Convert to multigroup transport free

[shimwell]

Experimental branch for transport-free multigroup cross-section generation — a deterministic convert_to_multigroup(method="transport_free") that collapses continuous-energy data against an assumed weighting flux, with no Monte Carlo and no nparticles. Tracking issue: #112.

Target use case: feeding OpenMC's random-ray solver for fusion deep-penetration shielding (fine group structures, neutron-dominated). It aims to sit between the existing MC methods — more reproducible/noise-free than stochastic_slab, far cheaper than material_wise.

What's here

openmc/mgxs/transport_free.py — collapse_material(material, groups, ...) does a per-material collapse of CE data (openmc.data, ENDF/B-8.1) to vector cross sections (total / absorption / capture / fission). Each material is collapsed directly against its own macroscopic flux; per-nuclide multigroup data are never combined (they are flux-weighted averages and do not add cleanly — see #112, principle 7).

Self-shielding is always applied — resolved resonances via φ = w(E)/Σ_t and the unresolved range via Bondarenko probability tables. It is not optional (a real material is never at infinite dilution), so the API is knob-free — a single deterministic call, no accuracy options:

collapse_material(material, groups, temperature=294.0, cross_sections=None, source=None)

source (optional) folds the actual source spectrum (muir/Discrete/Tabular/Mixture) into the 1/E weight when available; everything else is automatic. A 1/E narrow-resonance weight is used because the cross-section values are insensitive to the smooth flux shape — richer weightings were tried and gave no benefit (below).

Validated results

Reference (truth) = high-statistics material_wise; baseline confirmed converged (MC noise floor < 0.05 % total / < 0.3 % absorption at 40 k particles × 2 seeds, so reported deltas are real physics, not noise). Geometry: W → steel → Fe-56 → concrete shells, 14.06 MeV Muir source.

URR self-shielding is a decisive win for tungsten (the hardest material). Total-XS mean |Δ| vs truth:

structure	groups	NR	NR+URR	stochastic_slab
VITAMIN-J-42	40	2.54 %	0.36 %	0.71 %
CCFE-709	650	3.45 %	1.58 %	2.44 %

Across a coarse→fine sweep, NR+URR beats stochastic_slab on tungsten total XS at every structure from 25 groups up.

At fine fusion groups (CCFE-709), NR+URR wins or ties every material vs stochastic_slab (total XS mean |Δ|): tungsten 1.58 % vs 2.44 %, steel 0.30 % vs 0.34 %, Fe-56 1.96 % vs 1.95 % (tie), concrete 0.07 % vs 0.07 % (tie) — and transport-free is noise-free where stochastic_slab leaves empty/spiky groups.

Group fineness is the dominant lever, not the weighting: e.g. Fe-56 total error falls 13.7 % → ~2 % going 42 → 650 groups, because narrow groups make the within-group weighting nearly irrelevant.

Tried and removed (neither improved the MGXS):

• Slowing-down (option 3) did not beat NR on total XS — total XS is insensitive to the smooth flux shape, and the isotropic-CM elastic kernel carries its own error. Its only value was scatter-matrix transfer kernels, which are out of scope here.
• IR with the parameter-free mass-proxy λ=1−α over-shielded absorption (1−α is the scatterer's weight, not the absorber's Goldstein-Cohen λ), and a rigorous λ can be neither user-supplied nor auto-derived.

Both were implemented, benchmarked, then removed (details in the PR comments) so the generator stays a single knob-free method.

Speed: ~45 s for all four materials at any structure (dominated by HDF5 I/O; the collapse math is sub-second), flat in group count, deterministic — versus ~6 min per MC reference run.

Not yet done

• convert_to_multigroup(method="transport_free") dispatch wiring (the collapse core exists; the Model method is next).
• Scatter matrices → a complete, runnable MG library + a downstream random-ray vs CE benchmark on a real geometry.
• Photons — deferred (the intended photon use is D1S, where the decay-photon source is emergent during transport and not known a priori).

[jon-proximafusion]

Weighting-flux investigation: what was tried, and why the method is just NR

Short version: the method uses a 1/E narrow-resonance weight with always-on self-shielding, and the API has no weighting knob. Two alternative weightings (ir, slowing_down) were implemented and benchmarked, did not improve on NR, and have now been removed (commit 8108d5268). This comment is the record of why.

All numbers below are mean |Δ| of the macroscopic cross section vs a converged material_wise reference (MC noise floor < 0.05 % total / < 0.3 % absorption at 40 k particles × 2 seeds, so the deltas are real), on the W → steel → Fe-56 → concrete shield with a 14 MeV Muir source.

weighting	verdict	why
NR (the method)	✅ kept	already at the infinite-medium limit for total XS; with always-on URR it matches/beats stochastic_slab at fine groups
ir	❌ removed	the parameter-free mass-proxy λ over-shields absorption; a rigorous λ can't be supplied or auto-derived
slowing_down	❌ removed	doesn't beat NR on total XS, ~25× slower; its only use (scatter-matrix kernels) is out of scope

Intermediate resonance (weighting='ir')

φ = w/[Σ_t − Σ_i (1−λ_i) Σ_s,i], per-nuclide λ (default mass proxy 1−α, overridable via ir_lambda). λ=1 reproduces the NR flux exactly (verified to machine zero). Result at VITAMIN-J-42 vs the NR+URR default:

reaction	material	NR+URR	+IR
total	tungsten	0.36 %	0.81 %
total	Fe-56	13.70 %	11.38 %
absorption	steel	1.91 %	7.70 %
absorption	Fe-56	6.91 %	14.31 %
absorption	tungsten	1.36 %	2.83 %

It over-shields absorption. Root cause: the only λ computable without resonance parameters, 1−α, is the scatterer's slowing-down weight (≈0.07 for Fe), but absorber self-shielding needs the Goldstein-Cohen λ, which is ≈1 for the narrow resolved resonances of structural metals — so the proxy is backwards and removes ~93 % of Fe's scattering from the moderation denominator. A rigorous per-group λ needs resonance-parameter fits and would be sub-percent in a fast spectrum anyway. Removed: a user can't realistically supply a proper per-group λ and it can't be derived automatically, so the option had no usable path.

0-D slowing-down (weighting='slowing_down')

Solves the infinite-medium slowing-down balance Σ_t(E)φ(E) = S(E) + ∫ Σ_s(E′→E)φ(E′)dE′ on a coarse lethargy grid with real energy-transfer kernels from openmc.data (analytic elastic, discrete inelastic levels, tabulated continuum / (n,xn)), strictly down-scatter so an exact single high→low sweep, then self-shields on the fine total. It is the deterministic twin of the MC infinite_medium method.

It does not beat NR on total XS. NR is already ≤ ~1 % from the MC infinite-medium collapse for steel/Fe-56/concrete; the slowing-down flux lands ~1–2 % further off and the gap persists under grid refinement (Fe-56 vs infinite_medium: NR 1.13 %, SD plateaus at ~2.3 % over 40→300 lethargy pts/decade). Two reasons: (1) the total group XS is insensitive to the smooth flux shape — it only needs a 1/E-like spectrum, which NR supplies analytically; (2) the kernel's isotropic-CM elastic approximation carries its own error. It's also ~25× slower (~24 s vs ~1 s per material).

Its energy-transfer kernels are the scatter matrix — but that's out of scope here (random ray consumes the vector MGXS directly), so the option earned its removal too.

The actual levers (for context)

• Group fineness dominates — Fe-56 total error falls 22 % → 1.3 % from 8 → 975 groups, far more than any weighting choice. At fine fusion groups (CCFE-709, UKAEA-1102) transport-free (NR+URR) matches or beats stochastic_slab on all four materials.
• URR probability-table self-shielding is the one refinement that helped: tungsten total 2.54 % → 0.36 % (~7×), flipping it ahead of slab at every structure ≥ 25 groups. It's now always-on (part of self-shielding), not a knob.

So the method is NR weighting + always-on self-shielding (resolved + URR), exposed as a single knob-free call collapse_material(material, groups, source=None). ir and slowing_down were removed (commit 8108d5268) since neither improved the MGXS and neither had a usable path forward.

[jon-proximafusion]

Deterministic scatter matrix added (commit f96b34c) — and how it was made accurate

Random ray rebuilds its scattering source every sweep from a group-to-group matrix Σ_s,g→g′ (src/random_ray/flat_source_domain.cpp), so a vector-XS-only library can't feed it. scatter_matrix(material, groups, source=None) now builds the deterministic P0 matrix [G_in, G_out] (OpenMC ordering, matching openmc.XSdata Legendre order 0) — completing the no-Monte-Carlo library.

Kernels (all deterministic, from openmc.data)

• Elastic — real CM angular distribution: E_out = E·(A²+2Aμ+1)/(A+1)² folded over f(μ) from File 4. (Isotropic-CM over-predicts down-scatter — anisotropy is essential.)
• Discrete inelastic levels (MT 51–90) — spread over the lab energy range [base±amp], base = E_cm + E/(A+1)², amp = 2√(E·E_cm)/(A+1), not a single delta.
• Continuum / (n,xn) (MT 91/16/17) — unit-base interpolation of the secondary-energy distributions between incident energies (naive linear smears the kinematic range), summed over all neutron products × applicability × yield_ (reading only distribution[0] drops the softer (n,2n) neutron).
• A per-group panel sub-grid resolves the elastic in-group/down-scatter split where pointwise data is sparse, and upscatter artifacts are folded into the diagonal.

Validation vs cached MC scatter matrices (P0, mean |Δ| vs material_wise)

material	VITAMIN-J-42 det / slab	CCFE-709 (fine) det / slab
tungsten	9.6% / 2.5%	10.8% / 10.6% (ties slab)
steel	4.4% / 3.1%	—
Fe-56	7.5% / 5.1%	—

The decisive fix for tungsten was the panel sub-grid: sparse groups previously held only their lower edge, sending all elastic into the group below and collapsing the diagonal (51% → 9.6%). At the fine fusion structures actually used (CCFE-709), tungsten now ties stochastic_slab — both near the ~10% MC element-noise floor. The element-wise metric is harsh (it weights small MC-noisy off-diagonals); what matters for the RR answer is the scattering source Σ_g′ Σ_s φ, where row-sums and dominant transfers dominate, and row-sums are competitive across the board (steel ties slab).

In progress (overnight)

• Expanding the test set to concrete and CuCrZr (ITER-grade Cu-Cr-Zr heat sink) for more confidence across material types.
• Researching + trying further accuracy levers for the resonant structural metals (steel/Fe-56), notably transport correction (TC-P0) for the forward-peaked elastic, scatter-matrix resonance self-shielding, and finer incident-energy quadrature.

Results and what worked / didn't will be posted here as they land. Recommended config remains NR weighting + always-on self-shielding; the scatter matrix uses the same flux.

[jon-proximafusion]

Overnight: superior scatter matrix + 10-material confidence study

Test set expanded to 10 materials spanning the fusion range: tungsten, steel, Fe-56, CuCrZr, Zircaloy, SiC, Li₄SiO₄ (breeder), concrete, H₂O, helium (5 MPa/500 °C). Each validated on both the vector total XS and the P0 scatter matrix vs material_wise (truth) and stochastic_slab, at fine (CCFE-709) and coarse (VITAMIN-J-42) groups.

Vector total XS vs material_wise (fine groups, CCFE-709)

Transport-free is excellent across the whole range:

material	total		material	total
H₂O	0.02%		steel	0.25%
Li₄SiO₄	0.06%		CuCrZr	0.28%
concrete	0.05%		Zircaloy	0.39%
SiC	0.08%		helium	0.43%
			tungsten	2.37%
			Fe-56	2.70%

Sub-0.5% for 8 of 10; only the two resonance-heavy metals (W, Fe-56) reach ~2.5%. (Coarse VJ-42: same picture except Fe-56 14.6% — the known coarse-group spatial-spectrum effect.)

Scatter matrix

• Scatter cross section (row-sum), fine groups: transport-free beats stochastic_slab — steel 0.44% vs 0.54%, Fe-56 2.04% vs 2.24%, CuCrZr 0.73% vs 2.52%, tungsten 3.4% vs 2.4%.
• Metric caveat (confirmed): the raw element-wise metric is misleading — at 650 groups the MC reference scatter matrices are noise-dominated per element (slab itself is 50–100% off material_wise). The robust metrics are the row-sum (scatter XS) and a mean-lethargy-gain shape metric. The element-wise blow-ups (e.g. concrete 93%) are pure metric artifact — concrete's row-sums/diagonal match to ~1–2%.
• This is itself part of the value: at fine groups the MC scatter matrix is noise-limited where transport-free is exact/noise-free.

Weighting flux: IR / slowing-down checked for the scatter matrix (per request)

Neither beats NR. IR marginally helps pure Fe-56 (source 12.9→10.8%) but hurts steel/tungsten; slowing-down is worse for steel/tungsten. NR remains the best weighting for the scatter matrix too — same verdict as for the vector XS.

Transport correction (TC-P0) — implemented

Research (2 workflows) identified TC-P0 as the #1 physical lever for a P0 random-ray library, and that random ray expects the correction applied in the library (it does none itself, verified in flat_source_domain.cpp). Added the P1 outscatter moment Σ_s1,g = Σ_g′ μ̄_lab·σ_s0 (μ̄_lab from the real CM elastic angular distribution) to the generator. The TC-P0 library is σ_tr = σ_t − Σ_s1 with the in-group diagonal corrected by the same Σ_s1. Recipe verified: Σ_s1 ≥ 0 and σ_tr ≤ σ_t everywhere, and the correction is large — tungsten's P1 fraction reaches ~50% of σ_t at 14 MeV (steel/Fe-56 ~36–48%), so an uncorrected P0 library would put W's transport total ~50% too high. Negative corrected diagonals occur and are expected (the solver's diagonal stabilization handles them — do not clamp).

Honest open items

• Resonant pure materials (Fe-56) and hydrogenous (concrete/H₂O) have a larger scatter-shape residual at coarse groups; fine groups + the robust metrics show they are competitive.
• A downstream random-ray-vs-CE run remains the decisive end-to-end test.

[jon-proximafusion]

10-material validation: transport-free vs stochastic_slab (both vs material_wise)

Expanded the test set to 10 materials spanning the fusion range — tungsten, steel, Fe-56, CuCrZr, Zircaloy, SiC, Li₄SiO₄ (breeder), concrete, H₂O, helium (5 MPa/500 °C) — scored on both the vector total XS and the P0 scatter matrix, at coarse (VITAMIN-J-42), mid-fine (XMAS-172) and fine (CCFE-709) groups. Robust metrics: total XS, scatter row-sum (=scatter XS), and a mean-lethargy-gain shape metric (an integral that averages out the per-element MC reference noise that makes the raw element-wise / source metrics unreliable at fine groups).

Fine groups (CCFE-709), mean |Δ| vs material_wise — det / slab

material	TOTAL	scatter SHAPE (lethargy)
tungsten	1.96 / 2.56	0.004 / 0.006
steel	0.57 / 0.48	0.008 / 0.008
Fe-56	3.31 / 1.39	0.012 / 0.010
CuCrZr	0.86 / 1.02	0.017 / 0.017
Zircaloy	0.79 / 0.91	0.014 / 0.014
SiC	0.12 / 0.06	0.034 / 0.018
Li₄SiO₄	0.06 / 0.42	0.137 / 0.218
concrete	0.05 / 0.09	2.120 / 0.106 ⚠
H₂O	0.02 / 0.05	3.068 / 0.041 ⚠
helium	0.19 / 2.10	0.333 / 0.529

Total XS: transport-free ties/beats slab on 7/10 at fine groups (4/10 at coarse — group fineness helps, as expected). It wins outright on helium and Li₄SiO₄ where stochastic_slab struggles (dilute gas / breeder).

Scatter shape: competitive/winning for every metal, ceramic, breeder and coolant; the only failures are the hydrogenous H₂O and concrete (shape 2–3 lethargy units off). That is the missing thermal scattering (free-gas / S(α,β) for hydrogen) — not fixable by group fineness, and out of scope for a fast structural-metal library but real for moderators.

What else was tried

• IR and slowing-down weighting for the scatter matrix: neither beats NR (IR marginally helps pure Fe-56, hurts others; SD worse). NR remains the best weighting for the scatter matrix too — same verdict as for the vector XS.
• TC-P0 transport correction implemented (scatter_matrix(..., return_p1=True) → σ_tr = σ_t − Σ_s1, diagonal corrected) — the research's top lever for a P0 random-ray library; the correction is large (tungsten ~50% of σ_t at 14 MeV). Committed (65a553a).
• Metric lesson: raw element-wise / per-group source metrics blow up at fine groups because the MC reference scatter matrices are per-element noise-dominated (slab itself is 50–100% off material_wise). The robust quantities are the row-sum and the mean-lethargy-gain shape.

Honest bottom line

For the fusion-relevant materials (structural metals, ceramics, breeder, coolant) transport-free is as good as or better than stochastic_slab on both vector XS and the scatter matrix at fine groups, and is noise-free/deterministic. The documented limitation is hydrogenous moderators (H₂O, concrete), whose scatter matrices need a thermal scattering kernel. Next: a thermal kernel for moderators, and the decisive downstream random-ray-vs-CE benchmark.

[jon-proximafusion]

Free-gas thermal scattering kernel — tried and re-tested (commit 7b7164a)

The one remaining scatter-matrix weakness was the hydrogenous moderators (H₂O, concrete): their scatter shape was 2–3 lethargy units off material_wise because the kernel treated the target as at rest (no thermal motion / up-scatter). Added a free-gas (ideal-gas) thermal kernel: scatter_matrix(..., thermal=True) replaces the static elastic kernel below 5 eV for light nuclides (A≤20) with a target-Maxwellian quadrature (isotropic CM), preserving real thermal up-scatter. Verified against analytic Wigner-Wilkins (A=1) to ~1–2%.

Effect at fine groups (CCFE-709), scatter matrix vs material_wise

material	source err OFF→ON	shape (lethargy) OFF→ON	slab
H₂O	162% → 15%	3.07 → 1.43	0.041
concrete	148% → 29%	2.12 → 0.75	0.106
Li₄SiO₄	—	0.137 → 0.042 (beats slab 0.218)
SiC	—	0.034 → 0.024	0.018
Helium	—	0.333 → 0.117 (beats slab 0.529)
metals (W/steel/Fe/CuCrZr/Zr)	unchanged	unchanged	(A>20, correctly inert)

The thermal kernel cuts the hydrogenous scattering-source error 5–10× and the shape 2–3×, and is neutral for metals (the main use case). Helium and Li₄SiO₄ now beat stochastic_slab outright.

Honest residual

H₂O/concrete shape is still ~1.4 / 0.75 lethargy units vs slab's ~0.04 / 0.11. The free-gas kernel is correct (matches Wigner-Wilkins), so the residual is the harder full-thermalization match — bound S(α,β) for hydrogen in water/concrete (vs the free-gas the MC reference uses by default), plus per-fine-point (not per-group) thermal evaluation. That's the next refinement for moderators; for a fast fusion structural/ceramic/breeder/coolant library it's already in good shape.

Overall 10-material verdict (CCFE-709, fine groups)

• Total XS: transport-free ties/beats slab 7/10.
• Scatter shape: competitive/winning 8/10 — only H₂O/concrete remain (thermal).
• Plus the structural wins: noise-free, deterministic, instant.

[jon-proximafusion]

Elastic scatter binning fixed via angular CDF — the hydrogenous-shape residual was a binning bug, not thermal physics (commits 90af724, 103bd8e)

My previous comment blamed the H₂O/concrete scatter-shape residual on missing bound S(α,β). That diagnosis was wrong. Localising the H₂O error by energy showed the thermal region was already excellent (mean-lethargy-gain |Δ| 0.03 vs material_wise, up-scatter mass matched to 0.1%) — the whole residual sat in the fast/epithermal elastic range.

Root cause: the elastic kernel deposited one delta per angular-grid μ-point. For heavy nuclides E_out ≈ E (near-diagonal), so this is harmless — which is why every metal looked fine and the bug hid for months. But for hydrogen the down-scatter spans the full [0, E]; every μ ≈ −1 point maps to E_out ≈ 0 and piled into the lowest energy group. A 977 keV incoming group dumped 14 % of its scatter at the grid floor instead of spreading smoothly down the slowing-down range.

Fix (103bd8e): E_out is monotonic in μ, so the angular distribution is now spread smoothly across outgoing groups via its CDF, integrated over the group edges — no deltas. The P1/TC-P0 moment is spread consistently (row-sum unchanged, so the transport correction is unaffected).

Effect — scatter shape (mean lethargy gain) vs material_wise, CCFE-709

material	before	after fix	slab
H₂O	1.37	0.016	0.041
concrete	0.72	0.035	0.106
steel	0.008	0.008	0.008
SiC	0.024	0.023	0.018

H₂O and concrete now beat stochastic_slab on scatter shape (and the 14 % spurious dump is gone, 0 %). The earlier free-gas thermal kernel still does its job for genuine thermal up-scatter; the threshold for it was also corrected to 400·kT (OpenMC's settings.free_gas_threshold default) instead of a hardcoded 5 eV (90af724).

Full 10-material status — CCFE-709 (transport-free det / slab, both vs material_wise)

material	TOTAL d/slab	scatter ROWSUM d/slab	SHAPE d/slab
tungsten	1.96 / 2.56	3.60 / 2.36	0.004 / 0.006
steel	0.57 / 0.48	0.75 / 0.56	0.008 / 0.008
Fe-56	3.31 / 1.39	3.40 / 1.45	0.011 / 0.010
CuCrZr	0.86 / 1.02	1.34 / 1.12	0.017 / 0.017
Zircaloy	0.79 / 0.91	1.02 / 0.99	0.014 / 0.014
SiC	0.12 / 0.06	0.40 / 0.06	0.023 / 0.018
Li₄SiO₄	0.06 / 0.42	0.39 / 3.26	0.041 / 0.218
concrete	0.05 / 0.09	0.29 / 0.10	0.035 / 0.106
H₂O	0.02 / 0.05	0.17 / 0.05	0.016 / 0.041
helium	0.19 / 2.10	0.17 / 9.66	0.117 / 0.529

(All errors in %, except SHAPE in lethargy units. Bold = transport-free wins.)

Where it stands

• Scatter-matrix SHAPE: solved. With the binning fix, transport-free ties or beats slab on shape for all 10 materials — the moderators were the last holdout and now lead.
• Vector TOTAL XS: transport-free ties/beats slab 7/10.
• Remaining gaps are XS magnitude (TOTAL, scatter ROWSUM) on the deep resonant materials — Fe-56 above all (3.3 % vs slab 1.4 %), steel/SiC marginally. These are not a kernel issue: they are the spatial-spectrum limit. In this deliberately harsh 10-shell deep stack, slab's artificial mixed-material weighting flux happens to resemble the degraded local spectrum of a deep material better than any single near-source weighting can.

Weighting flux re-checked (again)

Confirmed the magnitude gaps are not closeable by changing the weighting flux: slowing-down weighting is worse than NR for exactly these materials (Fe-56 ROWSUM 3.6 % SD vs 3.4 % NR; tungsten 4.2 % vs 3.6 %), and IR was checked earlier with the same verdict. NR remains the best single weighting for both vector and scatter data. Closing the deep-resonant magnitude gap would require depth-/spectrum-aware data (multiple background σ₀ tiers or a degraded reference spectrum), which trades away the geometry-independence that makes this transport-free and instant.

Honest open items

• Deep resonant magnitude (Fe-56) is the one place slab still leads on a deep stack; for representative (non-deep-stack) spectra transport-free's per-material vector XS is sub-0.5 %.
• The decisive end-to-end test is a downstream random-ray-vs-CE shielding run — that, not the deep-stack collapse metric, is what determines whether transport-free is good enough to feed RR.

[jon-proximafusion]

Are we fooling ourselves? Monte-Carlo convergence check of the benchmark

stochastic_slab and material_wise are both Monte Carlo, so their error vs each other has a noise part (shrinks with particles) and a systematic part (never shrinks). Worry: maybe stochastic_slab is just under-converged, and 10× particles would let it overtake the deterministic method everywhere. I tested this directly.

Setup. Both references already run 100 active batches: material_wise at 40k/batch (4M histories), stochastic_slab at 12k/batch (1.2M). I regenerated stochastic_slab at 10× particles (120k/batch, 12M histories) and a second-seed material_wise (same N) to measure the truth's own noise. CCFE-709, 10-shell deep stack.

Q1 — does stochastic_slab improve with 10× particles? (error vs material_wise)

material	TOTAL 1× → 10×	ROWSUM 1× → 10×
tungsten	2.56 → 2.57	2.36 → 2.34
steel	0.48 → 0.48	0.56 → 0.51
Fe-56	1.39 → 1.41	1.45 → 1.47
CuCrZr	1.02 → 1.02	1.12 → 1.05
Zircaloy	0.91 → 0.91	0.99 → 1.00
SiC	0.06 → 0.05	0.06 → 0.06
Li₄SiO₄	0.42 → 0.06	3.26 → 0.07
concrete	0.09 → 0.07	0.10 → 0.07
H₂O	0.05 → 0.03	0.05 → 0.03
helium	0.39 → 0.04	10.70 → 2.30

For the dense/resonant metals where slab beats us (W, steel, Fe-56, CuCrZr, Zr) the error does NOT move with 10× particles — it is fully-converged systematic bias (the random-slab spectrum ≠ the deep-stack local spectrum), not noise. So the worry is unfounded there: more particles will not let slab pull ahead. Where slab was noise-limited is the low-density materials (Li₄SiO₄, helium): there its error collapses with 10× (Li₄SiO₄ row-sum 3.26 → 0.07). So our earlier "transport-free beats slab on Li₄SiO₄/He row-sum" was partly beating an under-converged slab — with a converged slab, slab is excellent on those too (helium row-sum is the one place we still lead, 0.17 vs 2.30).

Q2 — the bigger finding: the material_wise truth is itself noisy

Two independent material_wise seeds (same 4M histories) disagree by:

material	TOTAL seed1-vs-seed2	ROWSUM seed1-vs-seed2
Fe-56	2.82	2.45
Zircaloy	2.49	2.18
SiC	2.43	2.88
CuCrZr	2.13	1.83
tungsten	1.67	1.15
steel	1.14	0.95
Li₄SiO₄	1.93	1.72
helium	1.42	1.72
concrete	0.03	0.20
H₂O	0.01	0.02

The reference itself carries 1–3 % noise for the resonant metals (the deep shells get few tally scores). That means the fine-group per-material "errors" we quote for Fe-56/W/CuCrZr/Zr/SiC — for both transport-free and slab — are being measured against a ±2 % ruler. The transport-free-vs-slab gap on Fe-56 (3.3 vs 1.4) is partly comparing noise; both sit within the truth's uncertainty band.

Conclusions

1. The benchmark is sound for the materials with a clean ruler. H₂O and concrete have ~0.01–0.2 % reference noise — there transport-free wins cleanly and unambiguously (total 0.02/0.05, and it now leads on scatter shape after the elastic CDF fix).
2. slab is not under-converged where it matters — its lead on the resonant metals is real bias, not noise; 10× particles changes nothing there. We were not fooling ourselves in that direction.
3. But the resonant-metal comparison is reference-noise-limited. To cleanly resolve sub-2 % differences (Fe-56, W, CuCrZr, Zr) we need a much more converged material_wise (≈10–100× histories), or — better — to stop using the deep-stack collapse metric as the arbiter and run the downstream random-ray-vs-CE shielding calculation, which integrates over the spectrum and is the actual use case.

Next: a lit-review-guided self-shielding improvement to the scatter matrix (SCALE/AMPX Bondarenko f-factor on the elastic matrix), reported separately.

[jon-proximafusion]

Improving total / scatter row-sum: lit review + first implemented gain (commit ddee0a9)

A 6-angle, 21-source, adversarially-verified literature review of transport-free MG total/scatter generation for resonant structural materials, plus a code review, identified the techniques below. One is now implemented; the rest are a prioritized roadmap.

Implemented now — Bondarenko f-factor on the elastic matrix (URR)

The code review found the scatter matrix used the dilute elastic XS in the unresolved range while the vector total was already URR-shielded — an inconsistency. The lit review confirmed this is exactly what SCALE/AMPX corrects:

"resonance absorption may have an important effect on the elastic removal probability from a group containing one or more resonance, especially for heavy nuclides … Historically, Bondarenko self-shielding of the 2D elastic matrix has not been done in SCALE" — ORNL ESSM/SCALE. Ignoring it gave a −184 pcm bias that the f-factor reduced to +2 pcm.

I apply the probability-table band average to the elastic channel (a per-nuclide f_el(E)=⟨σ_el⟩/σ_el,dilute) before building the transfer matrix. CCFE-709 scatter row-sum vs material_wise:

material	before	f-factor	slab(10×)
tungsten	3.60	1.95	2.34
Zircaloy	1.02	0.94	1.00
steel / Fe-56 / SiC / Li₄SiO₄ / H₂O / concrete / He	—	unchanged	—

Tungsten and Zircaloy now beat slab on row-sum; nothing regresses.

Important caveat — the materials you named have no URR

Steel (Fe/Cr/Ni), Fe-56, and SiC (Si/C) have no unresolved range at all — they are fully resolved, so φ=1/Σ_t already self-shields them directly from the pointwise data. The URR f-factor does not touch them. Combined with the convergence check above (Fe-56 reference noise ≈ 2.8 %), their residual is currently below the noise of the material_wise ruler — we cannot reliably measure an improvement to steel/Fe-56/SiC without a much more converged reference (or the downstream RR-vs-CE test).

Roadmap for the resolved-range residual (verified techniques, in priority order)

1. Bondarenko f-factor on the elastic matrix in the resolved range (extend what's now done for the URR). SCALE recipe: σ_gg(σ₀)=∫_g σ_s(u)·p(u→g)·Φ(u,σ₀)du / ∫_g Φ(u,σ₀)du; shielded removal = σ_el,g(σ₀)−σ_gg(σ₀); renormalize the out-scatter row to it. This is the direct lever for steel/Fe-56 but needs σ₀-parameterised f-factors built from the pointwise resolved resonances.
2. IR (Goldstein-Cohen λ) weighting instead of pure NR for the collapse: φ(u)=C/(σ_a+λ_r·σ_rs,r+σ_b), σ_rs=σ_s−σ_p, with a per-nuclide λ found by iteration. λ corrects the removal (scattering) XS in the resonance range, not just absorption. (We tried a crude fixed-λ IR earlier — helped Fe-56, hurt others; the verified fix is per-nuclide iterated λ.)
3. Compound-material 0-D slowing-down weighting flux (FRENDY/MG): solve [σ₀+σ_t(E)]φ=C·σ₀+∫σ_s φ/((1−α)E′)dE′ on the whole material at once, capturing resonance interference among Fe/Cr/Ni — vs our single-nuclide 1/E. This is the principled replacement for the fixed weighting flux.
4. Per-Legendre-moment weighting for the transport XS (GROUPR): φ_l(E)=C/(σ_t+σ₀)^(l+1) — the P1/transport moment should use a steeper weight than P0; collapse Σ_t by the neutron current, not the scalar flux. Refines our TC-P0 σ_tr.
5. Deterministic resonance-elastic-upscatter kernel (RESK / deterministic DBRC) for the resolved-range elastic transfer matrix (NECP-Atlas, FRENDY) — a second-order Doppler effect; lowest priority for fast fusion spectra.

Honest takeaway

The scatter-matrix shape is solved (elastic CDF fix) and now the URR elastic magnitude is self-shielded (W, Zr fixed). The remaining total/row-sum gaps are on the fully-resolved resonant metals (steel, Fe-56, SiC), where (a) the proper fix is a σ₀-parameterised resolved-range f-factor or IR/compound-flux weighting, and (b) the current benchmark can't even measure sub-2 % gains there. The most informative next step is a converged reference or the downstream random-ray-vs-CE run, then lever (1)/(2).

[jon-proximafusion]

Consolidated scatter-matrix comparison (both fixes in) + correction to the noise figures

Definitive table at CCFE-709, 10-shell deep stack, transport-free with both fixes (elastic CDF-spread 103bd8ea6 + URR elastic self-shielding ddee0a93d), vs material_wise. The slab column is the converged 10× reference, and ref-noise is the material_wise seed-to-seed disagreement computed over the same well-populated rows (row-sum > 1e-3) used for the method errors — so it is directly comparable. A method-vs-slab difference smaller than ref-noise is not resolvable at this reference convergence.

Scatter row-sum (scatter XS magnitude), % error vs material_wise

material	det	slab(10×)	ref-noise	verdict
tungsten	1.95	2.34	2.35	tie (both ≲ noise); det lower
steel	0.75	0.51	1.40	tie (both ≪ noise)
Fe-56	3.40	1.47	0.52	slab wins — real, above noise
CuCrZr	1.34	1.05	2.86	tie (both ≪ noise)
Zircaloy	0.94	1.00	0.93	tie (≈ noise)
SiC	0.40	0.06	0.86	tie (both < noise)
Li₄SiO₄	0.39	0.07	0.80	tie (both < noise)
concrete	0.29	0.07	0.04	slab lower (real, both < 0.3 %)
H₂O	0.17	0.03	0.02	slab lower (real, both < 0.2 %)
helium	0.17	9.66	7.72	det wins (slab noise-limited)

Scatter shape (mean lethargy gain, |Δ| vs material_wise — lower better)

material	det	slab(10×)		material	det	slab(10×)
tungsten	0.004	0.004		SiC	0.023	0.017
steel	0.008	0.006		Li₄SiO₄	0.041	0.049
Fe-56	0.011	0.009		concrete	0.035	0.035
CuCrZr	0.017	0.011		H₂O	0.016	0.014
Zircaloy	0.014	0.013		helium	0.117	0.523

Verdict

• Shape: effectively a tie everywhere — both excellent (all ≤ 0.025 lethargy except helium, where det wins 4×). The elastic CDF fix closed the former moderator gap.
• Row-sum: most materials sit within the reference noise (an unresolvable tie at this convergence). The URR f-factor made tungsten (1.95 vs 2.34) and Zircaloy competitive.
• One real, above-noise loss remains — Fe-56 (3.40 vs 1.47, noise 0.52). It is the fully-resolved resonant metal the URR fix cannot touch; closing it needs the resolved-range Bondarenko f-factor / per-nuclide IR-λ lever from the roadmap.

Correction to the convergence-check comment above

That earlier comment quoted material_wise noise of ~2.5–2.8 % for Fe-56. Those figures over-stated the noise because they averaged relative error over all non-zero rows, including the near-empty deep rows that carry huge relative (but negligible absolute) scatter. Recomputed over the same row-sum > 1e-3 rows used for the method errors, Fe-56 row-sum noise is 0.52 %, not 2.5 %. The practical consequence: Fe-56's slab advantage is real, not a noise artifact — slab genuinely wins Fe-56's scatter row-sum. The qualitative convergence conclusions still hold (slab is converged for the metals; H₂O/concrete are clean-ruler; reference noise dominates the other metals), but Fe-56 specifically is a genuine gap, not a wash.

[jon-proximafusion]

Principled fix attempted: exact 0-D slowing-down weighting flux — definitive negative result

The roadmap's top within-material lever was to replace the narrow-resonance flux (φ=1/Σ_t) with the exact 0-D infinite-medium slowing-down flux, which captures wide/intermediate-resonance scattering with no Goldstein-Cohen λ. The earlier attempt at this was crippled (it solved the balance on a coarse 40-bin/decade grid, then divided a smooth source by the fine Σ_t — i.e. it was still essentially NR). This time it was done properly:

Σ_t(E)·φ(E) = S(E) + Σ_r ∫_E^{E/α_r} Σ_s,r(E′)/((1−α_r)E′)·φ(E′) dE′

solved by an exact high→low sweep on the full pointwise resonance grid (52k–323k points), so the scattering source carries φ's own resonance dips self-consistently. Fast (0.1–8 s/material).

It is implemented correctly — the wide-resonance physics IS captured

At Fe-56's scattering resonances the solved flux dips shallower than NR's 1/Σ_t (exactly the wide-resonance effect — neutrons scatter within the resonance and replenish the flux):

resonance	NR dip φ/φ(+30 %)	SD-proper dip
1.15 keV	0.199	0.224
27.7 keV	0.112	0.129
73.9 keV	0.255	0.259

(The two fluxes differ by >100 % in places — this is a genuinely different, correctly-solved flux, not a null.)

But it does not move the group-collapsed cross section

CCFE-709, vs material_wise:

material	TOTAL: NR / SD-proper / slab	ROWSUM: NR / SD-proper / slab
Fe-56	3.31 / 3.33 / 1.41	3.40 / 3.42 / 1.47
steel	0.57 / 0.59 / 0.48	0.75 / 0.77 / 0.51
SiC	0.12 / 0.12 / 0.05	0.40 / 0.40 / 0.06

The change is ≤ 0.02 % (and if anything marginally worse). Reason: at fine (5 %) groups the collapse ∫φσ/∫φ is dominated by the 1/Σ_t self-shielding factor that NR and SD share; the shallower-dip correction is a small, locally-cancelling effect on the numerator.

Conclusion — the weighting-flux avenue is now closed

NR ≈ crude IR (≈ +0.3 %) ≈ exact 0-D slowing-down flux (≈ 0). The within-material weighting flux is conclusively not the lever for the resonant-metal gap. Combined with the earlier evidence (per-material-sphere Fe-56 already ~2.7 %), the remaining Fe-56 deficit is the spatial-spectrum — the deep-stack local flux is softer/more degraded than any 0-D infinite-medium flux, and slab's random-slab geometry happens to reproduce that better. No assumed single weighting (NR/IR/SD) can capture a geometry-dependent local spectrum.

The only routes left for Fe-56's magnitude are therefore (a) finer groups (drives the collapse toward weighting-independence) or (b) spectrum/σ₀-aware data (multiple weightings / background tiers) — both trade away simplicity/geometry-independence. The decisive arbiter remains the downstream random-ray-vs-CE run, not the deep-stack collapse metric. The SD-flux solver is kept in the prototype for the record but is not added to the module — NR stays the verified-best weighting.

[jon-proximafusion]

Cross-talk benchmarks: when is NR+URR sufficient?

Built controlled 2-material adjacency benchmarks (concentric spheres, 14 MeV point source) to test whether the placement-BLIND NR+URR method here matches material_wise (the truth, which carries the real cross-talk). Two material pairs, both orderings; NR+URR vs the 1D deterministic transport prototype (#117) vs material_wise.

(Single-seed MC references at moderate statistics; estimated noise ~2% for the deep/resonant tallies — relevant below.)

W │ H₂O — resonant absorber + moderator

placement	material	NR total	NR scatter
W(0–6) ∣ H₂O(6–26)	W (front)	1.30	2.64
	H₂O (behind W)	0.02	0.20
H₂O(0–12) ∣ W(12–18)	H₂O (front)	0.02	0.20
	W (behind 12 cm water)	1.06	2.52

NR+URR is sufficient for both, in both placements. W behind 12 cm of water collapses to ~1% — at/below the MC noise floor (essentially exact), and no worse than W at the front. Moderating the spectrum does not break NR (water is smooth; it shifts the spectrum but carves no resonance dips at W's energies).

steel │ Fe-56 — two resonant metals (overlapping Fe resonances)

placement	material	NR total	TR total
steel(0–10) ∣ Fe-56(10–16)	steel (front)	0.46	0.20
	Fe-56 (behind steel)	3.01	0.66
Fe-56(0–10) ∣ steel(10–16)	Fe-56 (front)	3.25	0.90
	steel (behind Fe-56)	0.25	0.27

NR has a real ~3% gap on Fe-56 that transport closes to ~0.8%.

Key (and corrective) finding: the Fe-56 gap is NOT inter-material cross-talk

Fe-56's NR error is ~3% whether it sits behind steel or at the front, and material_wise's Fe-56 total XS changes only ≲ the MC noise (~0.4% mean) between the two placements. So the neighbour's identity barely matters. The gap is that NR's single 0-D weighting flux cannot represent the depth-varying slowing-down spectrum through a thick resolved-resonance region — whether the slowing-down is done by upstream steel or by Fe-56's own front layers. (Consistent with the earlier negative result that a 0-D slowing-down flux ≈ NR for Fe-56; you need the spatial dimension, which 1D transport supplies.)

Bottom line for this method

• Inter-material cross-talk (neighbour identity) is small (≲ noise). NR+URR's placement-blindness is not a major liability.
• NR+URR is sufficient for moderators and URR-dominated metals (H₂O, W): ~1% or better, placement-robust.
• The one genuine NR deficiency is thick resolved-resonance metals (Fe-56, ~3%) — and it's a spatial effect, not cross-talk.

Caveat for the end use (random ray)

material_wise bakes the test geometry's within-region spatial averaging into its MGXS; random ray resolves that spatial variation itself. So matching the region-averaged reference is not obviously the right target for RR — the decisive test remains RR-vs-CE, not this collapse metric.

[jon-proximafusion]

Wired transport_free into convert_to_multigroup + random-ray vs CE benchmark

Model.convert_to_multigroup(method="transport_free", ...) now generates the MGXS library
deterministically from this PR's module (NR+URR self-shielded vector XS + deterministic P0
scatter matrix; optional TC-P0 via correction="P0") — no Monte Carlo, no transport solve.
Added an end-to-end random-ray vs continuous-energy benchmark on a fusion shield
(examples/transport_free_random_ray/) to test the end use directly.

Geometry (neutronics-workshop tokamak materials)

14.1 MeV plasma source → tungsten armor → steel → lithium blanket → concrete bioshield.

Robustness — the clear win

• transport_free produced a library valid in every group (positive total XS, zero MC
  noise) → fed random ray with no fixups.
• stochastic_slab produced zero/negative total XS that random ray rejects (unpopulated
  deep-thermal groups + a TC-P0 artifact in lithium); it needed a positive-floor fixup to run
  at all. The harness applies the same fixup to both for fairness.
• The MC noise is visible directly: in the deep-thermal tail of the near-source regions slab
  swings over tens of decades of noise while transport_free is smooth (left panel below).

Accuracy vs CE — group-structure dependent

	deep-attenuation error vs CE	deepest-region (concrete) spectral error
VITAMIN-J-42 (40 grp)	slab 234% · tf 41%	slab 93% · tf 65%
CCFE-709 (650 grp)	slab 32% · tf 36%	slab 36% · tf 35%

At coarse groups transport_free is far better deep (slab's coarse XS badly mis-predict the
flux suppression). At fine fusion groups the two converge to near-parity — flux-weighted
spectral error within a few % in every region (slab a hair better at the first wall,
transport_free a hair better at the deepest point). This is exactly the collapse-metric finding:
finer groups make within-group weighting nearly irrelevant, so the methods agree.

In the physically-relevant (high-flux) energy range all three — CE, RR-slab, RR-tf — agree well;
the visible disagreement is the negligible deep-thermal tail (slab = MC noise, tf = smooth).

Bottom line

For the random-ray workflow, transport_free matches stochastic_slab's accuracy at the fine
groups used for fusion shielding, beats it at coarse groups, and is strictly more robust
(deterministic, noise-free, valid in every group — no negative-XS fixup needed).

Caveat: the RR solve here is moderately resolved (a few flat source regions per layer);
absolute errors vs CE (~20–40% flux-weighted at CCFE-709) are dominated by that and are common
to both methods. A higher-resolution RR solve (more source regions, linear source, more rays)
would sharpen the absolute verdict; the relative method comparison is unaffected.