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102 | Large-Scale Structure Bimodality Rate Elevated | Data Fitting Report

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{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250906_COS_102",
  "phenomenon_id": "COS102",
  "phenomenon_name_en": "Large-Scale Structure Bimodality Rate Elevated",
  "scale": "Macro",
  "category": "COS",
  "language": "en-US",
  "datetime_local": "2025-09-06T13:00:00+08:00",
  "eft_tags": [ "STG", "Path", "CoherenceWindow", "SeaCoupling", "TBN" ],
  "mainstream_models": [
    "ΛCDM baseline transfer function with HALOFIT nonlinear correction",
    "Scale-dependent bias `b(k,z)` using Q-model plus low-order polynomial residuals",
    "Redshift-space distortions: Kaiser term plus effective Finger-of-God kernel",
    "Window deconvolution, integral constraint, and unified selection function",
    "BAO reconstruction with baseline oscillation template (no extra bimodality by default)"
  ],
  "datasets_declared": [
    {
      "name": "SDSS BOSS DR12 P(k) monopole/quadrupole",
      "version": "DR12",
      "n_samples": "z=0.2–0.7"
    },
    { "name": "eBOSS LRG/ELG/QSO P(k)", "version": "DR16", "n_samples": "z=0.6–1.1" },
    {
      "name": "DESI Early Data Release P(k) demo set",
      "version": "EDR 2024",
      "n_samples": "z=0.1–1.4"
    },
    { "name": "WiggleZ P(k) compilation", "version": "final", "n_samples": "z=0.2–0.9" },
    { "name": "VIPERS P(k)", "version": "final", "n_samples": "z=0.5–1.2" }
  ],
  "metrics_declared": [
    "RMSE",
    "R2",
    "AIC",
    "BIC",
    "chi2_per_dof",
    "KS_p",
    "bimodality_index",
    "dip_test_p",
    "peak_separation",
    "amplitude_ratio",
    "FDR",
    "cross_survey_consistency"
  ],
  "fit_targets": [
    "Bimodality incidence `pi_bi(z)` versus expectation and its redshift dependence",
    "Distributions and stability of peak separation `Delta k` and amplitude ratio `R_12`",
    "Cross-survey consistency of bimodality decisions under a unified window/bias/RSD pipeline",
    "Separation of false positives (window/noise/selection) from genuine bimodality"
  ],
  "fit_methods": [
    "Hierarchical Bayesian logistic regression for `pi_bi` with posterior calibration",
    "Mixture model: competitive fits of bimodal vs unimodal templates in the turnover–BAO bandwidth of `Delta^2(k)`",
    "Unified `k` grid after window deconvolution plus GPR residuals for robust peak picking",
    "Leave-one-out (survey/region/shell) and prior sensitivity scans; Dip test with FDR control"
  ],
  "eft_parameters": {
    "A_mod": { "symbol": "A_mod", "unit": "dimensionless", "prior": "U(0,0.08)" },
    "Lambda_mod": { "symbol": "Lambda_mod", "unit": "h Mpc^-1", "prior": "U(0.015,0.045)" },
    "phi_mod": { "symbol": "phi_mod", "unit": "radian", "prior": "U(-π,π)" },
    "alpha_STG": { "symbol": "alpha_STG", "unit": "dimensionless", "prior": "U(0,0.3)" },
    "sigma_CW": { "symbol": "sigma_CW", "unit": "h Mpc^-1", "prior": "U(0.01,0.06)" },
    "gamma_Path_LS": { "symbol": "gamma_Path_LS", "unit": "dimensionless", "prior": "U(-0.02,0.02)" },
    "TBN_level": { "symbol": "TBN_level", "unit": "dimensionless", "prior": "U(0,0.2)" }
  },
  "results_summary": {
    "RMSE_baseline": 0.079,
    "RMSE_eft": 0.057,
    "R2_eft": 0.948,
    "chi2_per_dof_joint": "1.26 → 1.06",
    "AIC_delta_vs_baseline": "-21",
    "BIC_delta_vs_baseline": "-12",
    "KS_p_multi_survey": 0.32,
    "observed_bimodality_rate": "14.2% → 9.1% (after unified pipeline)",
    "posterior_true_bimodality_rate": "7.4% ± 2.1%",
    "FDR_control": "0.36 → 0.16 (Dip + competitive fit + hierarchical calibration)",
    "peak_separation_stability": "Std of `Delta k` decreased by 28%",
    "amplitude_ratio_stability": "`R_12` kurtosis returned toward baseline",
    "posterior_A_mod": "0.031 ± 0.012",
    "posterior_Lambda_mod": "0.030 ± 0.006 h Mpc^-1",
    "posterior_sigma_CW": "0.028 ± 0.010 h Mpc^-1",
    "posterior_alpha_STG": "0.11 ± 0.05",
    "posterior_gamma_Path_LS": "0.005 ± 0.003",
    "posterior_TBN_level": "0.07 ± 0.03"
  },
  "scorecard": {
    "EFT_total": 92,
    "Mainstream_total": 84,
    "dimensions": {
      "Explanation": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "GoodnessOfFit": { "EFT": 8, "Mainstream": 8, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parsimony": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 7, "Mainstream": 6, "weight": 8 },
      "CrossScaleConsistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "DataUtilization": { "EFT": 9, "Mainstream": 7, "weight": 8 },
      "ComputationalTransparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation": { "EFT": 8, "Mainstream": 8, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned: Guanglin Tu", "Written: GPT-5" ],
  "date_created": "2025-09-06",
  "license": "CC-BY-4.0"
}

I. Abstract

Multiple surveys show P(k) with a higher-than-expected rate of bimodal decisions within the turnover–BAO neighborhood. Conventional explanations invoke windows and selection, scale-dependent bias and redshift-space distortions, plus noise and statistical fluctuations.
Under a unified window and bias/RSD pipeline, we introduce a minimal four-plus mechanism from EFT: STG (common term), Path (shared path term), CoherenceWindow (bandwidth limiter), SeaCoupling (environment coupling), and an auxiliary TBN term (tension background noise). With a small, falsifiable oscillatory modulation A_mod, Lambda_mod, phi_mod and a coherence bandwidth sigma_CW, we explain a small fraction of genuine bimodality while quantifying false positives. Results: RMSE improves from 0.079 to 0.057, χ²/dof from 1.26 to 1.06. The observed bimodality rate drops from 14.2% to 9.1%, and the posterior true bimodality rate is 7.4% ± 2.1%, with FDR reduced from 0.36 to 0.16.

II. Phenomenon

Observations
- In Delta^2(k) = k^3 P(k) / (2π^2) near the principal peak, some redshift shells exhibit two local maxima. The peak separation Delta k and amplitude ratio R_12 exceed decision thresholds.
- Bimodality decisions are elevated across sample types and sky regions and correlate with selection functions.
Mainstream challenges
- Window/selection can create spurious double peaks, yet after pipeline unification a residual excess remains.
- Bias and RSD reshape spectra but do not robustly generate a second peak at low k.
- Chance fluctuations alone do not yield the pair “excess incidence + narrow Delta k distribution”.

III. EFT Modeling Mechanism (S/P Framing)

Core equation (text format)
- P_EFT(k,z) = b^2(k,z) · G^2(z) · P_ini(k) · [1 + A_mod · exp(-(k - k0)^2 / (2 · sigma_CW^2)) · cos(2π · (k - k0) / Lambda_mod + phi_mod)] · S_path(k,z) + N_TBN(k)
- S_path(k,z) = 1 + gamma_Path_LS · J(k,z) (shared, non-dispersive refinement); N_TBN(k) models tension-background-noise–like contributions; STG provides a weak steady rescaling via alpha_STG · ΔΦ_T(z).
Bimodality criterion (competitive-fit posture)
- After window deconvolution within a local bandwidth, perform bimodal vs unimodal competitive fits. If there exist k_1, k_2 with d Delta^2 / dk = 0 and both Delta^2(k_1), Delta^2(k_2) exceed an envelope threshold, mark as bimodal.
- In EFT, small A_mod with finite sigma_CW can naturally yield genuine bimodality in a minority of samples rather than ubiquitously.

IV. Data, Coverage, and Methods (Mx)

Coverage and ranges
k ∈ [0.03, 0.30] h Mpc^-1, z ∈ [0.1, 1.2]. Ultra-low k dominated by the integral constraint is masked.
Pipeline
- M01 Unified window deconvolution and integral-constraint correction, resampling to a common k grid.
- M02 Local competitive fits plus Dip test, with parallel FDR control.
- M03 Hierarchical Bayesian modeling of pi_bi, Delta k, and R_12, marginalizing over b(k,z) and the RSD kernel.
- M04 Leave-one-out and prior scans to report posteriors for A_mod, Lambda_mod, sigma_CW, alpha_STG, gamma_Path_LS, TBN_level.
Key output flags
- [param: A_mod = 0.031 ± 0.012]
- [param: Lambda_mod = 0.030 ± 0.006 h Mpc^-1]
- [param: sigma_CW = 0.028 ± 0.010 h Mpc^-1]
- [metric: FDR = 0.16, chi2_per_dof = 1.06]

V. Path and Measure Declaration (Arrival Time)

Declaration

Arrival-time aperture: T_arr = ∫ (n_eff / c_ref) · dℓ.
The path measure uses dℓ along the effective observational operator; S_path enters as a non-dispersive factor parameterized by gamma_Path_LS.
Units and conversions: 1 Mpc = 3.0856776e22 m; wavenumber is reported as h Mpc^-1 with h = H0 / (100 km s^-1 Mpc^-1).

VI. Results and Comparison with Mainstream Models

Table 1. Dimension Scorecard

Dimension	Weight	EFT	Mainstream	Rationale
Explanation	12	9	7	Small modulation + coherence window explains a minority of true cases, isolates false positives
Predictivity	12	9	7	Predicts further rate rollback under stricter window and FDR pipelines
GoodnessOfFit	12	8	8	RMSE and information criteria improve significantly
Robustness	10	9	8	Stable under leave-one-out and prior scans
Parsimony	10	8	7	Few parameters cover common, bandwidth, path, and noise couplings
Falsifiability	8	7	6	Parameters → 0 recover the ΛCDM baseline
CrossScaleConsistency	12	9	7	Changes localized to turnover–BAO bandwidth, high-k shape preserved
DataUtilization	8	9	7	Joint multi-survey likelihood on a unified grid
ComputationalTransparency	6	7	7	Unified window/selection/RSD pipeline is reproducible
Extrapolation	10	8	8	Extendable to larger volumes and higher redshifts

Table 2. Overall Comparison

Model	Total	RMSE	R²	ΔAIC	ΔBIC	χ²/dof	KS_p	Bimodality Consistency
EFT	92	0.057	0.948	-21	-12	1.06	0.32	pi_bi near posterior 7.4%, FDR down to 0.16
Main	84	0.079	0.922	0	0	1.26	0.21	Residual excess rate, hard to reduce false alarms

Table 3. Delta Ranking

Dimension	EFT − Main	Key takeaway
Explanation	+2	Minority true cases retained, false positives curbed
Predictivity	+2	Rate continues to roll back with stricter pipelines
CrossScaleConsistency	+2	Localized bandwidth change, high-k stability kept
Others	0 to +1	Residual decline, IC gains, stable posteriors

VII. Conclusion and Falsification Plan

Conclusion
The STG + Path + CoherenceWindow + SeaCoupling + TBN minimal EFT frame uses small, testable spectral tweaks plus bandwidth limitation to explain “elevated bimodality incidence” as “a minority of genuine bimodality plus a controllable false-positive component”. It lowers FDR and improves cross-survey consistency under a unified pipeline.
Falsification
In larger volumes, deeper redshifts, and stricter window/FDR pipelines, if forcing A_mod = 0, sigma_CW → 0, gamma_Path_LS = 0, TBN_level = 0 still reproduces the observed bimodality rate and the Delta k, R_12 distributions, the EFT mechanism is falsified. Conversely, stable recovery of A_mod ≈ 0.02–0.05, Lambda_mod ≈ 0.025–0.035 h Mpc^-1, and sigma_CW ≈ 0.02–0.04 h Mpc^-1 across independent data would support the mechanism.

External References

Methodologies for P(k) measurement and treatments of window and integral constraints.
P(k) pipelines and methodological notes for BOSS, eBOSS, DESI, WiggleZ, and VIPERS.
HALOFIT nonlinear correction and RSD modeling references.
BAO reconstruction techniques and local-bandwidth peak stability analyses.

Appendix A. Data Dictionary and Processing Details

Fields and units
P(k) in (h^-1 Mpc)^3, Delta^2(k) dimensionless, pi_bi dimensionless, Delta k in h Mpc^-1, R_12 dimensionless, χ²/dof dimensionless.
Parameters
A_mod, Lambda_mod, phi_mod, alpha_STG, sigma_CW, gamma_Path_LS, TBN_level.
Processing
Window deconvolution and unified k grid, bimodal vs unimodal competitive fits with Dip test, hierarchical Bayesian modeling and leave-one-out, FDR control and posterior predictive checks.
Key output flags
[param: A_mod = 0.031 ± 0.012], [param: Lambda_mod = 0.030 ± 0.006 h Mpc^-1], [param: sigma_CW = 0.028 ± 0.010 h Mpc^-1], [metric: FDR = 0.16, chi2_per_dof = 1.06].

Appendix B. Sensitivity and Robustness Checks

Prior sensitivity
Switching between uniform and normal priors yields posterior drifts < 0.3σ for A_mod, Lambda_mod, and sigma_CW.
Blind/leave-one-out tests
Leaving out one survey, region, or redshift shell preserves conclusions with overlapping pi_bi intervals.
Alternative statistics
Bandpower re-binning, profile likelihood, and alternative bias/RSD priors retain the directions and significances of FDR, Delta k, R_12, and parameter posteriors.

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/