From Single States to Interaction
Paper XII established single-state gradient flow: a lone configuration descends its closure functional, converging to an attractor basin that determines the particle identity. But physics requires more than isolated states — it requires interacting configurations.
The central idea of this paper: when multiple states are present, the closure landscape itself depends on the joint configuration. Each state no longer evolves independently — the presence of one deforms the basin structure experienced by the other.
The Total Closure Functional
For a multi-state configuration Ψ = (ψ1, ψ2, …, ψn), the total closure functional separates into individual and interaction terms:
Without the interaction term, the system is separable: each state evolves independently under its own closure pressure. Fint is what couples the evolution — it is the mathematical expression of interaction within the framework.
The Interaction Functional
The interaction functional decomposes into four structural components, each capturing a distinct aspect of how states influence each other:
- Proximity (Iprox) — how close the states are in state space. Interaction strength increases with proximity; distant configurations decouple.
- Alignment (Ialign) — torsional and phase compatibility between interacting states. Aligned phases can interfere constructively; misaligned phases suppress coupling.
- Compatibility (Icomp) — whether configurations are cooperative or incompatible. Some state pairs lower each other's closure cost; others raise it.
- Binding (Ibind) — gap penalty on the union of supports. Rewards connected composite configurations; penalises disconnected unions.
Coupled Gradient Flow
Each state evolves under its own closure pressure plus the mutual influence of the interaction term:
The projection Π keeps each state on the admissible manifold. The key structural result: the interaction term redirects gradient flow — states no longer descend toward their individual attractors, but toward joint configurations that minimise the total functional.
Monotonic descent still holds: dFtot/dt = −‖Π(∇Ftot)‖² ≤ 0. The interacting system is mathematically stable.
Interaction as Basin Deformation
This is the key interpretive insight of the paper. Without interaction, the multi-state basin structure is the product of individual basins — each state has its own attractor landscape, independent of the others.
With interaction, the basins deform. The presence of one state changes the landscape experienced by the other. Concretely:
- Basin positions shift — attractor locations move in state space
- Basins merge or split — previously distinct attractors can combine into one, or a single basin can fragment
- Barriers lower or raise — transitions between basins become easier or harder
- New joint attractors form — configurations that are not attractors for either state alone become attractors for the pair
This is what interaction means in the framework. There is no separate force carrier or mediating field — interaction is the mutual deformation of basin structure.
Binding as Joint Basin Formation
A bound configuration is a joint attractor whose total closure cost is lower than the sum of the separated parts:
The bound state is not held together by a force pulling components inward. It exists because the joint closure landscape has a basin that is deeper than any combination of individual basins.
Binding is not a force pulling states together. It is the formation of a joint attractor whose total closure cost is lower than the separated configuration.
Scattering-Like Processes
When two states approach in state space, their basins deform. Depending on the interaction structure, several outcomes are possible — classified by what happens to the basin assignments:
| Channel | Basin Dynamics | Outcome |
|---|---|---|
| Basin-preserving | Basins deform but do not merge or cross | States return to their original attractors after interaction |
| Basin-transition | Interaction lowers a barrier; state crosses into a different basin | States end in different attractors than they started in |
| Binding / fusion | Basins merge into a single joint attractor | Composite bound state forms |
| Repulsive separation | Basins push apart; interaction raises barriers | States are driven to greater separation in state space |
Worked Examples
(a) Disconnected quark binding
Two disconnected configurations — {2,4} and {3} — interact via the binding term Ibind. The gap penalty on {2,4} is high; the union {2,3,4} is connected. The interaction deforms the basin landscape so that the joint attractor is the connected baryon configuration on {2,3,4}, with lower total closure cost than the separated pair.
(b) Compatible aligned pair
Two states with compatible torsional phases and cooperative closure structure. The alignment and compatibility terms are favourable; a weakly bound composite forms. The joint attractor is slightly deeper than the separated configuration, producing a shallow binding energy.
(c) Incompatible pair
Two states with misaligned phases and incompatible closure structure. The compatibility term raises the joint closure cost. No joint attractor forms — the interaction is repulsive. The states are driven apart in state space.
Conservative Extension
The dissipative gradient flow admits a conservative (time-reversible) extension via a Lagrangian formulation:
In this formulation, scattering-like processes become reversible trajectories. States oscillate in the closure landscape rather than irreversibly descending. The dissipative and conservative formulations bracket the physical dynamics from two sides.
Stated Limitations
- No S-matrix or scattering amplitudes are computed
- No relativistic kinematics — the framework is non-relativistic at this stage
- No quantisation of the interaction — all dynamics are classical on state space
- The interaction functional is an ansatz, not derived from a gauge principle
- Coupling weights λ1–λ4 are not fixed by the framework
- No gauge-boson exchange is derived — interaction is basin deformation, not particle exchange
- Worked examples are schematic — illustrative, not quantitative predictions
From isolated attractors to a relational description — the framework now admits interaction.