energy; the two tables disagree. Every shell whose own jump the table resolves needs its own bracket found. Taking the steepest bracket in a window \(\pm5\%\) wide cannot find a weaker jump when a stronger one lies inside the same window.

1.6 Characteristic lines#

mc_characteristic_spectrum deposits a line when two conditions hold: the line energy exceeds the relaxation cutoff, and the line has a positive yield per vacancy after the cutoff mask. The cutoff is the largest recommended cutoff among the composition’s elements. The seed set must be the same set, taken per composition.

2. Diff against the implementation#

Check 1: seed rows and velocity convention match the kernels#

Item

Seeds (cxr_seeds.py)

Kernels

Match

energy field

E_repr_keV else E_keV (l. 159)

same (lines/_setup.py:255)

yes

velocity

beta_from_keV(E)[:,None]*v_hat (l. 166–167)

beta_all[:, None] * seg_v (_setup.py:262–263)

yes

electron subset

elec_id < electron_limit (l. 163–165); runner passes Ne (runner/cxr_grid.py:95)

seg_elec_id < Ne (_setup.py:261)

yes

orientation

_orientation_R(lattice, beam_uvw, azimuth_rad, recip_miscut_rad, surface_hkl=) (l. 187–189); same case/radiator fallbacks (l. 439–444)

same call (_setup.py:203–209); runner keys (runner/__init__.py:651–655, 672–692)

yes

row order

mosaic_rotation @ (rotation @ g_hkl), weight mosaic_weight (l. 194–204)

R_m @ (R_orient @ g_vec), wm (_per_hkl.py:661–673; _batched.py:231–243)

yes

mosaic quadrature

_mosaic_quadrature(case["mosaic_mc_fwhm_rad"], case["mosaic_mc_nodes"])

same helper and keys (_setup.py:466)

yes

resonance

\(\hbar c\,(\mathbf v\cdot\mathbf g)/(1-\mathbf v\cdot\hat{\mathbf n})\), vacuum (l. 168, 201)

same, then in-medium fixed point (_per_hkl.py:387–390; _kernels.py:460–520)

vacuum by design; see check 2

10 eV floor

resonance > _MIN_RESONANCE_EV \(=10\) (l. 72, 202)

E_res > 10.0 on the in-medium root (_per_hkl.py:399; _batched.py:431)

yes, up to the in-medium shift

intensity proxy

\(t_L^2=(L/\beta)^2\) (l. 169)

\(t_L\) = seg_L / beta_all (_setup.py:356)

yes

Numeric point (independent constants: \(\hbar c=1973.269804\) eV Å, \(m_ec^2=510.99895\) keV, and \(|\mathbf g_{002}|=4\pi/6.711\) Å\(^{-1}\)). The inputs are hopg (002), 100 keV, flight along \(\mathbf g\), and \(\hat{\mathbf n}\) at 60°. The seed centre and the closed form agree to \(2.2\times10^{-16}\) relative, both at 2790.5776 eV.

Differences that are disclosed or harmless:

  • The seeds read unclipped segments. The kernels first apply _clip_segments_to_cutoff with E_cut_lines_keV (default 5 keV), which drops rows below 5 keV and shortens terminal flights. The docstring discloses this. The resulting band is wider, but the quantile is not strictly conservative: an extra heavy interior weight moves \(q_\pm\) inward in relative terms of the true population. Not measured.

  • Under max_dE_frac substeps, the kernels add the rows of one flight coherently (_setup.py:264–290). The per-row \(t_L\) then overstates the width of the flight’s feature. Both feature_width_eV and narrowest_feature_width_eV are per-row. This is opt-in only (transport/api.py default max_dE_frac=0.0), and it is outside the measured cases.

Verdict: pass.

Check 2: vacuum root inside the tail margin of the in-medium root#

The implementation seeds from the vacuum root (l. 201), exactly as documented. The divergence is in the docstring’s justification (l. 119–123) and in the ledger Check. The derivation in §1.2 shows that the per-segment offset in units of the margin is not \(O(\delta)\). Measured results appear in §3. Verdict: fail as stated. Coverage is intact in every measured case, because the population band, not the margin, contains the shifted roots.

Check 3: dropped-reflection rule#

The rule is at l. 227–234: share == 0.0 or dropped_weight + share <= epsilon * total over the reflections sorted ascending, stopping at the first failure. The quantiles use searchsorted(..., side="left") (l. 80–85) at \((\varepsilon/2,\,1/2,\,1-\varepsilon/2)\) (l. 240–244). This is exactly the construction of §1.4, so \(U<2\varepsilon W\). A randomized check ran 300 synthetic populations with six reflections and \(\varepsilon\in\{10^{-3},10^{-2},0.05,0.2\}\), using a margin of essentially zero. The largest value of \(U/(2\varepsilon W)\) was 0.93. The bound held and was nearly tight. Verdict: pass, with the in-band normalization caveat of §1.4.

A robustness finding, not physics: _cxr_grid_policy accepts tail_widths = 0 (l. 345–347). A reflection whose radiating population is a single value, or all coincident values, then produces below_eV = above_eV = 0, and FeatureSeed raises has an empty window and no anchor. The failure was reproduced with tail_widths=0.0 in the randomized check.

Check 4: Chantler bracket versus xraydb edge energy#

The edge seeds read load_henke, which returns xraydb.chantler_energies and f2_chantler (materials/atomic.py:225–236). This is the same table that absorption_length_ang and the kernel’s native-node tabulation read. Both bracket nodes become anchors (l. 326–348). An independent recomputation with xraydb gives the steepest adjacent \(f_2\) ratio in \(\pm5\%\), and the xraydb energy lies above the bracket by:

Edge

xraydb (eV)

Chantler bracket (eV)

xraydb above lower / upper node

N K

409.9

401.41–401.60

2.11% / 2.07%

O K

543.1

531.71–531.99

2.14% / 2.09%

F K

696.7

685.00–685.39

1.71% / 1.65%

P L3

135.0

132.13–132.20

2.18% / 2.12%

C K

284.2

283.67–283.80

0.19% / 0.14%

The “up to 2.1%” statement holds when measured to the upper node. Measured to the lower node it is 2.18%, and the 5% search window absorbs either figure. The offset is not always positive: S L3 xraydb (162.5 eV) sits 1.3% below its bracket (164.66–164.76 eV). The symmetric search handles that case. Verdict: pass for the check as worded.

Divergence in claim (2), which says “the absorption jumps of the Chantler table”: the steepest bracket is chosen over the whole window (l. 308–315), and a bracket index that has already been seeded is skipped (l. 319–322). Suppose a shell’s own jump lies within 5% of a stronger jump. The search then returns the stronger bracket, and the shell is dropped. It does not appear in the seeds or in skipped. The docstring describes such brackets as “shared by two shells”, but the two shells have separate brackets in the table:

Shell

xraydb (eV)

Own Chantler jump (eV)

Modelled \(\mu\) step

Search returns

Se L2

1474.3

1474.16–1478.24 (ratios 1.13, 1.07, 1.11)

\(\times1.35\)

Se L3 bracket 1433.81

W M4

1872.0

1867.58–1878.66 (1.11, 1.09, 1.11)

\(\times1.36\)

W M5 bracket 1805.32

W L1

12100

12092.59–12107.01 (1.115)

\(\times1.125\)

W L2 bracket 11538.43

Mo L2

2625

2621.48–2628.72 (1.10, 1.11, 1.08)

—

Mo L3 bracket

Ga, As L2

1143.2, 1359.1

1140.8–1143.8, 1356.7–1360.5

—

L3 bracket

Ti L2

460.2

461.02–461.98 (1.17, 1.14)

—

Ti L3 bracket

The \(\mu\) steps come from absorption_length_ang across each window. For comparison, W L2 (\(f_2\) ratio 1.28) and Mo M5 (1.30) are seeded. The absorption_edge_seeds(["W", "Se"], 10, 20000) call returns only Se L3, Se K, W M5, W L3, and W L2. The wse2 bandwidths (10–1700, 50–3100, and 50–10200 eV at 30, 100, and 300 keV) contain Se L2 in every case and W M4 in two. The single-bracket threshold EDGE_MIN_F2_RATIO = 1.05 also skips W M3, whose jump is split over three nodes (1.041, 1.039, 1.032), about 12% in total. That behaviour is the stated policy.

Divergent term: argmax over the whole \(\pm5\%\) window combined with if index in seen: continue.

Check 5: characteristic cutoff#

The seeds use max(table.recommended_cutoff_eV) over the composition’s elements, then keep lines with line_yield_per_vacancy.any(axis=0) and centre > cutoff (l. 378–388). The kernel uses the same maximum (characteristic.py:764–766) and skips line_energy <= relaxation_cutoff. It also requires _xraydb_line_yields(table, cutoff) > 0, which is line_yield_per_vacancy with columns at or below the cutoff zeroed (l. 692–694, 817–827). The two sets agree for non-negative yields.

The compositions match as well. The kernel uses case["composition"] for one layer and abs_layers[i][2] for each layer (runner/emission.py:96–123). The seeds use abs_layers compositions when present and composition otherwise (l. 466–470).

An enumeration over all 26 catalog crystal elements compared the kernel’s line set with the seed set and found zero mismatches. The seeds’ extra condition fwhm > 0 removes no line.

Verdict: pass.

Limiting cases (build_window_plan)#

  • Single straight flight. The quantiles collapse to \(E_0\). The window is \(E_0\pm\tau w\) at spacing \(w/8\), and the reflection is never dropped, since \(W_r\le\varepsilon W_r\) is impossible for \(\varepsilon<1\). This is confirmed by the independent numeric point above (half-extents \(1.6=2\times0.8\) eV, spacing 0.1 eV) and by test_single_flight_window_is_centred_on_the_closed_form_resonance. Pass.

  • No seeds. The breakpoints are \(\{\texttt{start},\texttt{stop}\}\), giving one piece at the backbone spacing. coordinates() returns np.linspace(start, stop, resolution_num(...)) (_cxr_windows.py:159–165, 228–235). Across 2000 random \((\texttt{start},\texttt{stop},h)\) triples there were zero mismatches with np.array_equal. Pass.

Tests#

pyrite-dev test tests/energy-grid/test_line_seeds.py tests/energy-grid/test_line_windows.py gives 28 passed. No test covers a secondary shell near a stronger edge, and no test covers the in-medium offset.

3. Measured vacuum-to-in-medium offsets#

The script builds each case with build_ladder_case and one CPU transport (PYRITE_MC_BACKEND=cpu, seed 7, small \(N_e\)). It computes the kernel’s own in-medium root per segment with _in_medium_kinematics over refractive_index(...).real on a \(3\times10^1\)–\(1.2\,E_{\rm stop}\) geometric table. Seeds use feature_width_eV \(=w_\varepsilon\) from sinc_feature_spacing at \(\varepsilon=10^{-3}\).

The first-order formula of §1.1 reproduces the kernel’s shift to 1.1–1.5% of the largest shift.

Case (\(N_e\))

\(w_\varepsilon\) (eV)

max rel. shift, \(E_0>300\) eV

max shift / \(w_\varepsilon\) (worst per-reflection p99)

in-medium weight inside window vs vacuum

hopg 30 keV tilt 5 (60)

1.094

\(1.2\times10^{-3}\)

0.41 (0.27)

identical

wse2 30 keV tilt 85 (40)

10.76

\(3.8\times10^{-3}\)

0.11 (0.09)

identical

hopg 300 keV tilt 85 (20)

0.310

\(4.6\times10^{-3}\)

5.4 (4.2)

identical to \(<10^{-8}\)

wse2 300 keV tilt 5 (10)

2.206

\(2.7\times10^{-2}\)

3.8 (1.6)

differs by \(\le6\times10^{-6}\)

In the hopg 300 keV case, 0.7–2.7% of each reflection’s proxy weight above 300 eV has an in-medium root more than \(2w_\varepsilon\) from its vacuum root. Those segments are forward-going (\(\cos\Theta=0.76\)–\(0.99\)), lie at \(E_0=300\)–\(1300\) eV, and are all shifted downward, as §1.1 predicts. Median relative shifts are \(2\times10^{-5}\)–\(3\times10^{-4}\), and the maxima reach \(10^{-3}\)–\(10^{-2}\). “\(\sim10^{-5}\)” is therefore an underestimate.

Below 300 eV, the largest offset is 2.7–3.2 \(w_\varepsilon\) at hopg 30 keV and 18–35 \(w_\varepsilon\) at hopg 300 keV. The kernel rejects (returns NaN for) \(6\times10^{-5}\)–\(1\times10^{-2}\) of the vacuum-radiating weight.

Coverage survives all of this in these cases because the windows are very wide. The hopg 300 keV (002) window is 9.8–3381 eV at 0.039 eV spacing. Below 300 eV, the in-medium weight outside the window is 0 to \(9\times10^{-4}\) of that sub-band. That is at or below the vacuum figure in every reflection. The protection comes from the breadth of the population, not from the margin. A narrow population, such as a thin foil at low tilt, would not have it.

4. Acceptance evidence, read critically#

The row also requires the #101 window ladder. The error budget reports that the finest windowed rung of all twelve cases agrees with a uniform reference inside the window share. Worst cases: yield \(1.4\times10^{-4}\), FWHM \(8.7\times10^{-4}\). These numbers could not be reproduced here. The raw JSON (line_window_convergence_101b.json) is deliberately untracked and lives on the remote box. Reading the report and the harness (energy_grid/convergence_case.py):

  • The ladder is blind to seed placement. window_ladder rebuilds the seeds per rung, but only samples_per_feature changes. Window positions and extents are fixed, and edge spacing does not refine. The Richardson gate therefore cannot detect a mis-placed or missing window, including the two findings above. Only the comparison against the uniform reference can.

  • The reference is capped. reference_grid uses max(finest window spacing, span / 400000). For hopg 300 keV, with the band at 50–9100 eV, that gives about 0.023 eV. This is only 1.7× finer than the 8-sample rung it is compared against, since the 16- and 32-sample rungs were dropped. It is still about 13 samples per \(w_\varepsilon\), so it is a credible reference, but it is not “dense” relative to the rung. The report should state the reference spacing per case.

  • Windows degenerate to uniform at 100–300 keV. The report says so for hopg 100 keV tilt 5, and it is reproduced here for hopg 300 keV. In those cases the comparison tests a fine uniform grid, not the window/backbone transition. The evidence that windowing proper holds accuracy rests mostly on the 30 keV cases.

  • Per-case gate outcomes are not tabulated. The ledger Check asks that yield, centroid, and FWHM converge under window refinement. The report tabulates only the worst reference differences. It states one Richardson failure (wse2 100 keV tilt 85, line/background, which is not a required observable), and it states that hopg 300 keV rests on one 2/4/8 triple. Whether FWHM passes the gate in every case is asserted (“converges in … FWHM”) only for wse2 100/85.

  • The width-tail argument is consistent with these runs. samples_at_narrowest came out as exactly 8.0 in the hopg 30 keV (\(N_e=60\)) and hopg 300 keV (\(N_e=20\)) seeds here. It does not cover the substep case noted under check 1.

With these qualifications, the acceptance evidence is plausible. It does not test the two coverage findings, because the ladder cannot see them, and the reference comparison did not flag them in the cases run.

6. Fresh-context re-verification after remediation#

Fresh-context re-verification on 2026-09-17 started again from the cited source equations, units, assumptions, and limiting cases before inspecting the remediated implementation. No implementation file was modified.

  • Claim: line-window-seeding — montecarlo/spectrum/cxr_seeds.py::kinematic_cxr_seeds, ::absorption_edge_seeds, and ::characteristic_line_seeds. The governing equations are the in-medium resonance \(E_{\rm res}=\hbar c\,(\mathbf v\cdot\mathbf g)/ (1-\operatorname{Re}n(E_{\rm res})\,\mathbf v\cdot\hat{\mathbf n})\) and the finite-time first-zero width \(2\pi\hbar c/(D t_L)\), together with the Chantler \(f_2\) edge brackets and xraydb characteristic energies and widths.

  • Filters: units pass; limiting cases pass; signs and conventions pass, conditional on the repository’s inherited positive-\(\mathbf g\) convention. This claim does not resolve the separate line-energy-dispersion harmonic-sign discrepancy.

  • Re-derivation: matches. Production passes the case band and composition into kinematic seeding; seeding builds the kernels’ padded native-node table, calls their _in_medium_kinematics, rejects the same unsettled roots, and derives feature widths from the same denominator as a_width. The independent pointwise-Brent regression pins the in-medium centre.

  • Secondary-edge remediation: matches. Shells are processed strongest first, but each weaker shell selects the strongest qualifying bracket outside already claimed native-node windows. The Se L2 regression pins the previously omitted nearby edge; independent probes also found W M4/L1 and Mo/Ga/As/Ti L2 anchors within 0.24% of their nominal edges.

  • Verdict: rederived; existing regression anchors advance the ledger state to anchored. Human sign-off remains required.

Focused verification reported 54 passing seed/window tests and five passing documentation tests. The built page contained rendered math notranslate nodes with no surviving dollar delimiters. A supplementary Z=3–92 catalog sweep was incomplete because the current xraydb/SciPy stack rejects Cs’s non-strictly-increasing Chantler coordinates; this does not affect the targeted secondary-edge checks above.