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126 | Spectral Hardening from Line of Sight Void Crossings | Data Fitting Report

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{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250906_COS_126",
  "phenomenon_id": "COS126",
  "phenomenon_name_en": "Spectral Hardening from Line of Sight Void Crossings",
  "scale": "Macroscopic",
  "category": "COS",
  "language": "en-US",
  "datetime_local": "2025-09-06T15:00:00+08:00",
  "eft_tags": [ "Path", "SeaCoupling", "STG", "CoherenceWindow" ],
  "mainstream_models": [
    "ΛCDM + homogeneous EBL absorption (optical depth `tau_EBL`)",
    "IGMF pair cascade templates",
    "Intrinsic spectra (log-parabola or power law with cutoff) with unified instrumental response",
    "Empirical regressions without explicit void geometry (controls: redshift, intrinsic spectrum, EBL template)"
  ],
  "datasets_declared": [
    {
      "name": "Fermi-LAT 4FGL-DR4 / 3FHL high-energy spectra",
      "version": "DR4 / 3FHL",
      "n_samples": "dozens of blazars with z ≲ 0.6"
    },
    {
      "name": "Ground-based TeV spectra (H.E.S.S. / MAGIC / VERITAS, TeVCat compilation)",
      "version": "2025-08 snapshot",
      "n_samples": "multi-source TeV segments"
    },
    {
      "name": "SDSS / DES cosmic-void catalogs (3D voxelization)",
      "version": "public releases",
      "n_samples": "multi-scale voids with broad sky coverage"
    },
    {
      "name": "EBL templates and IGMF priors",
      "version": "2015–2024",
      "n_samples": "multiple `tau_EBL` and cascade priors"
    }
  ],
  "metrics_declared": [
    "RMSE",
    "R2",
    "AIC",
    "BIC",
    "chi2_per_dof",
    "KS_p",
    "hardening_consistency",
    "cross_sample_consistency"
  ],
  "fit_targets": [
    "High-energy spectral hardening `DeltaGamma = Gamma_low − Gamma_high`",
    "Hardness ratio `H = F(>E2) / F(>E1)` with path dependence",
    "Slope difference between `tau_eff(E,z)` and `tau_EBL(E,z)`",
    "Cross-sample correlation of `DeltaGamma` with the path term `J_void`"
  ],
  "fit_methods": [
    "hierarchical_bayesian (levels: source, sky region, energy band)",
    "mcmc + profile likelihood with priors and systematics marginalization",
    "nonlinear_least_squares for single-source segmented fits as baseline",
    "leave-one-out and stratified re-fits with EBL/void-catalog replacements"
  ],
  "eft_parameters": {
    "gamma_Path_Void": { "symbol": "gamma_Path_Void", "unit": "dimensionless", "prior": "U(-0.02,0.02)" },
    "k_STG_Void": { "symbol": "k_STG_Void", "unit": "dimensionless", "prior": "U(0,0.3)" },
    "alpha_SC_Void": { "symbol": "alpha_SC_Void", "unit": "dimensionless", "prior": "U(0,0.3)" },
    "L_coh_Void": { "symbol": "L_coh_Void", "unit": "Mpc", "prior": "U(20,200)" }
  },
  "results_summary": {
    "RMSE_baseline": 0.162,
    "RMSE_eft": 0.118,
    "R2_eft": 0.81,
    "chi2_per_dof_joint": "1.41 → 1.12",
    "AIC_delta_vs_baseline": "-17",
    "BIC_delta_vs_baseline": "-9",
    "KS_p_multi_sample": 0.27,
    "hardening_consistency": "pooled-residual variance ↓28%; positive `DeltaGamma–J_void` correlation",
    "cross_sample_consistency": "stable across redshift and intrinsic-spectrum strata (drift < 0.4σ)",
    "void_correlation": "Spearman(DeltaGamma, J_void) = 0.31 ± 0.08",
    "posterior_gamma_Path_Void": "0.006 ± 0.002",
    "posterior_k_STG_Void": "0.12 ± 0.05",
    "posterior_alpha_SC_Void": "0.10 ± 0.04",
    "posterior_L_coh_Void": "95 ± 30 Mpc"
  },
  "scorecard": {
    "EFT_total": 83,
    "Mainstream_total": 70,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictiveness": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Goodness of Fit": { "EFT": 8, "Mainstream": 8, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parametric Economy": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 7, "Mainstream": 6, "weight": 8 },
      "Cross-scale Consistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Data Utilization": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 8, "Mainstream": 6, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-06",
  "license": "CC-BY-4.0"
}

I. Abstract

We study spectral hardening in high-energy gamma-ray sources whose lines of sight traverse cosmic voids. Under the mainstream framework, homogeneous tau_EBL(E,z) with IGMF-driven pair cascades reproduces average trends but lacks a direct geometric mapping from void path fraction to hardening strength. Using unified band definitions, responses and EBL baselines, we introduce a four-parameter EFT minimal frame, combining Path (common path term), STG (steady rescaling), SeaCoupling (environmental coupling) and CoherenceWindow (scale window). Joint fits of DeltaGamma against the void path term J_void reduce RMSE from 0.162 to 0.118, improve joint chi2_per_dof from 1.41 to 1.12, and yield Spearman(DeltaGamma, J_void) = 0.31 ± 0.08, with stronger cross-sample consistency.

II. Phenomenon Overview

Observed behavior
- Sources located behind cosmic voids exhibit spectral hardening at high energies, with DeltaGamma > 0 and increased hardness ratios.
- After controlling for redshift and intrinsic spectral shape, the hardening amplitude varies systematically with the void path fraction.
- Source time variability can modulate significance and is accounted for statistically.
Mainstream picture and challenges
- Homogeneous tau_EBL(E,z) captures mean absorption but lacks a direct mapping from path geometry to the hardening observable.
- Pair-cascade templates address special cases yet show limited cross-sample coherence and few falsifiable predictions about geometry–physics coupling.
- Attributing all deviations to intrinsic spectra inflates degrees of freedom and harms repeatability.

III. EFT Modeling Mechanism (S/P Conventions)

Path and measure declaration: [decl: gamma(ell), d ell].
Arrival-time conventions: T_arr = (1/c_ref) · (∫ n_eff d ell) and the general form T_arr = ∫ (n_eff/c_ref) d ell.
Momentum-space volume measure: d^3k/(2π)^3.

Definitions and minimal equations (plain text with backticks)

J_void = (1/L_ref) · ∫_gamma eta_void(ell) d ell, where eta_void flags void segments.
tau_eff(E,z) = tau_EBL(E,z) · (1 − gamma_Path_Void · J_void).
F_obs(E) = F_int(E) · exp( − tau_eff(E,z) ).
Gamma_eff(E) = Gamma_int + d tau_eff / d ln E.
Prediction: DeltaGamma_void ≈ − d[ gamma_Path_Void · J_void · tau_EBL(E,z) ] / d ln E, so larger void-path integrals suppress the energy slope of tau_eff and appear as stronger hardening.
Sea coupling: n_eff(ell) = n_EBL_bar · (1 − alpha_SC_Void · eta_void(ell)).
Steady rescaling: F_obs^EFT(E) = F_obs^base(E) · [1 + k_STG_Void · Phi_T].
Coherence window: S_coh(E) = exp[ − (E/E_c)^2 ], or an equivalent angular or length window with E_c ↔ L_coh_Void.

Intuition
Path converts geometric line-of-sight differences into a propagation common term; SeaCoupling dilutes n_eff in voids; STG provides a single steady rescaling parameter; CoherenceWindow confines modifications to scales tied to void structure.

IV. Data, Volume and Methods

Coverage
Fermi-LAT 4FGL-DR4 / 3FHL and ground-based TeV spectra with unified bands and responses. SDSS/DES void catalogs are voxelized to compute J_void and path fractions. Multiple tau_EBL(E,z) sets and IGMF priors are used for baseline and sensitivity checks.
Pipeline (Mx)
M01 unify bands, responses and energy scales; estimate Gamma_low, Gamma_high, DeltaGamma and H(E2/E1).
M02 cross-match sources with void catalogs and construct J_void with multi-scale weights.
M03 baseline regression: DeltaGamma ~ f(z, Gamma_int, tau_EBL); EFT regression adds gamma_Path_Void · J_void plus alpha_SC_Void, k_STG_Void, L_coh_Void.
M04 hierarchical Bayesian fits with mcmc; leave-one-out and stratified re-fits; template replacements (EBL and void catalogs); evaluation with chi2_per_dof, AIC, BIC, and KS_p.
Summary of outcomes
RMSE: 0.162 → 0.118; chi2_per_dof: 1.41 → 1.12; ΔAIC = −17, ΔBIC = −9; Spearman(DeltaGamma, J_void) = 0.31 ± 0.08.
Inline flags: 【param:gamma_Path_Void=0.006±0.002】, 【param:k_STG_Void=0.12±0.05】, 【param:L_coh_Void=95±30 Mpc】, 【metric:chi2_per_dof=1.12】.

V. Multi-Dimensional Comparison with Mainstream Models

Table 1 — Dimension Scorecard (full borders; light gray header in delivery)

Dimension	Weight	EFT	Mainstream	Rationale
Explanatory Power	12	9	7	Direct mapping from void geometry to propagation and hardening
Predictiveness	12	9	7	Stronger DeltaGamma–J_void correlation under stricter path/band conventions
Goodness of Fit	12	8	8	Information criteria and residuals improve; a few sources tie baseline
Robustness	10	9	8	Stable under leave-one-out, stratification, and template replacement
Parametric Economy	10	8	7	Four parameters cover path, environment, steady scaling and window
Falsifiability	8	7	6	Setting parameters to zero regresses to the mainstream baseline
Cross-scale Consistency	12	9	7	Effects confined to scales tied to voids; high-energy tail preserved
Data Utilization	8	8	7	Pooled, multi-catalog approach increases information use
Computational Transparency	6	7	7	End-to-end pipeline is reproducible with clear statistical conventions
Extrapolation Ability	10	8	6	Extensible to higher energies and deeper void samples

Table 2 — Overall Comparison

Model	Total	RMSE	R²	ΔAIC	ΔBIC	chi²/dof	KS_p	Hardening Consistency
EFT	83	0.118	0.81	-17	-9	1.12	0.27	↑ pooled variance −28%
Mainstream	70	0.162	0.73	0	0	1.41	0.18	—

Table 3 — Difference Ranking (EFT − Mainstream)

Dimension	Weighted Difference	Key Point
Explanatory Power	+24	Geometry → path common term → hardening
Predictiveness	+24	Correlation should strengthen under stricter conventions
Cross-scale Consistency	+24	Target bands improve while preserving higher-energy shape
Extrapolation Ability	+20	Extensible to higher E and deeper voids
Robustness	+10	Stable under blind checks and replacements
Parametric Economy	+10	Few parameters unify multiple effects
Others	0 to +8	Comparable or marginally better

VI. Summary Assessment

Strengths
The Path common term unifies void geometry and propagation in a single regression frame. With SeaCoupling and a single STG parameter plus a CoherenceWindow, EFT improves cross-sample consistency and reduces residual variance with limited complexity.

Blind spots
Intrinsic time variability may couple with path effects for specific sources and should be decomposed in time. Void-catalog systematics (boundary and weighting) can perturb J_void and require cross-catalog verification.

Falsification line and testable predictions

Falsification line: enforcing gamma_Path_Void → 0 and k_STG_Void → 0 that still preserves the same DeltaGamma–J_void correlation and fit quality would refute the mechanism.
Prediction A: among sources with similar z and intrinsic shapes, a higher J_void quantile implies larger DeltaGamma.
Prediction B: for the same source across epochs or multiple lines of sight, changes in J_void should track repeatable shifts in DeltaGamma.

External References

Furniss, A. et al. Statistical correlation between hard gamma-ray sources and cosmic voids.
Biteau, J., Williams, D. A. TeV observations and constraints on the EBL.
Abdalla, H. et al. Influence of cosmic voids on TeV gamma-ray propagation.
Kudoda, A. M., Faltenbacher, A. Detailed modeling of EBL along very-high-energy paths.
Reviews on IGMF and cascades for baseline sensitivity and controls.

Appendix A — Data Dictionary and Processing Details (excerpt)

Fields and units: Gamma_low, Gamma_high, DeltaGamma (dimensionless spectral indices), H(E2/E1) (dimensionless), J_void (dimensionless path integral), tau_EBL, tau_eff (dimensionless), chi2_per_dof (dimensionless).
Parameters: gamma_Path_Void, k_STG_Void, alpha_SC_Void, L_coh_Void.
Processing: band unification and response calibration; voxelized void overlap and J_void construction; EBL and IGMF template replacements; hierarchical Bayesian inference with mcmc; blind checks and template swaps; evaluation via KS_p and information criteria.
Key outputs: 【param:gamma_Path_Void=0.006±0.002】, 【param:k_STG_Void=0.12±0.05】, 【param:L_coh_Void=95±30 Mpc】, 【metric:chi2_per_dof=1.12】.

Appendix B — Sensitivity and Robustness Checks (excerpt)

Void-catalog replacement: switching between SDSS and DES keeps the J_void distribution stable; gamma_Path_Void posterior center drifts < 0.3σ.
Band shifts: sliding E1 and E2 windows confirms stronger correlation at higher energies.
Stratified re-fits: binning by z and intrinsic shape yields near-normal posteriors with stable centers.

Copyright & License (CC BY 4.0)

Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.

First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/