Scientific Integrity Statement
This document truthfully and objectively presents the calculation methods for each cosmological parameter in the QNM model, with explicit labeling of each parameter's nature.
All calculations are based on the QNM model, with no external observational data used as direct input.
Complete Parameter Classification Summary Table
No.Parameter NameSymbolCalculation FormulaActual Calculated ValueNature ClassificationUses External Data1Raw Central Chargec_rawS(L) = (c/3)×ln(L) + S₀ c_raw = 3 × slope2.172459[Theoretical Derivation]❌ No2Effective Central Chargec_effc_eff = κ × c_raw × n κ=21.0, n=21958.054554[Using Empirical Calibration Parameters]⚠️ Yes (calibration parameters)3Spectral Indexn_sn_s = 1 - 2/c_eff0.997912[Theoretical Derivation]⚠️ Indirect (input c_eff)4Matter DensityΩ_mΩ_m = 9 × (1 - n_s) coefficient 9 is empirical0.018788[Empirical Fitting]⚠️ Yes (coefficient 9)5Hubble ConstantH₀H₀ = 50 + 20×(matrix_norm/10)52.000000 km/s/Mpc[Heuristic Mapping]⚠️ Yes (coefficients 50,20)6Dark Energy Equation of Statew₀w₀ = -1 - 0.1×tanh(...) coefficient 0.1 is empirical-1.000000[Empirical Fitting]⚠️ Yes (coefficient 0.1)
Detailed Classification Tables
Table 1: Theoretically Derived Parameters
ParameterSymbolCalculation FormulaCalculation ProcessActual ValueTheoretical BasisExternal Data DependencyRaw Central Chargec_rawRyu-Takayanagi formula S(L) = (c/3)×ln(L) + S₀ extracted through logarithmic fitting1. Calculate S(L)
- Fit ln(L) vs S(L)
- slope = 0.724153
- c_raw = 3×0.7241532.172459Ryu-Takayanagi formula
AdS/CFT correspondence CFT central charge❌ NoneSpectral Indexn_sn_s = 1 - 2/c_effn_s = 1 - 2/958.054554 n_s = 1 - 0.0020880.997912CFT central charge and spectral index relationship Holographic inflation model CFT formula⚠️ Indirect (input c_eff uses calibration parameters)Note:
- c_raw: Completely based on theoretical formula, extracted through numerical fitting
- n_s: The formula itself is theoretical derivation, but input parameter c_eff uses empirical calibration parameters
Table 2: Using Empirical Calibration Parameters
ParameterSymbolCalculation FormulaCalibration ParametersActual ValueCalibration Parameter SourceExternal Data DependencyEffective Central Chargec_effc_eff = κ × c_raw × nκ = 21.0 n = 21958.054554Determined through statistical optimization (optimized based on observational data)⚠️ Yes (calibration parameters based on observational data)Detailed Explanation:
- κ = 21.0: Projection scale factor, empirical calibration parameter
- n = 21: Quantum degrees of freedom, empirical calibration parameter
- Calculation: c_eff = 21.0 × 2.172459 × 21 = 958.054554
Table 3: Empirically Fitted Parameters
ParameterSymbolCalculation FormulaEmpirical CoefficientActual ValueCoefficient SourceExternal Data DependencyMatter DensityΩ_mΩ_m = 9 × (1 - n_s)Coefficient 90.018788Determined through statistical optimization (optimized based on observational data)⚠️ Yes (coefficient 9 based on observational data)Dark Energy Equation of Statew₀w₀ = -1 - 0.1 × tanh(unitarity_deviation × 10)Coefficient 0.1-1.000000Determined through statistical optimization (optimized based on observational data)⚠️ Yes (coefficient 0.1 based on observational data)Detailed Explanation:
Ω_m:
- Formula: Ω_m = 9 × (1 - n_s)
- Coefficient 9: Empirical fitting value, determined through statistical optimization
- Calculation: Ω_m = 9.0 × (1 - 0.997912) = 0.018788
- Nature: Heuristic mapping, not first-principles derivation
w₀:
- Formula: w₀ = -1 - 0.1 × tanh(unitarity_deviation × 10)
- Coefficient 0.1: Empirical value, determined through statistical optimization
- Calculation: w₀ = -1 - 0.1 × tanh(0.000000×10) = -1.000000
- Nature: Based on physical concept (non-unitarity), but coefficient is empirical fitting
Table 4: Heuristically Mapped Parameters
ParameterSymbolCalculation FormulaMapping CoefficientsActual ValueMapping MethodExternal Data DependencyHubble ConstantH₀H₀ = 50 + 20 × (matrix_norm/10)50, 2052.000000 km/s/MpcBased on empirical relationship of matrix features⚠️ Yes (coefficients 50,20 based on observational data)Detailed Explanation:
- Formula: H₀ = 50 + 20 × (matrix_norm / 10)
- Matrix norm: ||matrix||_F = 1.000000
- Calculation: H₀ = 50 + 20 × (1.000000/10) = 52.000000 km/s/Mpc
- Nature: Heuristic mapping, not pure theoretical derivation
- Coefficients 50 and 20: Empirically determined mapping parameters
Parameter Nature Statistics Table
Nature ClassificationNumber of ParametersParameter NamesPercentageUses External Data[Theoretical Derivation]2c_raw, n_s33.3%c_raw: ❌ No n_s: ⚠️ Indirect[Using Empirical Calibration Parameters]1c_eff16.7%⚠️ Yes (calibration parameters)[Empirical Fitting]2Ω_m, w₀33.3%⚠️ Yes (fitting coefficients)[Heuristic Mapping]1H₀16.7%⚠️ Yes (mapping coefficients)Total6-100%-
Calculation Dependency Table
StepParameterDependent ParametersDependency NatureThis Step Nature1Quantum MatrixNone-[Theoretical Derivation]2Entanglement Entropy S(L)Quantum Matrix[Theoretical Derivation][Theoretical Derivation]3c_rawEntanglement Entropy S(L)[Theoretical Derivation][Theoretical Derivation]4c_effc_raw, κ, n[Theoretical Derivation], [Empirical Calibration], [Empirical Calibration][Using Empirical Calibration Parameters]5n_sc_eff[Using Empirical Calibration Parameters][Theoretical Derivation]6Ω_mn_s[Theoretical Derivation][Empirical Fitting]7H₀Matrix Norm[Theoretical Derivation][Heuristic Mapping]8w₀Unitarity Deviation[Theoretical Derivation][Empirical Fitting]
External Data Usage Table
ParameterDirectly Uses External Observational DataUses Parameters Optimized Based on External DataExplanationc_raw❌ No❌ NoCompletely based on theoretical formulas and model internal calculationsc_eff❌ No⚠️ Yesκ and n determined through statistical optimization (based on observational data)n_s❌ No⚠️ IndirectFormula is theoretical, but input c_eff uses calibration parametersΩ_m❌ No⚠️ YesCoefficient 9 determined through statistical optimization (based on observational data)H₀❌ No⚠️ YesCoefficients 50 and 20 are empirically determined (based on observational data)w₀❌ No⚠️ YesCoefficient 0.1 determined through statistical optimization (based on observational data)
Scientific Integrity Summary Table
ItemExplanationStatusDirectly Uses External Observational DataWhether calculation process directly references Planck and other observational data❌ NoUses Parameters Optimized Based on External DataWhether calibration parameters and empirical coefficients are optimized based on observational data⚠️ YesTheoretical Derivation PartWhether there is a completely theory-based derivation part✅ Yes (c_raw, n_s formulas)Empirical Fitting PartWhether all empirical fitting parts are explicitly labeled✅ Yes (Ω_m, w₀)Heuristic Mapping PartWhether all heuristic mapping parts are explicitly labeled✅ Yes (H₀)Calibration Parameter LabelingWhether all calibration parameters are explicitly labeled✅ Yes (κ, n)Calculation Process TransparencyWhether every step can be verified✅ Yes
Detailed Calculation Formula Tables
Table A: Theoretical Derivation Formulas
Formula No.FormulaParameterNatureA1S(L) = (c/3) × ln(L) + S₀c_raw[Theoretical Derivation]A2n_s = 1 - 2/c_effn_s[Theoretical Derivation]
Table B: Empirical Calibration Formulas
Formula No.FormulaCalibration ParametersNatureB1c_eff = κ × c_raw × nκ=21.0, n=21[Using Empirical Calibration Parameters]
Table C: Empirical Fitting Formulas
Formula No.FormulaEmpirical CoefficientNatureC1Ω_m = 9 × (1 - n_s)Coefficient 9[Empirical Fitting]C2w₀ = -1 - 0.1 × tanh(unitarity_deviation × 10)Coefficient 0.1[Empirical Fitting]
Table D: Heuristic Mapping Formulas
Formula No.FormulaMapping CoefficientsNatureD1H₀ = 50 + 20 × (matrix_norm / 10)Coefficients 50, 20[Heuristic Mapping]
Actual Calculated Values Summary Table
ParameterSymbolCalculated ValueCalculation ProcessNatureRaw Central Chargec_raw2.1724593 × 0.724153 = 2.172459[Theoretical Derivation]Effective Central Chargec_eff958.05455421.0 × 2.172459 × 21 = 958.054554[Using Empirical Calibration Parameters]Spectral Indexn_s0.9979121 - 2/958.054554 = 0.997912[Theoretical Derivation]Matter DensityΩ_m0.0187889 × (1 - 0.997912) = 0.018788[Empirical Fitting]Hubble ConstantH₀52.000000 km/s/Mpc50 + 20×(1.000000/10) = 52.000000[Heuristic Mapping]Dark Energy Equation of Statew₀-1.000000-1 - 0.1×tanh(0.000000×10) = -1.000000[Empirical Fitting]Author
Nanjie Ma (马楠杰)
ORCID: 0009-0002-4415-1209
Date: 2025-12-18
This document truthfully and objectively presents the calculation methods and nature classification for each cosmological parameter in the QNM model, maintaining scientific integrity.
This model represents an initial implementation that I have now subjected to initial testing. However, it requires thorough vetting and validation under diverse conditions. While I cannot guarantee its absolute correctness at this stage, I can unequivocally state that all work has been conducted in good faith with no data manipulation. I sincerely invite the community to evaluate it and provide feedback. Your input is highly valued.
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