Analytical Residual Topology and Systematic Bias Decomposition in Evaluated Nuclear Cross-Section: A Transfer-Constrained Mathematical Framework for Chromium Isotopes ^†

Abstract

Systematic inconsistencies between evaluated nuclear data libraries and experimental measurements may reflect structured, non-stochastic deviations rather than random error. This study develops a residual-learning framework for the neutron total cross-sections of Cr-50, Cr-53, and Cr-54 using TENDL and EXFOR data. Logarithmic and percentage residuals are modeled using a nonlinear analytical basis with weighted regularization. The analysis reveals heavy-tailed, skewed residual structures and persistent mid-energy underestimation. Residual corrections are strongly regime-dependent, while cross-isotopic transfer remains only partially conserved. Overall, the framework provides an interpretable approach to systematic bias decomposition and physically consistent correction of evaluated nuclear cross-section data.

1. Introduction

Accurate neutron cross-sections are essential to nuclear science and engineering, but evaluated libraries can show persistent discrepancies from experimental measurements [1,2,3]. These deviations may arise from systematic, energy-dependent effects related to model assumptions, resonance treatment, and parameterization [4,5,6]. Reliable assessment therefore requires an analysis of the residual structure beyond aggregate error metrics [7].

Chromium isotopes provide a useful test case because of their complex energy-dependent behavior and sufficient experimental coverage [8]. The evaluated-to-experimental differences can be treated as a structured residual field governed by physical and mathematical constraints [9], while conventional validation methods may obscure this structure and limit cross-isotopic generalization [10].

This study develops an analytical residual-learning framework for Cr-50, Cr-53, and Cr-54 using TENDL and EXFOR data [11]. Residuals are represented through an energy-dependent feature space and nonlinear analytical basis with weighted regularization to characterize systematic bias and its energy-resolved topology [12].

The main contributions of this study are as follows:

• An analytical residual-learning formulation is introduced to characterize evaluated-to-experimental discrepancies while retaining an interpretable representation of systematic bias.
• Energy-resolved residuals exhibit pronounced skewness and heavy-tailed behavior, supporting the presence of systematic and non-Gaussian deviation patterns.
• A persistent mid-energy underestimation regime is identified across the investigated chromium isotopes, indicating a shared region of disagreement between evaluated and experimental data.
• Cross-isotopic transfer analysis indicates that the residual structure is only partially conserved across neighboring chromium isotopes, limiting direct generalization.
• The residual formulation provides an interpretable pathway for bias decomposition and physically consistent multiplicative correction while allowing for comparison with nonlinear regression models.

Collectively, these observations motivate an explicitly energy-resolved treatment of discrepancies in evaluated neutron cross-sections and provide a framework for examining their structure, learnability, and transferability.

2. Methodology

This study compares TENDL and EXFOR neutron total cross-sections for Cr-50, Cr-53, and Cr-54 over their common energy ranges [13]. TENDL energies were converted to MeV, duplicate points were averaged, and non-physical values with an E ≤ 0 or σ ≤ 0 were removed.

Ω^(a) = [max(min E_t^(a), min E_x^(a)), min(max E_t^(a), max E_x^(a))]

(1)

where E_t^(a) and E_x^(a) are the TENDL and EXFOR energy values, respectively, and Ω^(a) is their common interval. The TENDL cross-section was interpolated onto the EXFOR energy grid as

σ̃_t^(a)(E_i) = $ℐ$[σ_t^(a)](E_i), E_i ∈ Ω^(a)

(2)

where σ̃_t^(a) is the interpolated TENDL cross-section, and $ℐ$ denotes linear interpolation. The evaluated-to-experimental mismatch was formulated as

σ_x^(a)(E) = σ_t^(a)(E) exp($ℛ$^(a)(E))

(3)

where σ_x^(a) and σ_t^(a) denote the EXFOR and TENDL cross-sections, respectively, and $ℛ$^(a) is the residual correction. The logarithmic residual was defined as

r_i^(a) = log σ_x^(a) (E_i) − log σ̃_t^(a) (E_i)

(4)

where r_i^(a) is the logarithmic residual at energy E_i. The corresponding percentage residual was calculated as

p_i^(a) = 100 [σ̃_t^(a)(E_i) − σ_x^(a)(E_i)]/σ_x^(a)(E_i)

(5)

where p_i^(a) is the percentage residual. Residual learning used the transformed feature vector

x_i^(a) = [log E_i, log σ̃_t^(a)(E_i), ∂log σ̃_t^(a) / ∂log E, A, Z, N]

(6)

where A, Z, and N are the mass number, atomic number, and neutron number, respectively. A nonlinear basis expansion Φ(x) was used to capture curvature and interaction effects, giving

r̂_i^(a) = Φ(x_i^(a))ᵀβ

(7)

where β* denotes the optimized coefficient vector, w_i denotes the observation weight, λ denotes the regularization parameter, and i indexes the observations.

β* = arg minβ {Σ_i w_i[r_i − Φ(x_i)ᵀβ]² + λ‖β‖₂²}, w_i = (1 + |r_i|)⁻¹

(8)

where w_i is the observation weight, and λ is the regularization parameter. The corrected cross-section was then reconstructed as

σ̂_c = σ̃_t exp(r̂)

(9)

where σ̂_c denotes the corrected cross-section, and r̂ denotes the estimated residual. This formulation preserves positivity and multiplicative consistency. Thepooled chromium dataset was split into 80% training and 20% test sets. Transferability was evaluated by training on Cr-50 and Cr-53 and testing on unseen Cr-54. Performance was assessed using RMSE, MAE, MAPE, bias, bounded-error coverage, skewness, and kurtosis.

3. Results and Discussion

The TENDL–EXFOR discrepancies for the chromium isotopes form a structured, energy-dependent residual field rather than random scatter. As shown in Table 1, Cr-53 provides the closest agreement with the experimental data, with the lowest RMSE of 1.590 b, MAE of 1.092 b, and MAPE of 20.91%. Cr-50 and Cr-54 show larger deviations.

Table 1. Global residual statistics for TENDL relative to EXFOR across chromium isotopes.

Isotope	RMSE (b)	MAE (b)	MAPE (%)	Bias (%)	Std Dev (%)	Skewness	Kurtosis	|e| < 20%
Cr-50	2.538	1.344	43.24	−5.47	68.44	2.79	12.16	37.97
Cr-53	1.590	1.092	20.91	−6.16	29.73	2.33	14.83	59.67
Cr-54	2.311	1.456	46.19	−24.27	58.60	3.10	16.19	27.64

Cr-53 shows better agreement, with a narrower residual envelope, lower variance, and higher bounded-error coverage. In contrast, Cr-50 and Cr-54 show broader dispersion and larger residual excursions, indicating stronger disagreement with the experimental data.

The discrepancy is strongly energy-dependent. As shown in Table 2, the largest deviation occurs at 0.249–0.269 MeV, where the mean bias reaches −60.61%, the mean T/E ratio falls to 0.394, and only 1.5% of points remain within the 20% error threshold. Lower-energy regions are comparatively more stable. Figure 1 shows the energy-dependent residual envelopes and residual-cloud structures across the three chromium isotopeThe energy–mass residual topology in Figure 2 reveals ridge-like, low-dimensional structures across the investigated isotopes. The residuals follow organized pathways rather than arbitrary dispersion, suggesting a shared structural component in the evaluated data. This alignment is the most evident in the mid-energy region, where the strongest underestimation occurs.

Table 2. Representative energy-resolved residual behavior for Cr-50.

Energy (MeV)	Mean Bias (%)	Median (%)	Std (%)	Mean T/E	Median T/E	|e| < 20%	|e| < 50%
0.100–0.108	7.40	3.18	41.46	1.074	1.032	0.968	0.987
0.146–0.158	−18.71	−29.77	53.54	0.813	0.702	0.142	0.748
0.249–0.269	−60.61	−69.10	55.47	0.394	0.309	0.015	0.061
0.290–0.313	9.70	6.44	47.12	1.097	1.064	0.444	0.733

Figure 1. Bias atlas showing mean and median residual envelopes with uncertainty bands and residual-cloud structure across chromium isotopes.

Figure 2. Energy–mass residual topology illustrating ridge-like manifold structures of systematic bias across chromium isotopes.

The pooled held-out results in Table 3 confirm that the residual field is learnable. Random Forest performs the best, with R² = 0.872, RMSE = 1.794 b, MAE = 0.733 b, and MAPE = 21.06%. XGBoost also performs competitively with R² = 0.849, while Gradient Boosting and SVR are less effective. Broader validation across isotopes and reaction channels is still needed.

Table 3. Residual-learning model performance on the pooled held-out test subset after correction.

Model	RMSE (b)	MAE (b)	MAPE (%)	Bias (%)	Std Dev (%)	R2	|e| < 20%	|e| < 50%
Random Forest	1.794	0.733	21.06	−2.31	28.42	0.872	68.75	91.20
XGBoost	4.239	0.769	25.91	−3.85	35.17	0.849	61.32	88.54
Gradient Boosting	1.991	0.847	27.49	−4.72	39.63	0.563	54.18	84.91
SVR	2.067	0.854	38.84	−6.95	45.72	0.388	42.07	79.33

Overall, the TENDL–EXFOR discrepancies form a coherent residual manifold characterized by regime-dependent bias, heavy-tailed asymmetry, and partial cross-isotopic consistency. These patterns indicate that the deviations are systematic, interpretable, and suitable for physics-informed correction.

4. Conclusions

This study shows that TENDL–EXFOR discrepancies in chromium cross-sections form a coherent, energy-dependent residual manifold rather than random variability. Mid-energy underestimation is accompanied by asymmetric, heavy-tailed residuals, while stable bias envelopes and ridge-like energy–mass structures indicate that these deviations are systematic and partly shared across isotopes.

Ensemble-based residual learning further shows that the bias field is learnable and correctable. Random Forest achieved the best performance with R² = 0.872 on the pooled held-out test subset. Overall, the results support a systematic, regime-dependent interpretation of evaluated nuclear data discrepancies. Future work will examine broader isotopic chains and additional reaction channels.