Appendix AH — Derivation 34: Composite Modes and Coherence Binding

In PBG, composite particles—such as baryons, mesons, atoms, and molecules—arise from bound modal structures whose coherence envelopes remain mutually anchored. These structures are not fundamental in themselves, but emerge from modal aggregates that collectively minimise anchoring cost while preserving phase structure.

1. Composite Anchoring Cost

The total coherence cost for a system of interacting modes is:

C_{total} = \sum_{i} \int ρ_{i}^{2} (α | \nabla ϕ_{i} |^{2} + β B^{2} [ψ_{i}]) d^{3} x + \sum_{i < j} C_{int} [ψ_{i}, ψ_{j}]

where:

( \rho_i ) is the coherence density of mode ( i )
( \phi_i ) is its internal phase
( B[\psi_i] ) is its emitted anchoring field
( \mathcal{C}_{\text{int}} ) is the overlap cost from mutual anchoring

The mutual coherence constraint requires that overlapping regions preserve total phase continuity. This introduces stabilising terms under coherence preservation and destabilising terms when overlap becomes incoherent.

2. Anchoring-Driven Binding

Stable composite systems minimise the total anchoring cost while preserving phase compatibility. The optimal separation and orientation of constituent modes arises from the balance between:

Self-coherence anchoring
Mutual coherence gain from shared anchoring
Phase preservation constraints

This explains:

Stable baryons (quarks in mutual anchoring structures)
Neutral atoms (electron phase structure compatible with nuclear envelope)
Molecules (anchoring overlaps minimising intermodal cost)

3. Saturation and Composite Integrity

Composite systems can saturate if the anchoring cost exceeds the benefit of coherence overlap. Saturation occurs when:

δ C_{int} > δ C_{self}

At this point, the system tends to split, collapse, or emit a subcomponent to restore a lower-energy coherence configuration.

4. Summary

Composite modes in PBG are:

Not fundamental particles
Coherence-bound modal structures
Stabilised by anchoring overlap and coherence compatibility

Their structure can be fully derived from anchoring cost minimisation without invoking separate interaction terms.

Appendix AG | [Index](./Appendix Master) | Appendix AI