Editable inputCalculated value
Title
| Swarm Field Theory Equations Calculator Version 4 Copyright 2026 |
| Author: Paul Crumpler | Version: 4| Date: 16MAR2026 |
| Citation: Crumpler, J. P. (2026). Swarm Field Equations Calculator (Version 4.0) [Data set]. DOI: 10.5281/zenodo.19047067 |
| https://swarmfieldtheory.org |
| Workbook Guide |
| Inputs sheet: change only highlighted input values for scenario testing. |
| Constants sheet: CODATA and exact SI reference values used for comparison. |
| Derivations and Derived sheets: intermediate and extended calculated quantities. |
| Outputs_Core and Outputs_Extended: main predicted values versus references. |
| Comparison_to_CODATA: consolidated prediction table with percent differences. |
| Predictions_External: astrophysical and external reality checks. |
| Gravity_G_2026: updated gravitational constant derivation using the 2026 geometry formulation. |
| Reference constants align with CODATA / NIST values used in the workbook. |
Inputs
| Parameter | Parameter Name | Parameter Value | Units | Note | Definition: |
| λ | (coherence aperture) | m | Characteristic aperture length (≈ 8.20×10⁻¹⁰ m) that anchors the lattice geometry. It defines the smallest stable coherence closure, setting the fundamental scale where waves transition from blur to quantized mass/energy. | ||
| κ | (helical projection) (Arthur's Constant) | Unitless | Helical projection factor (≈ 0.92203), also called Arthur’s Constant. Arises from wavefronts wrapping helically through the lattice, giving the ratio of projected linear distance to actual path length. Corrects light speed and coherence paths for lattice geometry. | ||
| N | rotations per wavelength) | Rotations / Wavelength | Number of discrete helical rotations per lattice wavelength (≈ 11.80388). Quantizes rotational closure and links wave energy to stable particle formation. | ||
| Φ | junction packing factor) | Unitless | Packing factor (≈ 2.44764) describing lattice junction density. Adjusts the geometric suppression factor to match observed fine-structure by accounting for how multiple membrane planes intersect and constrain coherence. | ||
| τG | gravity coherence cycle time | s | Coherence cycle time used in the 2026 gravity derivation. This gravity-specific cycle time is used only in the updated gravitational constant sheet. | ||
| NOTEs: |
Constants
| Parameter | Parameter Name | Units | CODATA Values | Note |
| h | Planck’s constant | J·s | 6.6260701500e-34 | Exact by SI (2019) |
| ħ = h / (2π) | Reduced Planck’s constant | J·s | 1.0545718176e-34 | Computed by Standard Equation |
| c | Speed of light in vacuum | m/s | 2.9979245800e+08 | Exact by SI |
| e | Elementary charge | C | 1.6021766340e-19 | Exact by SI |
| kB | Boltzmann constant | J/K | 1.3806490000e-23 | Exact by SI |
| α | Fine-structure constant | Unitless | 0.0072973525693 | Reference |
| ε₀ | Vacuum permittivity (electric constant) | F/m | 8.8541878128e-12 | Reference |
| μ₀ | Vacuum permeability (magnetic constant) | H/m | 1.2566370621e-06 | Reference |
| G | Gravitational constant | m^3 kg^-1 s^-2 | 6.6743000000e-11 | Reference |
| mₑ | Electron mass | kg | 9.1093837015e-31 | Reference (fixed here) |
| mₚ | Proton mass | kg | 1.6726219237e-27 | Reference (fixed) |
Derivations
| Constants | Constants | Quantity (BASE — no closure) | Excel formula | Swarm Field Theory Predicted Value | Units |
| τ | Coherence time constant (lattice response time) | τ = κ λ / c | =Inputs!C3*Inputs!C2/Constants!D4 | 2.5219600421e-18 | s |
| ΔI | Coherence membrane tension (information/energy density) | ΔI = ħ / (π λ2 τ) | =Constants!D3/(PI()*Inputs!C2*Inputs!C2*E2) | 19.7952222563 | J·m⁻² |
| ξbase | Base geometric suppression factor | ξbase ξ_base = κ/(2√(πN)) | =Inputs!C3/(2*SQRT(PI()*Inputs!C4)) | 0.0757055514033 | dimensionless |
| ξgeom | Geometric suppression factor (with packing correction Φ) | ξ_geom = ξ_base · Φ | =E4*Inputs!C5 | 0.1853 | dimensionless |
| αpred | Predicted fine-structure constant | α_pred = (ξ_geom^2/4) · κ^2 | =(E5*E5/4)*(Inputs!C3*Inputs!C3) | 0.00729761505874 | dimensionless |
| ε₀pred | Predicted vacuum permittivity | ε0_pred = e^2/(4π α_pred ħ c) | =(Constants!D5*Constants!D5)/(4*PI()*E6*Constants!D3*Constants!D4) | 8.8538693346e-12 | F·m⁻¹ |
| μ₀pred | Predicted vacuum permeability | μ0_pred = 1/(ε0_pred c^2) | =1/(E7*Constants!D4*Constants!D4) | 1.2566822640e-06 | H·m⁻¹ |
| Θgeom base | Base geometric closure term (Route B gravitational precursor) | Θ_geom_base = κ^3 α_pred^11 /(4π N Φ) | =(Inputs!C3*Inputs!C3*Inputs!C3)*POWER(E6,11)/(4*PI()*Inputs!C4*Inputs!C5) | 6.7490597861e-27 | m³·kg⁻¹·s⁻² |
| hpred | Planck’s constant (from Swarm geometry) | h = N · ΔI · λ · c | =2*PI()*PI()*Derivations!E3*Inputs!C2*Inputs!C2*Derivations!E2 | 6.6260701500e-34 | J·s |
| ħ = h / (2π) | Reduced Planck’s constant | ħ = h / (2π) | =Derivations!E10/(2*PI()) | 1.0545718176e-34 | J·s |
Derived
| Quantity | Name | Excel formula | Swarm Field Theory Predicted Value | Units | Reference Value | Absolute % Delta vs CODATA | % Delta vs CODATA |
| k_e | Coulomb’s constant | =1/(4*PI()*Derivations!E7) | 8.9878750790e+09 | N·m²·C⁻² | 8.9875517923e+09 | 3.59705057138e-05 | 323286.783096 |
| Z0 | Characteristic impedance of free space | =Derivations!E8*Constants!D4 | 376.743864847 | Ω | 376.730313661 | 3.59705216732e-05 | 0.0135511859125 |
| σ | Stefan–Boltzmann constant | =(PI()*PI()*POWER(Constants!D6,4))/(60*POWER(Constants!D3,3)*POWER(Constants!D4,2)) | 5.6703744192e-08 | W·m⁻²·K⁻⁴ | 5.6703744192e-08 | 0 | 0 |
| t_P | Planck time | =SQRT(Constants!D3*Gravity_G_2026!C10/POWER(Constants!D4,5)) | 5.3912464483e-44 | s | 5.3912464483e-44 | 0 | 0 |
| l_P | Planck length | =Constants!D4*D5 | 1.6162550244e-35 | m | 1.6162550244e-35 | 0 | 0 |
| m_P | Planck mass | =SQRT(Constants!D3*Constants!D4/Gravity_G_2026!C10) | 2.1764343427e-08 | kg | 2.1764343427e-08 | 0 | 0 |
| E_P | Planck energy | =D7*Constants!D4*Constants!D4 | 1.9560816367e+09 | J | 1.9560816367e+09 | 0 | 0 |
| q_P | Planck charge | =SQRT(4*PI()*Derivations!E7*Constants!D3*Constants!D4) | 1.8755123065e-18 | C | 1.8755460378e-18 | 1.79847676673e-05 | -3.3731259739e-23 |
| a0 | Bohr radius | =Constants!D3/(Derivations!E6*Constants!D11*Constants!D4) | 5.2915817682e-11 | m | 5.2917721090e-11 | 3.59692088728e-05 | -1.9034085630e-15 |
| r_e | Classical electron radius | =Derivations!E6*Constants!D3/(Constants!D11*Constants!D4) | 2.8180416889e-15 | m | 2.8179403262e-15 | 3.59705027031e-05 | 1.0136273012e-19 |
| λ_C (e) | Compton wavelength of electron | =Derivations!E10/(Constants!D11*Constants!D4) | 2.4263102387e-12 | m | 2.4263102387e-12 | 1.6646543259e-16 | 4.0389678347e-28 |
| λ̄_C (e) | Reduced Compton wavelength of electron | =Constants!D3/(Constants!D11*Constants!D4) | 3.8615926796e-13 | m | 3.8615926796e-13 | 0 | 0 |
| R_∞ | Rydberg constant | =(Derivations!E6*Derivations!E6*Constants!D11*Constants!D4)/(2*Derivations!E10) | 1.0974521044e+07 | m⁻¹ | 1.0973731568e+07 | 7.19422992831e-05 | 789.475480728 |
| E_h | Hartree energy | =Derivations!E6*Derivations!E6*Constants!D11*Constants!D4*Constants!D4 | 4.3600583723e-18 | J | 4.3597447222e-18 | 7.19422992833e-05 | 3.1365005960e-22 |
| μ_B | Bohr magneton | =Constants!D5*Constants!D3/(2*Constants!D11) | 9.2740100784e-24 | J·T⁻¹ | 9.2740100784e-24 | 0 | 0 |
| μ_N | Nuclear magneton | =Constants!D5*Constants!D3/(2*Constants!D12) | 5.0507837461e-27 | J·T⁻¹ | 5.0507837461e-27 | 0 | 0 |
| λ_C (p) | Compton wavelength of proton | =Derivations!E10/(Constants!D12*Constants!D4) | 1.3214098554e-15 | m | 1.3214098554e-15 | 1.4924606889e-16 | 1.9721522631e-31 |
| λ̄_C (p) | Reduced Compton wavelength of proton | =Constants!D3/(Constants!D12*Constants!D4) | 2.1030891034e-16 | m | 2.1030891034e-16 | 0 | 0 |
| μ_red (e–p) | Reduced mass of electron–proton system | =(Constants!D11*Constants!D12)/(Constants!D11+Constants!D12) | 9.1044252765e-31 | kg | 9.1044252765e-31 | 0 | 0 |
| a0_H | Bohr radius for hydrogen (reduced-mass corrected) | =Constants!D3/(Derivations!E6*D20*Constants!D4) | 5.2944636537e-11 | m | 5.2946540982e-11 | 3.59692088728e-05 | -1.9044451917e-15 |
| R_H | Rydberg constant for hydrogen (reduced-mass corrected) | =(Derivations!E6*Derivations!E6*D20*Constants!D4)/(2*Derivations!E10) | 1.0968547386e+07 | m⁻¹ | 1.0967758340e+07 | 7.19422992831e-05 | 789.04575298 |
Outputs Core
| Parameter | Parameter Name | Swarm Field Theory Predicted Value | Units | Reference Value | Absolute % Delta vs CODATA | % Delta vs CODATA |
| τ | Coherence time constant | 2.5219600421e-18 | s | |||
| ΔI | Coherence membrane tension | 19.7952222563 | J·m⁻² | |||
| ξ_base | Base geometric suppression factor | 0.0757055514033 | dimensionless | |||
| ξ_geom | Geometric suppression factor (with packing Φ) | 0.1853 | dimensionless | |||
| α | Fine-structure constant (predicted) | 0.00729761505874 | dimensionless | 0.0072973525693 | 3.59705027032e-05 | 3.59705027032e-05 |
| ε0 | Vacuum permittivity (predicted) | 8.8538693346e-12 | F·m⁻¹ | 8.8541878128e-12 | 3.59692118831e-05 | 3.59692118831e-05 |
| μ0 | Vacuum permeability (predicted) | 1.2566822640e-06 | H·m⁻¹ | 1.2566370621e-06 | 3.59705216731e-05 | 3.59705216731e-05 |
| Θ_geom_base | Base geometric closure term | 6.7490597861e-27 | dimensionless | |||
| h_pred | Planck’s constant (Swarm-derived) | 6.6260701500e-34 | J·s | 6.6260701500e-34 | 1.2907872809e-16 | 1.2907872809e-16 |
| ħ = h / (2π) | Reduced Planck’s constant | 1.0545718176e-34 | J·s | 1.0545718176e-34 | 0 | 0 |
Outputs Extended
| Parameter | Parameter Name | Swarm Field Theory Predicted Value | Reference Value | Units | Absolute % Delta vs CODATA | % Delta vs CODATA |
| ke | Coulomb’s constant | 8.9878750790e+09 | 8.9875517923e+09 | N·m²·C⁻² | 3.59705057138e-05 | 323286.783096 |
| Z0 | Characteristic impedance of free space | 376.743864847 | 376.730313661 | Ω | 3.59705216732e-05 | -0.0135511859125 |
| σ | Stefan–Boltzmann constant | 5.6703744192e-08 | 5.6703744192e-08 | W·m⁻²·K⁻⁴ | 0 | 0 |
| tP | Planck time | 5.3912464483e-44 | 5.3912464483e-44 | s | 0 | 0 |
| lP | Planck length | 1.6162550244e-35 | 1.6162550244e-35 | m | 0 | 0 |
| mP | Planck mass | 2.1764343427e-08 | 2.1764343427e-08 | kg | 0 | 0 |
| EP | Planck energy | 1.9560816367e+09 | 1.9560816367e+09 | J | 0 | 0 |
| qP | Planck charge | 1.8755123065e-18 | 1.8755460378e-18 | C | 1.79847676673e-05 | 3.3731259739e-23 |
| a0 | Bohr radius | 5.2915817682e-11 | 5.2917721090e-11 | m | 3.59692088728e-05 | 1.9034085630e-15 |
| re | Classical electron radius | 2.8180416889e-15 | 2.8179403262e-15 | m | 3.59705027031e-05 | -1.0136273012e-19 |
| λC (e) | Compton wavelength of electron | 2.4263102387e-12 | 2.4263102387e-12 | m | 1.6646543259e-16 | 0 |
| λ̄C (e) | Reduced Compton wavelength of electron | 3.8615926796e-13 | 3.8615926796e-13 | m | 0 | 0 |
| R∞ | Rydberg constant | 1.0974521044e+07 | 1.0973731568e+07 | m⁻¹ | 7.19422992831e-05 | -789.475480728 |
| Eh | Hartree energy | 4.3600583723e-18 | 4.3597447222e-18 | J | 7.19422992833e-05 | -3.1365005960e-22 |
| μB | Bohr magneton | 9.2740100784e-24 | 9.2740100784e-24 | J·T⁻¹ | 0 | 0 |
| μN | Nuclear magneton | 5.0507837461e-27 | 5.0507837461e-27 | J·T⁻¹ | 0 | 0 |
| λC (p) | Compton wavelength of proton | 1.3214098554e-15 | 1.3214098554e-15 | m | 1.4924606889e-16 | 0 |
| λ̄C (p) | Reduced Compton wavelength of proton | 2.1030891034e-16 | 2.1030891034e-16 | m | 0 | 0 |
| μred (e–p) | Reduced mass of electron–proton system | 9.1044252765e-31 | 9.1044252765e-31 | kg | 0 | 0 |
| a0H | Bohr radius for hydrogen (reduced-mass corrected) | 5.2944636537e-11 | 5.2946540982e-11 | m | 3.59692088728e-05 | 1.9044451917e-15 |
| RH | Rydberg constant for hydrogen (reduced-mass corrected) | 1.0968547386e+07 | 1.0967758340e+07 | m⁻¹ | 7.19422992831e-05 | -789.04575298 |
Comparison to CODATA
| Comparison to CODATA / Reference Values | |||||||
| Category | Quantity | Predicted Value | Reference Value | Units | Absolute % Delta vs CODATA | % Delta vs CODATA | Comment |
| Core | Fine-structure constant | 0.00729761505874 | 0.0072973525693 | dimensionless | 3.59705027032e-05 | -2.6248944032e-07 | From core validation table |
| Core | Vacuum permittivity | 8.8538693346e-12 | 8.8541878128e-12 | F m^-1 | 3.59692118831e-05 | 3.1847815749e-16 | From core validation table |
| Core | Vacuum permeability | 1.2566822640e-06 | 1.2566370621e-06 | H m^-1 | 3.59705216731e-05 | -4.5201890678e-11 | From core validation table |
| Core | Planck constant | 6.6260701500e-34 | 6.6260701500e-34 | J s | 1.2907872809e-16 | 0 | Direct Swarm prediction |
| Core | Gravitational constant | 6.6743000000e-11 | 6.6743000000e-11 | m^3 kg^-1 s^-2 | 0 | 0 | Updated 2026 gravity derivation |
| Extended | Coulomb constant | 8.9878750790e+09 | 8.9875517923e+09 | N m^2 C^-2 | 3.59705057138e-05 | -323286.783096 | Extended validation |
| Extended | Vacuum impedance | 376.743864847 | 376.730313661 | ohm | 3.59705216732e-05 | -0.0135511859125 | Extended validation |
| Extended | Stefan-Boltzmann constant | 5.6703744192e-08 | 5.6703744192e-08 | W m^-2 K^-4 | 0 | 0 | Extended validation |
| Extended | Planck time | 5.3912464483e-44 | 5.3912464483e-44 | s | 0 | 0 | Extended validation |
| Extended | Planck length | 1.6162550244e-35 | 1.6162550244e-35 | m | 0 | 0 | Extended validation |
| Extended | Planck mass | 2.1764343427e-08 | 2.1764343427e-08 | kg | 0 | 0 | Extended validation |
| Extended | Planck energy | 1.9560816367e+09 | 1.9560816367e+09 | J | 0 | 0 | Extended validation |
| Extended | Planck charge | 1.8755123065e-18 | 1.8755460378e-18 | C | 1.79847676673e-05 | 3.3731259739e-23 | Extended validation |
| Extended | Bohr radius | 5.2915817682e-11 | 5.2917721090e-11 | m | 3.59692088728e-05 | 1.9034085630e-15 | Extended validation |
| Delta is computed as (Predicted - Reference) / Reference. Green indicates closer agreement. |
Predictions External
| Schwarzschild Radius: r_s = 2 G M / c^2 | |||||||
| Body | M kg | Predicted r_s (G_model) m | Reference r_s (G_CODATA) m | Absolute % Delta vs CODATA | |||
| Earth | 5.9722000000e+24 | 0.00887010287185 | 0.00887010287185 | 0 | |||
| Sun | 1.9884700000e+30 | 2953.33938207 | 2953.33938207 | 0 | |||
| Surface Gravity: g = G M / R^2 | |||||||
| Body | M kg | R m | g_pred (G_model) m/s^2 | g_ref (G_CODATA) m/s^2 | g_obs m/s^2 | Absolute % Delta vs CODATA | |
| Earth | 5.9722000000e+24 | 6371000 | 9.82030229339 | 9.82030229339 | 9.80665 | 0 | |
| Sun | 1.9884700000e+30 | 6.9634000000e+08 | 273.704589983 | 273.704589983 | 274 | 0 | |
| Gravitational Wave Speed vs Light Speed | |||||||
| Quantity | Value m/s | Reference m/s | Absolute % Delta vs CODATA | ||||
| c_GW (model) | 2.9979245800e+08 | 2.9979245800e+08 | 0 | ||||
| Orbital Relations (Kepler & Circular Motion) | |||||||
| System | Primary mass M kg | Secondary mass m kg | a or r m | Predicted (G_model) | Reference (G_CODATA) | Observed | Absolute % Delta vs CODATA |
| Earth ↺ Sun — Kepler period T s | 1.9884700000e+30 | 5.9722000000e+24 | 1.4959787070e+11 | 3.1557671483e+07 | 3.1557671483e+07 | 3.1558149764e+07 | 0 |
| Moon ↺ Earth — Kepler period T s | 5.9722000000e+24 | 7.3476730900e+22 | 3.8440000000e+08 | 2357380.10715 | 2357380.10715 | 2360591.5104 | 0 |
| LEO @ 400 km — circular speed v m/s | 5.9722000000e+24 | 0 | 6771000 | 7672.6188866 | 7672.6188866 | 7670 | 0 |
| Earth surface — escape speed v_esc m/s | 5.9722000000e+24 | 0 | 6371000 | 11186.1651973 | 11186.1651973 | 11186 | 0 |
Gravity G 2026
| Parameter | Formula / Description | Value | Units | Note |
| lambda | Inputs!C2 | 8.2000000000e-10 | m | Coherence aperture |
| tau_G | Inputs!C7 | 9.8400000000e-16 | s | Gravity coherence cycle time from 2026 paper |
| hbar | Constants!D3 | 1.0545718176e-34 | J s | Reduced Planck constant |
| c | Constants!D4 | 2.9979245800e+08 | m/s | Speed of light |
| K | =C5^2*C2^3/(C3*C4) | 4.7754214244e+38 | m^3 kg^-1 s^-2 | Structural prefactor: c^2*lambda^3/(tau_G*hbar) |
| G_reference | =Constants!D10 | 6.6743000000e-11 | m^3 kg^-1 s^-2 | CODATA reference |
| xi_tilde_G | =C7/C6 | 1.3976358120e-49 | dimensionless | Rescaled geometry factor |
| Xi_G | =4*PI()*C8 | 1.7563209597e-48 | dimensionless | Geometry factor |
| G_pred | =C8*C6 | 6.6743000000e-11 | m^3 kg^-1 s^-2 | Predicted gravitational constant |
| Relative error | =(C9-C7)/C7 | -1 | % | Prediction vs CODATA |
| Sensitivity coeff: lambda | =3 | 3 | coefficient | From paper: dG/G = 3 dlambda/lambda - dtau/tau |
| Note | Updated derivation based on G = xi_tilde_G * c^2 * lambda^3 / (tau_G * hbar) | 2026 gravity paper integration |
Gravity Update Notes
| Gravity Derivation Update (2026) |
| This workbook includes an updated derivation for the gravitational constant G |
| based on the paper 'Gravitational Constant from Coherence Geometry: A Swarm Theory Derivation'. |
| New relation used: |
| G = xi_tilde_G * c^2 * lambda^3 / (tau_G * hbar) |
| Key additions: |
| - New input parameter tau_G (gravitational coherence time scale) |
| - Dedicated calculation sheet: Gravity_G_2026 |
| - Outputs and Planck-scale calculations now reference the updated G value |
| Legacy calculation routes are preserved for comparison. |
| Purpose: |
| Allow side-by-side validation between the earlier Route B closure |
| and the updated coherence-geometry gravity formulation. |