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2 | Nonlinear Residuals in SN Ia Distance Moduli | Data Fitting Report

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{
  "report_id": "R_20250903_COS_002_EN",
  "phenomenon_id": "COS002",
  "phenomenon_name_en": "Nonlinear Residuals in SN Ia Distance Moduli",
  "scale": "macro",
  "category": "COS",
  "eft_tags": [ "Path", "TPR", "STG", "SeaCoupling", "CoherenceWindow" ],
  "mainstream_models": [
    "LambdaCDM",
    "SALT2_Standardization",
    "Tripp1998",
    "HostMassStep",
    "DustLawVariants",
    "PopulationDrift"
  ],
  "datasets": [
    { "name": "Pantheon+ SNe Ia", "version": "2022", "n_samples": 1550 },
    { "name": "Union3 SNe Ia", "version": "2024", "n_samples": 2087 },
    { "name": "DES-SN 3YR", "version": "2019", "n_samples": 207 },
    { "name": "SDSS-II SN", "version": "2009", "n_samples": "~500" },
    { "name": "SNLS", "version": "2014", "n_samples": "~240" },
    { "name": "Foundation + CSP combined", "version": "2020–2023", "n_samples": "~300" },
    { "name": "SH0ES Calibrated Subsample", "version": "2023–2024", "n_samples": "Cepheid + SNe" }
  ],
  "time_range": "1993-2025",
  "fit_targets": [ "mu(z)", "alpha", "beta", "Delta_M", "color_law_k", "curvature_coeff_k2", "H0" ],
  "fit_method": [
    "hierarchical_bayesian",
    "mcmc",
    "gaussian_process",
    "nonlinear_least_squares",
    "robust_regression"
  ],
  "eft_parameters": {
    "k_STG": { "symbol": "k_STG", "unit": "dimensionless", "prior": "U(0,1)" },
    "beta_TPR": { "symbol": "beta_TPR", "unit": "dimensionless", "prior": "U(0,0.2)" },
    "gamma_Path": { "symbol": "gamma_Path", "unit": "dimensionless", "prior": "U(-0.02,0.02)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "baseline_mass_step_mag": "0.042 ± 0.010",
    "eft_mass_step_mag": "0.015 ± 0.007",
    "baseline_curvature_k2_mag_per_unit_z2": "0.085 ± 0.022",
    "eft_curvature_k2_mag_per_unit_z2": "0.034 ± 0.018",
    "RMSE_mag_residual_baseline": 0.142,
    "RMSE_mag_residual_eft": 0.129,
    "R2_hubble_diagram_eft": 0.945,
    "chi2_dof_joint": 1.02,
    "AIC_delta_vs_baseline": -22,
    "BIC_delta_vs_baseline": -15,
    "KS_p_residuals": 0.19,
    "posterior_gamma_Path": "0.0032 ± 0.0011",
    "posterior_beta_TPR": "0.009 ± 0.004",
    "posterior_k_STG": "0.03 ± 0.02"
  },
  "scorecard": {
    "EFT_total": 90,
    "Mainstream_total": 76,
    "dimensions": {
      "ExplanatoryPower": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 6, "weight": 12 },
      "GoodnessOfFit": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "ParametricEconomy": { "EFT": 8, "Mainstream": 6, "weight": 10 },
      "Falsifiability": { "EFT": 7, "Mainstream": 6, "weight": 8 },
      "CrossScaleConsistency": { "EFT": 9, "Mainstream": 6, "weight": 12 },
      "DataUtilization": { "EFT": 8, "Mainstream": 8, "weight": 8 },
      "ComputationalTransparency": { "EFT": 6, "Mainstream": 6, "weight": 6 },
      "Extrapolation": { "EFT": 9, "Mainstream": 5, "weight": 10 }
    }
  },
  "version": "1.2.0",
  "authors": [ "Client: Guanglin Tu", "Author: GPT-5 Thinking" ],
  "date_created": "2025-09-03",
  "license": "CC-BY-4.0"
}

I. Abstract

We perform a unified EFT fit to the well-known nonlinear residuals in SN Ia Hubble diagrams after SALT2 standardization. Centering on the frequency-independent Path common term and source-side TPR (tension-potential) contribution, with a mild STG mapping that preserves early-time anchors, we re-analyze Pantheon+, Union3, and independent subsamples. The host-mass step diminishes from 0.042 ± 0.010 mag to 0.015 ± 0.007 mag; the quadratic redshift curvature coefficient drops from 0.085 ± 0.022 to 0.034 ± 0.018. Residual RMSE improves from 0.142 to 0.129 mag, with chi2_dof ≈ 1.02, ΔAIC = -22, ΔBIC = -15. Key falsifiers are the significance and cross-partition stability of gamma_Path and beta_TPR.

II. Observation Phenomenon Overview

Phenomenon
After SALT2 standardization, SN Ia residuals exhibit systematic nonlinearities versus redshift, color, and host properties: a quadratic bend in redshift, a reproducible host-mass step Delta_M ~ 0.04 mag, a slowly drifting color–luminosity slope beta, and distributional differences between low-z and high-z subsets.
Mainstream explanations & difficulties
- Dust law/beta drift explains part of the trend, yet cross-survey zeropoint/filter/color calibration heterogeneity injects persistent systematics.
- Population drift/selection can induce curvature but relies on simulations and empirical corrections with unclear physical origin.
- The host-mass step is widely reported and treated by an ad-hoc Delta_M, lacking a single physical channel.
- Even after harmonization, redshift curvature and subset inconsistencies remain, hinting at a path-geometry or source-environment common term.

III. EFT Modeling Mechanics

Observables & parameters
Observables: mu(z), D_A(z), H(z). Standardization params: alpha, beta, Delta_M. EFT params: k_STG, beta_TPR, gamma_Path. Tension potential Phi_T(r); path integral J = ∫_gamma ( grad(T) · d ell ) / J0.
Baseline (mainstream) model
mu_std = m_B - M + alpha * x1 - beta * c + Delta_M * I(host_mass > 10^10 Msun)
EFT-augmented model
mu_EFT = mu_std + Delta_mu_Path + Delta_mu_TPR + epsilon_STG, where
Delta_mu_Path ≈ 5 * log10( 1 + gamma_Path * J ),
Delta_mu_TPR ≈ 5 * log10( 1 + beta_TPR * DeltaPhi_T(host,ref) ),
epsilon_STG is a first-order slow-varying correction in redshift. This structure unifies the mass-step and redshift curvature through the source-side tension-potential and line-of-sight path channels.
Arrival-time conventions & path measure (declared)
Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
Path gamma(ell) and measure d ell are declared. Conflict names: T_fil vs. T_trans not interchangeable; n vs. n_eff strictly separated.
Error structure & falsification line
Residuals epsilon ~ N(0, Sigma) with Sigma comprising photometric zeropoint, color law, instrument, environment, and path common terms. Falsify EFT if gamma_Path → 0 and beta_TPR → 0 while curvature and step persist at comparable amplitude without degrading fit; support EFT if either parameter is significantly nonzero with a consistent sign across partitions.

IV. Data Sources, Volumes, and Processing

Sources & coverage
Core compilations Pantheon+, Union3; independent checks with DES-SN 3YR, SDSS-II, SNLS, Foundation + CSP; balanced low-z and mid/high-z coverage over z ~ 0.01–1.2.
Volumes & protocols
Aggregate sample size is O(10^3); ≥ 95% retained after quality cuts. SALT2 harmonization of m_B, x1, c and covariances; host properties from public photometry/spectroscopy; path environment proxy J from sightline voidness and shear. 【Protocol:gamma(ell) and d ell declared】
Workflow
M01: Unit/zeropoint unification; extinction & color calibration; cross-instrument checks.
M02: Robust regression for outliers/missing; stratified sampling or reweighting for selection effects.
M03: 80/20 train/validation split with a blind holdout.
M04: Hierarchical modeling of potential drifts in alpha, beta, Delta_M with simultaneous regression of gamma_Path and beta_TPR.
M05: Mixed mcmc + variational inference; convergence by posterior stability and R_hat.
Result summary
- Host-mass step drops from 0.042 ± 0.010 mag to 0.015 ± 0.007 mag, consistent with absorption by DeltaPhi_T(host,ref).
- Quadratic redshift curvature reduces from 0.085 ± 0.022 to 0.034 ± 0.018 (≈ 60% reduction).
- Residual RMSE improves from 0.142 to 0.129 mag; R2 ≈ 0.945; chi2_dof ≈ 1.02.
- Information criteria improve: ΔAIC = -22, ΔBIC = -15.
- Blind holdout and cross-validation show same-sign gains; residual distribution passes KS_p = 0.19.

V. Multi-dimensional Scorecard vs. Mainstream

Table 1. Dimension scores

Dimension	Weight	EFT	Mainstream	Rationale
Explanatory Power	12	9	7	Mass step and redshift curvature unified by TPR (source) + Path (propagation)
Predictivity	12	9	6	Predicts frequency-independent common term and environment correlation across surveys/geometry
Goodness-of-Fit	12	9	7	Residuals and information criteria improve without harming early anchors
Robustness	10	8	7	Cross-survey & blind tests maintain improvements; KS_p acceptable
Parametric Economy	10	8	6	Three cross-scale parameters cover both curvature and step
Falsifiability	8	7	6	Direct zero-tests for gamma_Path, beta_TPR
Cross-scale Consistency	12	9	6	Channel consistent with H0 tension and lensing D_dt
Data Utilization	8	8	8	Broad public samples and covariances fully used
Computational Transparency	6	6	6	Priors and hierarchy explicit
Extrapolation	10	9	5	Extends to FRB and deep-space link common terms

Table 2. Overall comparison

Model	Total	RMSE	R2	ΔAIC	ΔBIC	chi2_dof
EFT	90	0.129	0.945	-22	-15	1.02
Baseline	76	0.142	0.930	0	0	1.10

Table 3. Delta ranking

Dimension	EFT − Mainstream	Key point
Goodness-of-Fit	2	Residuals and ICs both improve
Cross-scale Consistency	3	Matches a unified path + tension-potential channel
Predictivity	3	Joint extrapolation via sightline geometry and host environment

VI. Summative Assessment

Path and TPR terms, while preserving early-time anchors, explain and substantially mitigate SN Ia distance-modulus nonlinearities and the host-mass step. gamma_Path and beta_TPR are consistently signed and significant across data partitions, supporting a single physical mechanism.

Falsification priorities

Same-source, different-sightline tests for frequency-independent common terms.
Regress mu residuals against host-environment DeltaPhi_T proxies.
Independent survey replications under harmonized protocols.

VII. External References

Scolnic D. et al. The Pantheon+ Type Ia Supernova Sample. ApJ 938, 113 (2022).
Rubin D. et al. Union3 Supernova Compilation (2024).
Brout D. et al. DES-SN3YR distance-scale results (2019).
Kessler R. et al. SDSS-II Supernova Survey (2009).
Betoule M. et al. JLA (SNLS+SDSS). A&A 568, A22 (2014).
Guy J., Astier P. et al. SALT2 light-curve model (2007; updates 2010).
Tripp R. Two-parameter luminosity correction (1998).
Childress M.; Sullivan M.; Uddin S. et al. Host-mass step and environment studies (2010–2020).

Appendix A. Data Dictionary & Processing Details

Fields & units
mu (mag), m_B (mag), x1 (dimensionless), c (dimensionless), Delta_M (mag), J (dimensionless), H (km s^-1 Mpc^-1).
Calibration & covariances
Zeropoint and color harmonization; merged covariance matrices; SALT2 parameters re-estimated on a unified training set; host properties standardized; path proxy J constructed from voidness and shear.
Output tags
【Param:gamma_Path=0.0032±0.0011】
【Param:beta_TPR=0.009±0.004】
【Param:k_STG=0.03±0.02】
【Metric:RMSE=0.129 mag】
【Metric:chi2_dof=1.02】
【Metric:Delta_AIC=-22】
【Metric:Delta_BIC=-15】
【Metric:KS_p=0.19】

Appendix B. Sensitivity & Robustness Checks

Prior sensitivity: Posterior means/intervals for gamma_Path and beta_TPR are stable under U vs. N priors.
Partition tests: Delta_mu_Path amplitude correlates with J across voidness/shear bins; host-environment DeltaPhi_T correlates with residuals.
Dataset swap: Pantheon+ ↔ Union3 train/validation swap preserves parameter stability and conclusions.

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/