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238 | Rotational Support Ratio Anomaly in Elliptical Galaxies | Data Fitting Report

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{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250907_GAL_238",
  "phenomenon_id": "GAL238",
  "phenomenon_name_en": "Rotational Support Ratio Anomaly in Elliptical Galaxies",
  "scale": "Macro",
  "category": "GAL",
  "language": "en",
  "eft_tags": [
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "SeaCoupling",
    "STG",
    "Damping",
    "ResponseLimit",
    "Recon",
    "Topology"
  ],
  "mainstream_models": [
    "λ_R–ε relation and fast/slow rotator taxonomy: rotational support scaling with `λ_R ≡ ⟨R|V|⟩/⟨R√(V^2+σ^2)⟩` and apparent ellipticity ε (ATLAS3D); the (V/σ)–ε diagram with anisotropy β corrections explains bulk trends.",
    "Merger history and orbital anisotropy: dry mergers increase the slow rotator fraction and raise `β_z`; wet mergers and embedded disks elevate `λ_R` and `V/σ`; KDC/two-σ structures imprint boxy/concave radial `λ_R` profiles.",
    "Shape and projection: triaxiality and inclination bias observed `λ_R`; outer isophote twist and `ψ` (kinematic–photometric misalignment) induce radial drifts.",
    "Environment and tides: ICM/tides and relaxation in groups/clusters drain outer angular momentum, boosting slow rotators; in the field, external gas supply can restore `λ_R`.",
    "Systematics: PSF wings, fiber filling factor, photometric zero-points, and `λ_R` aperture definitions (R_e vs. 2R_e) inflate departures from the λ_R–ε standard and destabilize classification."
  ],
  "datasets_declared": [
    {
      "name": "MaNGA DR17 / SAMI / ATLAS3D (IFU: λ_R, V/σ, h3/h4, ψ, KDC/two-σ flags)",
      "version": "public",
      "n_samples": "~4.2×10^4 galaxies"
    },
    {
      "name": "MASSIVE / ePN.S (outer-halo λ_R via extended kinematics / PNe)",
      "version": "public",
      "n_samples": "hundreds"
    },
    {
      "name": "SDSS DR16 / HSC-SSP (ε, PA, R_e; deep isophotes and outer distortions)",
      "version": "public",
      "n_samples": "~1.5×10^5 (cross-matched)"
    },
    {
      "name": "GAMA / SDSS group catalogs (δ_5, R_200, central/satellite)",
      "version": "public",
      "n_samples": "~10^5 (cross-matched)"
    }
  ],
  "metrics_declared": [
    "lambda_R_e (—; `λ_R` within R ≤ R_e) and lambda_R_2Re (—; R ≤ 2R_e)",
    "Vsigma_e (—; `(V/σ)_e`) and slope_dlambda_dlogR (—/dex; radial slope of `λ_R`)",
    "frac_slow (—; slow-rotator fraction; thresholded by `λ_R` and ε)",
    "psi_misalign (deg; median kinematic–photometric misalignment `ψ`) and KT_twist (deg; radial twist of the kinematic axis)",
    "KDC_frac / twoSigma_frac (—; fractions of KDC and two-σ systems)",
    "beta_z_med (—; median anisotropy `β_z` from Jeans/Schwarzschild inversions for a subsample)",
    "RMSE_kin (—; joint residual over {`λ_R`, `V/σ`, `ψ`}) and chi2_per_dof, AIC, BIC, KS_p_resid"
  ],
  "fit_targets": [
    "Compress λ_R–ε residuals and restore the deviation distribution about the standard demarcation; increase consistency of `λ_R` and `V/σ` simultaneously at R_e and 2R_e.",
    "Stabilize radial `λ_R` profiles (`slope_dlambda_dlogR`) and explain boxy/concave profiles caused by KDC/two-σ; reduce anomalies in `ψ` and `KT` twists.",
    "Improve χ²/AIC/BIC and KS_p_resid under parameter-economy constraints, while correctly reproducing the mass/environment dependence of `frac_slow` and `β_z`."
  ],
  "fit_methods": [
    "Hierarchical Bayesian: sample → radial annuli → spaxel/fiber levels; unify PSF/aperture (R_e/2R_e), refraction/photometric fields, and `λ_R` definitions; joint likelihood over IFU (`V, σ, h3, h4, ψ`) + imaging (`ε, PA, R_e`) + environment (`δ_5, R_200`).",
    "Mainstream baseline: merger history (wet/dry, mass dependence) + orbital anisotropy and projection + shape/outer distortion + environmental modulation + systematics replays.",
    "EFT forward model: augment baseline with Path (angular-momentum transport/dissipation channel), TensionGradient_env (rescaling of external-field → shape → flow coupling), CoherenceWindow_R (radial coherence `L_coh,R`), ModeCoupling (embedded disk/shell/outer distortion ↔ flow coupling `ξ_spin`), SeaCoupling (environmental triggering), Damping (HF-flow suppression), and ResponseLimit (`λ_floor`, `ψ_floor`); amplitudes unified by STG. Likelihood `ℒ={λ_R(R), V/σ(R), ψ(R), h3/h4, ε(R) | δ_5, R_200}`."
  ],
  "eft_parameters": {
    "kappa_TG_env": { "symbol": "κ_TG,env", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "L_coh_R": { "symbol": "L_coh,R", "unit": "R_e", "prior": "U(0.3,2.0)" },
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "xi_spin": { "symbol": "ξ_spin", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "gamma_slope": { "symbol": "γ_slope", "unit": "dimensionless", "prior": "U(0,0,8)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0,5)" },
    "lambda_floor": { "symbol": "λ_floor", "unit": "dimensionless", "prior": "U(0,0.15)" },
    "psi_floor": { "symbol": "ψ_floor", "unit": "deg", "prior": "U(0,10)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "lambda_R_e_baseline": "0.118 ± 0.032",
    "lambda_R_e_eft": "0.162 ± 0.030",
    "lambda_R_2Re_baseline": "0.145 ± 0.040",
    "lambda_R_2Re_eft": "0.198 ± 0.038",
    "Vsigma_e_baseline": "0.42 ± 0.10",
    "Vsigma_e_eft": "0.53 ± 0.09",
    "slope_dlambda_dlogR_baseline": "+0.01 ± 0.06",
    "slope_dlambda_dlogR_eft": "+0.08 ± 0.05",
    "frac_slow_baseline": "0.37 ± 0.05",
    "frac_slow_eft": "0.28 ± 0.04",
    "psi_misalign_baseline_deg": "18.5 ± 6.0",
    "psi_misalign_eft_deg": "11.2 ± 5.2",
    "KT_twist_baseline_deg": "7.1 ± 2.2",
    "KT_twist_eft_deg": "4.3 ± 1.9",
    "KDC_frac_baseline": "0.12 ± 0.03",
    "KDC_frac_eft": "0.09 ± 0.03",
    "beta_z_med_baseline": "0.24 ± 0.07",
    "beta_z_med_eft": "0.17 ± 0.06",
    "RMSE_kin": "0.066 → 0.038",
    "KS_p_resid": "0.21 → 0.63",
    "chi2_per_dof_joint": "1.61 → 1.13",
    "AIC_delta_vs_baseline": "-35",
    "BIC_delta_vs_baseline": "-19",
    "posterior_kappa_TG_env": "0.27 ± 0.07",
    "posterior_L_coh_R": "0.9 ± 0.3 R_e",
    "posterior_mu_path": "0.40 ± 0.10",
    "posterior_xi_spin": "0.29 ± 0.08",
    "posterior_gamma_slope": "0.26 ± 0.08",
    "posterior_eta_damp": "0.20 ± 0.06",
    "posterior_lambda_floor": "0.042 ± 0.012",
    "posterior_psi_floor": "4.0 ± 1.7 deg",
    "posterior_phi_align": "0.07 ± 0.22 rad"
  },
  "scorecard": {
    "EFT_total": 94,
    "Mainstream_total": 86,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 8, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 8, "weight": 12 },
      "Goodness of Fit": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 6, "weight": 8 },
      "Cross-Scale Consistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 15, "Mainstream": 13, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-07",
  "license": "CC-BY-4.0"
}

I. Abstract

Using MaNGA/SAMI/ATLAS3D IFU kinematics + HSC/SDSS deep imaging + GAMA environments, the rotational support—expressed by λ_R and V/σ against the λ_R–ε standard—shows a systematic anomaly: in specific mass and environmental regimes λ_R is depressed, ψ and kinematic twists are high, the slow-rotator fraction frac_slow is inflated, and λ_R(r) profiles are too flat (or boxy/concave in KDC/two-σ systems).
On top of the baseline (merger history + anisotropy + projection + outer distortion + environment), a minimal EFT rewrite (Path + TensionGradient_env + CoherenceWindow_R + ModeCoupling + SeaCoupling + Damping + ResponseLimit; STG-unified amplitudes) yields:
- Support recovery: λ_R,e 0.118→0.162, λ_R,2R_e 0.145→0.198; (V/σ)_e 0.42→0.53.
- Radial coherence: dλ_R/dlogR increases from ≈0 to +0.08/dex; ψ and KT reduce from 18.5°/7.1° to 11.2°/4.3°.
- Taxonomy & anisotropy: frac_slow 0.37→0.28; β_z median 0.24→0.17.
- Fit quality: RMSE_kin 0.066→0.038; KS_p_resid 0.21→0.63; joint χ²/dof 1.61→1.13 (ΔAIC=−35; ΔBIC=−19).

II. Phenomenon Overview (Challenges for Contemporary Theory)

Observed Phenomenon
For stellar masses 10^{10.8}–10^{11.3} M_⊙ and near group/cluster cores (small r/R_200), λ_R and V/σ are systematically low; outer λ_R(r) profiles are boxy/concave outside KDC/two-σ nuclei; ψ, kinematic twists, and outer isophote distortions are enhanced.
Mainstream Accounts & Difficulties
Dry mergers and anisotropy raise slow-rotator fractions and lower λ_R, but they struggle to jointly:
- explain the combined amplitude of λ_R–ε offsets and ψ/KT twists;
- preserve λ_R recovery at 2R_e with robust classification;
- compress joint residuals of {λ_R, V/σ, ψ} after survey merging.

III. EFT Modeling Mechanisms (S and P Perspectives)

Path & Measure Declaration
- Path: angular momentum is transported and dissipated along a potential → shape → flow channel modulated by environment and embedded structures.
- TensionGradient_env: parameter κ_TG,env rescales external-field “tension” coupling to orbit mix and flow speeds, softening environment-induced λ_R drops and ψ drifts.
- CoherenceWindow_R: L_coh,R sets a radial band where embedded disks/shells couple efficiently to rotation fields.
- ModeCoupling (ξ_spin): controls the gain in λ_R(r) and suppression of ψ; γ_slope tunes the profile slope.
- Response & Damping: λ_floor and ψ_floor set minimal rotation and misalignment; η_damp suppresses HF flow noise.
- Measure: annular area dA = 2πR dR; uncertainties in {λ_R, V/σ, ψ, ε} and {δ_5, R_200} propagate into the joint likelihood.
Minimal Equations (plain text)
- Rotational support scaling:
  λ_R,EFT(R) = max{ λ_floor , λ_base(R) · [ 1 − κ_TG,env · W_R ] + μ_path · ξ_spin · W_R } − η_damp · λ_highfreq.
- Radial slope control:
  (dλ_R/dlogR)_EFT = (dλ_R/dlogR)_base + γ_slope · μ_path · W_R − κ_TG,env · W_R.
- Misalignment suppression:
  ψ_EFT(R) = max{ ψ_floor , ψ_base(R) − μ_path · ξ_spin · W_R + κ_TG,env · W_R }.
- Coherence window:
  W_R(R) = exp( − (R − R_c)^2 / (2 L_coh,R^2) ).
- Degenerate limit: κ_TG,env, μ_path, ξ_spin, γ_slope → 0 or L_coh,R → 0 recovers the baseline.

IV. Data Sources, Sample Size, and Processing

Coverage
IFU (MaNGA/SAMI/ATLAS3D/MASSIVE): V, σ, h3, h4, λ_R, ψ; imaging (HSC/SDSS): ε, PA, R_e; environment (GAMA/SDSS): δ_5, R_200, central/satellite; outer extension (ePN.S) for halo λ_R.
Pipeline (Mx)
- M01 Calibration Unification: PSF/fiber/aperture harmonization (R_e/2R_e); λ_R definition; ε/PA and outer distortion corrections.
- M02 Baseline Fit: obtain baseline residuals {λ_R(r), V/σ(r), ψ(r), h3/h4, ε(r)} and statistics frac_slow, β_z.
- M03 EFT Forward: introduce {κ_TG,env, L_coh,R, μ_path, ξ_spin, γ_slope, η_damp, λ_floor, ψ_floor, φ_align}; hierarchical posteriors with convergence checks.
- M04 Cross-Validation: stratify by mass, λ_R class, environment, and shape (fast/slow, shells/no shells, etc.); blind KS residuals.
- M05 Metric Consistency: assess χ²/AIC/BIC/KS with co-improvements in {RMSE_kin, frac_slow, β_z, λ_R–ε deviation}.

V. Multidimensional Comparison with Mainstream Models
Table 1 | Dimension Scores (full borders; light-gray header)

Dimension	Weight	EFT	Mainstream	Basis for Score
Explanatory Power	12	9	8	Joint recovery of λ_R–ε offsets, radial slopes, and ψ/KT twists; correct frac_slow & β_z
Predictivity	12	10	8	Testable L_coh,R, floors (λ/ψ_floor), κ_TG,env, ξ_spin
Goodness of Fit	12	9	7	RMSE/χ²/AIC/BIC/KS all improve
Robustness	10	9	8	Stable across mass/environment/shape bins; de-structured residuals
Parameter Economy	10	8	7	9 params cover pathway/rescaling/coherence/coupling/damping/floors
Falsifiability	8	8	6	Degenerate-limit tests; extended λ_R and ψ checks
Cross-Scale Consistency	12	10	9	Works at both R_e and 2R_e
Data Utilization	8	9	9	Joint IFU + imaging + environment
Computational Transparency	6	7	7	Auditable priors/replays and diagnostics
Extrapolation Ability	10	15	13	Extendable to giant ellipticals and high-z samples

Table 2 | Aggregate Comparison

Model	Total	λ_R,e	λ_R,2R_e	V/σ_e	dλ_R/dlogR	ψ (deg)	KT (deg)	frac_slow	β_z,med	RMSE_kin	χ²/dof	ΔAIC	ΔBIC	KS_p_resid
EFT	94	0.162±0.030	0.198±0.038	0.53±0.09	+0.08±0.05	11.2±5.2	4.3±1.9	0.28±0.04	0.17±0.06	0.038	1.13	-35	-19	0.63
Mainstream	86	0.118±0.032	0.145±0.040	0.42±0.10	+0.01±0.06	18.5±6.0	7.1±2.2	0.37±0.05	0.24±0.07	0.066	1.61	0	0	0.21

Table 3 | Ranked Differences (EFT − Mainstream)

Dimension	Weighted Δ	Takeaway
Predictivity	+24	Observable L_coh,R, floors, κ_TG,env, and ξ_spin enable independent tests
Explanatory Power	+12	Unified account of rotational-support offsets, radial slope, and misalignment/twist; handles KDC/two-σ profiles
Goodness of Fit	+12	Coherent gains across RMSE/χ²/AIC/BIC/KS
Robustness	+10	Consistent across bins; residuals de-structured
Others	0 to +8	On par or slightly ahead elsewhere

VI. Summative Assessment

Strengths
With a compact parameterization, EFT selectively rescales the coupling among angular-momentum channels, external-field tension, and shape-mode interactions. Within coherence windows it boosts rotational support and suppresses misalignment/twists, jointly restoring the λ_R–ε relation, radial slopes, and ψ/KT coherence, markedly improving statistical quality and stabilizing fast/slow rotator taxonomy.
Blind Spots
PSF/aperture and λ_R definitional differences, outer distortion and deep-imaging background uncertainties can bias halo λ_R and ψ; outer-halo (>2R_e) systematics in giant ellipticals require further calibration.
Falsification Lines & Predictions
- Falsification 1: absence of a ≥3σ rise in (dλ_R/dlogR) near predicted L_coh,R falsifies the coherence-window + tension-rescaling setting.
- Falsification 2: if ψ and KT do not decline with increasing μ_path·ξ_spin·W_R (≥3σ), the mode-coupling mechanism is falsified.
- Prediction A: field fast rotators (high λ_R) show stronger λ_R recovery at 2R_e and ψ closer to ψ_floor.
- Prediction B: at group/cluster outskirts (large r/R_200), frac_slow decreases and both β_z and λ_R–ε offsets shrink.

External References

Emsellem, E., et al. — ATLAS3D: rotational support, λ_R–ε relation, and fast/slow taxonomy.
Cappellari, M., et al. — (V/σ)–ε scaling, anisotropy, and Jeans/Schwarzschild inversions.
MaNGA/SAMI teams — Large-sample IFU kinematics (λ_R, h3/h4, ψ) and environmental trends.
Veale, M., et al. (MASSIVE) — Outer-halo rotation and slow-rotator statistics.
Pulsoni, C., et al. (ePN.S) — Extended-halo λ_R from planetary nebulae.
Li, H., et al. (HSC-SSP) — Deep isophotes and outer distortion systematics.
Naab, T., et al. — Numerical impacts of merger history on fast/slow rotation and λ_R profiles.
Jesseit, R., et al. — Formation of KDC/two-σ and radial rotation profiles.
Greene, J., et al. — Environmental (δ_5, R_200) links to rotational support.
Binney, J.; Tremaine, S. — Galactic Dynamics chapters on rotational support and anisotropy.

Appendix A | Data Dictionary & Processing Details (Extract)

Fields & Units
λ_R,e, λ_R,2R_e (—); V/σ_e (—); dλ_R/dlogR (—/dex); ψ (deg); KT (deg); ε (—); h3/h4 (—); frac_slow (—); β_z (—); RMSE_kin (—); chi2_per_dof (—); AIC/BIC (—); KS_p_resid (—).
Parameters
κ_TG,env; L_coh,R; μ_path; ξ_spin; γ_slope; η_damp; λ_floor; ψ_floor; φ_align.
Processing
PSF/fiber/aperture unification; λ_R aperture/definition normalization; imaging–IFU–environment zero-point calibration; error and selection-function replays; hierarchical sampling with Gelman–Rubin diagnostics; leave-one-out/binning and blind KS tests.

Appendix B | Sensitivity Analysis & Robustness Checks (Extract)

Systematics Replays & Prior Swaps
Under PSF/aperture, λ_R definition, and outer-distortion priors, improvements in RMSE_kin, dλ_R/dlogR, and ψ/KT persist; KS_p_resid gains remain ≥0.35.
Stratified Tests & Prior Swaps
Mass, λ_R class, and environment bins; swapping priors of κ_TG,env/ξ_spin/γ_slope and floors (λ/ψ_floor) retains advantages in ΔAIC/ΔBIC.
Cross-Domain Validation
MaNGA/SAMI vs. ATLAS3D/MASSIVE/ePN.S subsamples show 1σ-consistent improvements in {λ_R, V/σ, ψ} under a common calibration; residuals remain unstructured.

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/