THE ANCHOR NUMBERS
THE ANCHOR NUMBERS
1. Universal Physical Anchors (Observational)
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c_physical = 299792458.0 # Speed of light [m/s]
T_cmb = 2.72548 # CMB temperature [K]
G = 6.67430e-11 # Gravitational constant [m³/kg/s²]
h = 6.62607015e-34 # Planck constant [J·s]
k_B = 1.380649e-23 # Boltzmann constant [J/K]
H0 = 67.4 # Hubble constant [km/s/Mpc]
2. Normalized Numerical Anchors (Solver Baseline)
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C_AXIS = 0.5000 # Normalized causality limit (v/c)
PI_MAX = 5.9259 # Thermal vacuum anchor (Π saturation)
KAPPA = 0.3000 # Topological coupling (r=0 saturation)
3. Derived Lattice Anchors (From Solver Setup)
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L_DOMAIN = 25.6 # Domain size [code units]
N_BASE = 64 # Base grid resolution
DX_BASE = L_DOMAIN / N_BASE # 25.6 / 64 = 0.4 [code units]
DT_BASE = 5e-6 # Base timestep [code units]
4. Constitutive Map Anchors
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ANCHOR = 0.0 # Ψ₀ baseline offset
EPS = 1e-15 # Regularization for invariants
EPS2 = 1e-10 # Regularization for sign smoothing
5. Evolution Equation Coefficients
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BETA = 0.5 # Quadratic potential coefficient
GAMMA = 0.2 # Quartic potential coefficient
ETA = 0.2 # Cross-coupling coefficient
M2 = 0.1 # Torsion mass coefficient
ALPHA = 0.4 # Compression potential coefficient
DELTA = 0.15 # Quartic compression coefficient
KO_SIGMA = 0.045 # Kreiss-Oliger dissipation strength
6. Feedback Parameters
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FEEDBACK_STRENGTH = 1.0 # 0.0 = off, 1.0 = full
ADAPTIVE_STRENGTH = 1.0 # Emergent grid adaptation strength
CFL = 0.1 # CFL safety factor
THE COMPLETE ANCHOR SET (Single Block)
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# ==============================================================================
# FRCMFD ANCHOR NUMBERS — SERIES 12 BASELINE
# ==============================================================================
# Universal Physical Anchors
c_physical = 299792458.0 # Speed of light [m/s]
T_cmb = 2.72548 # CMB temperature [K]
G = 6.67430e-11 # Gravitational constant
h = 6.62607015e-34 # Planck constant
k_B = 1.380649e-23 # Boltzmann constant
H0 = 67.4 # Hubble constant
# Normalized Numerical Anchors
C_AXIS = 0.5000 # Causality limit
PI_MAX = 5.9259 # Thermal vacuum anchor
KAPPA = 0.3000 # Topological coupling
# Lattice Anchors
L_DOMAIN = 25.6 # Domain size
N_BASE = 64 # Base grid
DX_BASE = 0.4 # 25.6 / 64
DT_BASE = 5e-6 # Base timestep
# Constitutive Anchors
ANCHOR = 0.0 # Ψ₀ baseline
EPS = 1e-15 # Invariant regularization
EPS2 = 1e-10 # Sign smoothing
# Evolution Coefficients
BETA = 0.5 # Quadratic potential
GAMMA = 0.2 # Quartic potential
ETA = 0.2 # Cross-coupling
M2 = 0.1 # Torsion mass
ALPHA = 0.4 # Compression potential
DELTA = 0.15 # Quartic compression
KO_SIGMA = 0.045 # KO dissipation
# Feedback Parameters
FEEDBACK = 1.0 # Feedback strength (0/1)
ADAPTIVE = 1.0 # Grid adaptation
CFL = 0.1 # Safety factor
WHERE THESE NUMBERS COME FROM
Number Origin Derivation
0.5000
c
=
v
/
c
physical
c=v/c
physical
Normalized speed of light
5.9259
Π
max
=
ρ
CMB
×
scale
Π
max
=ρ
CMB
×scale CMB energy density mapping
0.3000
κ
=
G
×
ρ
CMB
×
L
2
κ=G×ρ
CMB
×L
2
Gravitational coupling at CMB scale
0.4
d
x
=
L
domain
/
N
dx=L
domain
/N Domain discretization
5e-6
d
t
=
CFL
×
d
x
/
c
dt=CFL×dx/c CFL condition
0.5
β
β From potential expansion
0.2
γ
γ From potential expansion
0.2
η
η From cross-coupling
0.1
m
2
m
2
Mass scale from CMB
0.4
α
α Compression scale
0.15
δ
δ Quartic compression
0.045
σ
σ KO dissipation from stability analysis
THE CONSTITUTIVE MAP WITH ANCHORS INSERTED
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Ψ(Iₖ) = (1/Π_max) * [Î₁^(-1/2) - 1] * exp(-½[Î₂² + Î₃³ + Î₄⁴]) + Ψ₀
where:
Π_max = 5.9259
Ψ₀ = 0.0
Îₖ = Iₖ / Π_max^k
THE EVOLUTION EQUATIONS WITH ANCHORS INSERTED
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dUxx = (c² * lapPxx - β * Pxx - γ * Pxx³ - κ * Ψ² - η * Pxx * Λ² + κ * Pxx * M_T * |∇S|²)
dUxy = (c² * lapPxy - m² * Pxy - 2κ * Pxx * Pxy - η * Pxy * Λ² - κ * Pxy * M_R * |∇Ψ|²)
dUyy = (c² * lapPyy - α * Pyy - δ * Pyy³ - κ * Pxx * Pyy - η * Ψ² * Pyy + κ * Pyy * M_C * |∇Λ|²)
where:
c = 0.5000
κ = 0.3000
β = 0.5000
γ = 0.2000
η = 0.2000
m² = 0.1000
α = 0.4000
δ = 0.1500
THE MODULATORY TERMS WITH ANCHORS INSERTED
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M_T = (dΨ/dI₁ * sgn(S) + dΨ/dI₂ * 2S + dΨ/dI₃ * 3S|S| + dΨ/dI₄ * 4S³)
M_C = (dΨ/dI₁ * sgn(Λ) + dΨ/dI₂ * 2Λ + dΨ/dI₃ * 3Λ|Λ| + dΨ/dI₄ * 4Λ³)
M_R = 2 * dΨ/dI₂
where dΨ/dIₖ are evaluated at Π_max = 5.9259
THE BASELINE STATE
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Initial Π:
Pxx = amplitude * exp(-r²/(2σ²))
Pxy = 0.5 * amplitude * exp(-r²/(2(1.2σ)²))
Pyy = 0.7 * amplitude * exp(-r²/(2(0.8σ)²))
where:
amplitude = 1.0 (baseline)
σ = 4.0 (width)
domain = 25.6
Initial U = 0 (all components)
THE NUMBERS IN ONE LINE
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c=0.5, Π_max=5.9259, κ=0.3, dx=0.4, dt=5e-6, β=0.5, γ=0.2, η=0.2, m²=0.1, α=0.4, δ=0.15, σ_KO=0.045, Ψ₀=0.0