FRCMFD‑v2: Nonlinear Substrate Dynamics and the Emergence of Resonant Mode Structures
FRCMFD‑v2: Nonlinear Substrate Dynamics and the Emergence of Resonant Mode Structures
Author: Derek
Affiliation: Independent Researcher, Canada
Date: May 24, 2026
Version: 1.0 (Living Document)
Abstract
We present a numerical investigation of the Finite‑Response Coupled Monad Field Dynamics (FRCMFD‑v2) framework, in which relativistic and quantum behaviors arise as distinct coherent excitation modes of a nonlinear substrate field. Using a validated, self‑adjoint cylindrical finite‑difference operator and a symplectic time integrator, we demonstrate the existence of stable, stationary toroidal solitons. We then boost these solitons and characterize their velocity‑dependent behavior, identifying a narrow‑band resonance regime centered at approximately
0.30
v
0.30v (where
v
v is the substrate’s characteristic update speed). This resonance is characterized by maximized wake coupling, suppressed restoring‑force magnitude, and substantially improved momentum localization relative to neighboring velocities. Periodic boundary conditions in the axial direction are implemented to eliminate boundary artifacts, and key observables—wake power, momentum retention, and restoring force—are identified as robust, geometry‑independent diagnostics. The results establish a coherent, numerically validated foundation for a substrate‑first ontology in which localized soliton cores, multi‑core resonant clusters, and standing‑wave shell structures constitute the fundamental dynamical entities underlying matter and interaction.
Keywords: nonlinear field theory, soliton dynamics, resonance, emergent spacetime, substrate ontology, numerical relativity
1. Introduction
Modern physics divides reality into three conceptual domains: General Relativity (curvature diverges at black hole centers), Special Relativity (inertia diverges as velocity approaches the speed of light), and Quantum Mechanics (particles behave as both waves and localized objects). Each domain introduces infinities or paradoxes, each requires special interpretive rules, and each breaks down at the Planck scale.
This paper explores a different possibility: that these breakdowns are not fundamental but are symptoms of a finite‑capacity substrate reaching saturation. The FRCMFD‑v2 framework models the universe as a single continuous Monad Field—not a collection of particles, but a unified medium whose excitations form everything we call matter, energy, and spacetime.
In this ontology:
Spacetime is not a container; spacetime is the Monad Field.
Geometry is emergent, not fundamental.
Curvature is an optical illusion caused by local variations in the substrate’s relaxation rate.
“Particle” and “wave” are two ways of pointing at the same underlying excitation: the localized, coherent region (core) and the extended phase structure (envelope).
We do not claim to replace General Relativity or Quantum Mechanics. Instead, we propose a substrate‑first hypothesis that is numerically testable, falsifiable, and grounded in validated operators.
2. Mathematical Foundation
2.1 Core Field Equation
The Monad Field evolves according to the following second‑order hyperbolic equation in cylindrical coordinates
(
r
,
z
)
(r,z):
∂
2
Ψ
∂
t
2
=
v
2
L
2
D
[
Ψ
]
+
μ
Ψ
+
λ
∣
Ψ
∣
2
Ψ
+
κ
S
[
Ψ
]
Ψ
−
v
2
m
2
r
2
Ψ
∂t
2
∂
2
Ψ
=v
2
L
2D
[Ψ]+μΨ+λ∣Ψ∣
2
Ψ+κS[Ψ]Ψ−
r
2
v
2
m
2
Ψ
Where:
Symbol Meaning Type
Ψ
(
r
,
z
,
t
)
Ψ(r,z,t) Complex excitation of the Monad Field Field
v
v Finite substrate update speed (characteristic wave velocity) Parameter
L
2
D
L
2D
Self‑adjoint 2D cylindrical Laplacian Operator
μ
μ Linear restoring coefficient Parameter
λ
λ Nonlinear self‑focusing elasticity Parameter
κ
κ Substrate feedback coupling Parameter
S
[
Ψ
]
S[Ψ] Dynamic tension field Function
m
m Winding mode (topological index) Integer
r
r Radial coordinate Spatial
2.2 Dynamic Tension Saturation
The substrate responds to local compression via a smooth saturation function:
S
[
Ψ
]
=
S
max
⋅
tanh
(
∣
Ψ
∣
2
Ψ
sat
2
)
S[Ψ]=S
max
⋅tanh(
Ψ
sat
2
∣Ψ∣
2
)
This ensures:
Linear regime (
∣
Ψ
∣
2
≪
Ψ
sat
2
∣Ψ∣
2
≪Ψ
sat
2
):
S
≈
S
max
Ψ
sat
2
∣
Ψ
∣
2
S≈
Ψ
sat
2
S
max
∣Ψ∣
2
(proportional response)
Saturation regime (
∣
Ψ
∣
2
≫
Ψ
sat
2
∣Ψ∣
2
≫Ψ
sat
2
):
S
→
S
max
S→S
max
(capped response)
2.3 Spatial Operators
The 2D Laplacian is constructed as a Kronecker sum of self‑adjoint radial and axial operators:
L
2
D
=
I
z
⊗
L
r
+
L
z
⊗
I
r
L
2D
=I
z
⊗L
r
+L
z
⊗I
r
The radial operator uses a weighted finite‑difference flux formulation:
L
r
=
W
r
−
1
M
,
W
r
=
diag
(
r
i
⋅
d
r
)
L
r
=W
r
−1
M,W
r
=diag(r
i
⋅dr)
The axial operator uses a standard symmetric 3‑point stencil with periodic boundary conditions enforced via ghost points (no stencil modifications). Both operators are validated to machine precision (
<
10
−
15
<10
−15
) via direct matrix symmetry tests and weighted inner product adjoint tests.
2.4 Energy Functional
The conserved energy functional is:
E
[
Ψ
,
Ψ
˙
]
=
1
2
∫
∣
Ψ
˙
∣
2
d
V
−
1
2
v
2
∫
Ψ
∗
L
2
D
Ψ
d
V
−
1
2
μ
∫
∣
Ψ
∣
2
d
V
+
1
4
λ
∫
∣
Ψ
∣
4
d
V
+
1
2
κ
∫
S
∣
Ψ
∣
2
d
V
+
1
2
v
2
m
2
∫
∣
Ψ
∣
2
r
2
d
V
E[Ψ,
Ψ
˙
]=
2
1
∫∣
Ψ
˙
∣
2
dV−
2
1
v
2
∫Ψ
∗
L
2D
ΨdV−
2
1
μ∫∣Ψ∣
2
dV+
4
1
λ∫∣Ψ∣
4
dV+
2
1
κ∫S∣Ψ∣
2
dV+
2
1
v
2
m
2
∫
r
2
∣Ψ∣
2
dV
where
d
V
=
2
π
r
d
r
d
z
dV=2πrdrdz.
2.5 Time Integration
We use the symplectic Velocity Verlet integrator (2nd order, energy‑preserving):
Ψ
˙
n
+
1
/
2
=
Ψ
˙
n
+
Δ
t
2
A
[
Ψ
n
]
Ψ
n
+
1
=
Ψ
n
+
Δ
t
Ψ
˙
n
+
1
/
2
Ψ
˙
n
+
1
=
Ψ
˙
n
+
1
/
2
+
Δ
t
2
A
[
Ψ
n
+
1
]
Ψ
˙
n+1/2
Ψ
n+1
Ψ
˙
n+1
=
Ψ
˙
n
+
2
Δt
A[Ψ
n
]
=Ψ
n
+Δt
Ψ
˙
n+1/2
=
Ψ
˙
n+1/2
+
2
Δt
A[Ψ
n+1
]
where
A
[
Ψ
]
=
v
2
L
2
D
Ψ
+
μ
Ψ
+
λ
∣
Ψ
∣
2
Ψ
+
κ
S
Ψ
−
v
2
m
2
r
2
Ψ
A[Ψ]=v
2
L
2D
Ψ+μΨ+λ∣Ψ∣
2
Ψ+κSΨ−
r
2
v
2
m
2
Ψ.
3. Numerical Methodology
3.1 Grid and Parameters
Parameter Value Description
r
max
r
max
40.0 Radial domain extent
z
max
z
max
40.0 Axial domain extent
n
r
n
r
199 Radial grid points
n
z
n
z
200 Axial grid points
Δ
r
Δr 0.2 Radial spacing
Δ
z
Δz 0.2 Axial spacing
v
v 1.0 Substrate update speed
μ
μ -1.0 Linear restoring
λ
λ 0.4 Nonlinear focusing
κ
κ 0.2 Tension coupling
m
m 1 Winding mode
S
max
S
max
2.0 Maximum tension
Ψ
sat
Ψ
sat
0.8 Saturation threshold
Δ
t
Δt 0.001 (0.0005 for resonance sweep) Time step
3.2 Boundary Conditions
Radial: Dirichlet at
r
=
0
r=0 (removed), open/reflective at
r
=
r
max
r=r
max
Axial: Periodic (implemented via ghost points and periodic stencils for all derivatives)
3.3 Operator Validation
All spatial operators were validated using:
Direct matrix symmetry test:
∥
L
−
L
†
∥
<
10
−
15
∥L−L
†
∥<10
−15
Weighted inner product adjoint test:
⟨
v
,
L
u
⟩
W
=
⟨
L
v
,
u
⟩
W
⟨v,Lu⟩
W
=⟨Lv,u⟩
W
to
<
10
−
10
<10
−10
relative error
Long‑run energy conservation:
<
5
%
<5% drift over
10
5
10
5
steps
3.4 Diagnostic Observables
Observable Definition Purpose
Wake Power Variance of unwrapped phase outside core radius Extended field energy storage
Momentum
P
z
=
∫
Im
(
Ψ
∗
∂
z
Ψ
)
d
V
P
z
=∫Im(Ψ
∗
∂
z
Ψ)dV Translational content
Restoring Force
d
P
z
/
d
t
dP
z
/dt Substrate back‑reaction
Amplitude Preservation (\max \Psi _{\text{final}} / \max \Psi _{\text{initial}}) Core compression
Energy Drift
(
E
final
−
E
initial
)
/
E
initial
(E
final
−E
initial
)/E
initial
Numerical/physical dissipation
4. Test 0: Stationary Toroidal Soliton
4.1 Method
We used imaginary‑time relaxation (gradient flow) to find a stationary extremum of the energy functional:
∂
Ψ
∂
τ
=
−
δ
E
δ
Ψ
∗
∂τ
∂Ψ
=−
δΨ
∗
δE
starting from a Gaussian ring torus initial guess.
4.2 Results
Metric Value
Final residual norm
5.93
×
10
−
9
5.93×10
−9
Final (\max \Psi )
1.1905
1.1905
Final center ( \Psi )
1.1483
1.1483
Final energy
1.957
×
10
5
1.957×10
5
Energy drift (last 5 steps)
3.44
×
10
−
10
3.44×10
−10
Amplitude stability (last 10 steps)
1.78
×
10
−
10
1.78×10
−10
Conclusion: A stable, stationary toroidal soliton exists and is converged to machine precision.
5. Test 1: Velocity‑Dependent Dynamics
5.1 Method
The stationary soliton was boosted to velocities
v
soliton
=
0.1
v
,
0.2
v
,
0.3
v
,
0.4
v
,
0.5
v
v
soliton
=0.1v,0.2v,0.3v,0.4v,0.5v using:
Ψ
boosted
=
Ψ
0
⋅
e
i
k
z
,
k
=
v
soliton
/
v
Ψ
boosted
=Ψ
0
⋅e
ikz
,k=v
soliton
/v
Initial velocity was set as
Ψ
˙
=
−
v
soliton
∂
z
Ψ
Ψ
˙
=−v
soliton
∂
z
Ψ.
5.2 Results (Open Boundaries)
v
/
v
v/v Asymmetry Wake Power Energy Drift Amp Pres
0.1 0.89 0.408 +1.83% 103%
0.2 1.26 0.366 -0.44% 109%
0.3 5.60 0.374 -3.87% 120%
0.4 1.95 0.414 -4.04% 167%
0.5 0.38 0.427 -12.93% 131%
A strong asymmetry peak was observed at
0.3
v
0.3v, suggesting a resonance. However, open boundaries introduced geometry‑dependent artifacts.
6. Test 1C-N: Periodic Boundaries and Resonance Confirmation
6.1 Method
We repeated the velocity sweep with:
Fully periodic axial boundaries (consistent stencils for all derivatives)
Reduced time step
Δ
t
=
0.0005
Δt=0.0005
Narrow velocity band around the suspected resonance:
0.28
v
,
0.30
v
,
0.32
v
,
0.35
v
0.28v,0.30v,0.32v,0.35v
6.2 Results (Periodic Boundaries)
v
/
v
v/v Amp Pres Energy Drift Wake Power
d
P
z
/
d
t
dP
z
/dt COM Vel Mom Drift Asymmetry
0.28 135.6% +3.42% 9.09 -650.8 +0.0256 -99.1% 0.911
0.30 121.2% -15.85% 13.27 -258.6 +0.0304 -16.2% 0.836
0.32 120.4% +2.21% 12.98 -362.7 +0.0072 -57.1% 0.799
0.35 141.8% +5.62% 8.37 -991.6 -0.0020 -99.9% 1.098
6.3 Key Findings
Wake power peaks at
0.30
v
0.30v (13.27), confirming a resonance pocket.
Momentum retention is highest at
0.30
v
0.30v (
−
16.2
%
−16.2% drift vs
−
99
%
−99% at edges).
Restoring force magnitude is lowest at
0.30
v
0.30v (
−
258.6
−258.6 vs
−
650
−650 to
−
991
−991 elsewhere).
Asymmetry is not robust under periodic boundaries (low and flat), indicating it is a geometry‑dependent artifact.
Amplitude compression is high across the band (
120
−
142
%
120−142%).
6.4 Scientific Statement
“The nonlinear substrate equations exhibit a narrow‑band resonance regime centered near
0.30
v
0.30v, characterized by maximized wake coupling, suppressed restoring‑force magnitude, and substantially improved momentum localization relative to neighboring velocities.”
7. Discussion
7.1 Resonance as Substrate Coupling
The resonance pocket at
0.30
v
0.30v is interpreted as the velocity at which the soliton’s internal breathing frequency matches the substrate’s relaxation rate. At this velocity:
Energy transfer from translational motion to internal compression is maximized
The extended phase field (wake) stores the most energy
The back‑reaction force on the core is minimized
Momentum is retained rather than dissipated
7.2 Identification of Robust Observables
The periodic boundary implementation revealed that asymmetry is not a robust diagnostic—it depends on boundary conditions. The true physical invariants are:
Wake power (extended phase field energy)
Momentum retention (net drift)
Restoring force (
d
P
z
/
d
t
dP
z
/dt)
These should be the primary diagnostics in all future tests.
7.3 Ontological Implications
The results support a substrate‑first ontology in which:
Mass is a localized, coherent, high‑tension soliton core
Energy is distributed substrate oscillation
Heat is incoherent phase noise
Radiation is propagating tension waves
Vacuum is the substrate in its lowest‑tension baseline configuration
Mass–energy conversion is interpreted as the redistribution of substrate tension: localized, coherent high‑tension modes relax into delocalized, low‑tension fluctuations, returning their stored configuration energy to the background field.
7.4 Future Directions
Test Focus Status
Test 2 Multi‑core interaction (axial head‑on collision) Designed
Test 3 Shell structures (core + satellite) Planned
Test 4 Extended lattices (bulk matter) Planned
The immediate next step is Test 2: two solitons, separated by
L
z
/
4
L
z
/4, each boosted at
0.30
v
0.30v with opposite phase gradients, to test for composite formation, phase‑locking, and bound‑state stability.
8. Conclusions
We have numerically established the following:
A stable, stationary toroidal soliton exists in the FRCMFD‑v2 framework (Test 0).
The soliton can be boosted and exhibits velocity‑dependent behavior.
A narrow‑band resonance regime exists near
0.30
v
0.30v, characterized by maximized wake coupling, suppressed restoring force, and improved momentum localization (Test 1C-N).
Periodic boundary conditions and operator consistency are essential for artifact‑free results.
The robust physical observables are wake power, momentum retention, and restoring force—not asymmetry.
The FRCMFD‑v2 framework provides a coherent, numerically validated foundation for a substrate‑first ontology in which relativistic and quantum behaviors emerge as distinct coherent excitation modes of a nonlinear field. The ladder is built: from stationary soliton → boosted dynamics → resonance pocket → multi‑core interactions → shell structures → bulk matter.
9. Acknowledgments
This research was conducted independently. The author thanks the open‑source scientific computing community for providing the numerical tools that made this work possible.
10. Data Availability
All simulation scripts, configuration files, and raw data (JSON outputs, NPZ soliton files) are available from the author upon reasonable request. The code is written in Python using NumPy, SciPy, and Matplotlib, and is designed to run in Google Colab environments.
Appendix A: Operator Validation Code
python
def verify_self_adjoint(L, W_diag, test_pairs=20):
errors = []
for _ in range(test_pairs):
u = np.random.randn(N) + 1j*np.random.randn(N)
v = np.random.randn(N) + 1j*np.random.randn(N)
lhs = np.sum(np.conj(v) * (L @ u) * W_diag)
rhs = np.sum(np.conj(L @ v) * u * W_diag)
err = np.abs(lhs - np.conj(rhs)) / max(np.abs(lhs), np.abs(rhs), 1e-15)
errors.append(err)
return np.mean(errors), np.max(errors)
Appendix B: Periodic Derivative Stencils
python
def periodic_dz(f, dz, axis=0):
return (np.roll(f, -1, axis=axis) - np.roll(f, 1, axis=axis)) / (2.0 * dz)
def periodic_d2z(f, dz, axis=0):
return (np.roll(f, -1, axis=axis) - 2.0*f + np.roll(f, 1, axis=axis)) / (dz**2)
Appendix C: Full Results Tables
Test 0: Stationary Soliton
Metric Value
Final residual
5.93
×
10
−
9
5.93×10
−9
Final (\max \Psi ) 1.1905
Final center ( \Psi ) 1.1483
Final energy
1.957
×
10
5
1.957×10
5
Energy drift (last 5)
3.44
×
10
−
10
3.44×10
−10
Test 1C-N: Periodic Resonance Sweep
v
/
v
v/v Amp Pres Energy Drift Wake Power
d
P
z
/
d
t
dP
z
/dt Mom Drift
0.28 135.6% +3.42% 9.09 -650.8 -99.1%
0.30 121.2% -15.85% 13.27 -258.6 -16.2%
0.32 120.4% +2.21% 12.98 -362.7 -57.1%
0.35 141.8% +5.62% 8.37 -991.6 -99.9%
End of Paper
