Appendix AI — Derivation 35: Modal Decay and Anchoring Instability

In PBG, modal structures persist only while they can maintain anchoring to their coherence field. When saturation occurs, or coherence is no longer sustainable, modes undergo structural collapse—observed as decay.

1. Saturation and Instability

Each anchored mode has a maximum coherence capacity defined by its saturation threshold ( \sigma_0 ). As the internal structure intensifies (e.g. from binding, energy input, or environmental tension), the required anchoring cost grows.

Decay occurs when:

This imbalance breaks the coherence envelope, allowing the mode to fragment into lower-cost structures.

2. Anchoring Collapse Mechanism

As anchoring stress increases:

• Phase structure becomes unstable
• Coherence envelope cannot accommodate internal phase topology
• The mode splits into substructures that are easier to anchor individually

This appears as:

• Particle decay
• Emission of a photon or neutrino
• Fission of a bound composite

3. Decay Products from Anchoring Pathways

The allowed decay paths correspond to coherence-conserving transformations. For example:

• A muon decays into an electron + neutrinos because the resulting modes have compatible coherence with the lost structure
• A neutron decays into a proton + electron + antineutrino via anchoring redistribution and phase rebalance

The bias landscape determines which decay channels are allowed, not a fixed force law.

4. Anchoring Degeneracy and Decay Width

If a structure has multiple nearby coherence configurations with similar cost, it exhibits instability:

• The system ‘wanders’ between configurations
• Coherence fluctuations increase
• Eventually, a transition occurs

This explains natural decay widths and modal half-lives without invoking uncertainty principles.

5. Summary

• Modal decay in PBG is a structural collapse
• Caused by exceeded anchoring cost
• Decay products emerge from coherence-preserving transitions
• No force fields, just anchoring instability

Appendix AH | [Index](./Appendix Master) | Appendices/Appendix AJ