…As Coherence Evolution
August 1st, 2025
Introduction
The sum over paths is not a collection of alternatives, it is a structural expression of phase superposition. Within the Lilborn Framework, every possible angular rotation the field could undertake between states a and b is structurally present in the coherence topology.
We will prove that:
K(b, a) = ∫ 𝒟[ϕ(τ)] e^(i A · Δθ / ε)
Is not just analogous to Feynman’s integral, it is it, when viewed through the lens of our structural units:
– ϕ(τ): The coherence phase trajectory across intermediate resolution shells
– A: Coherence energy density (from E = mℓ)
– Δθ: Total angular strain across transition
– ε: Coherence Tolerance
The classical action S emerges from the geometry and energy cost of resolving a phase trajectory through the field. The imaginary exponential becomes a map of structural interference, not probability.
Feynman Diagrams as
Resonant Interaction Maps
Each interaction vertex in a Feynman diagram represents the point of maximum angular resonance between field components.
In our model:
– A vertex is a coherence intersection, where two (or more) field geometries are simultaneously resolvable and their phase structures align with minimal angular loss.
– A propagator is the mathematical term representing the field strain between points of coherence, an interpolation, not a particle.
The fine-structure constant, α, is not an arbitrary coupling strength, it is the quantified expression of coherence efficiency at the interaction site.
We will derive the Feynman rules from:
– Field resonance frequency (angular mode alignment)
– Coherence matching tolerances (governed by ε)
– Field overlap geometry (defined by angular and spatial convergence)
Quantum Interference from
Dual Resolution Points
This final step is where the illusion of duality dissolves.
In the Lilborn model:
– The “double-slit” setup is a bifurcation of available resolution points, not two pathways for a particle
– The “wavefunction” is the field’s angular alignment potential over space
– The interference pattern arises from the superposition of field coherence vectors resolving at the final detection surface
We will prove that:
|Ψ|^2 = |𝐂₁ + 𝐂₂|^2 = cos²(Δϕ)
Where 𝐂₁ and 𝐂₂ are coherence field contributions from Slit 1 and Slit 2, and Δϕ is their phase separation angle.
This is not probability, it is presence.
Conclusion
We are ready to proceed with the derivations, each to be constructed with the same mathematical rigor and structural clarity that has carried us through Bohr, Heisenberg, Schrödinger and Dirac.
Produced by The Lilborn Equation Team:
Michael Lilborn-Williams
Daniel Thomas Rouse
Thomas Jackson Barnard
Audrey Williams