Appendix AH — Derivation 34: Composite Modes and Coherence Binding

# Appendix AH — Derivation 34

Composite Modes and Coherence Binding

Composite “particles” in PBG are phase-locked aggregates of
individual modes.
They bind only when their mutual anchoring lowers the total
coherence cost.

0 Symbols

Symbol	Meaning	Units
$ρ_{i}$	envelope amplitude of mode i	dimensionless
$ϕ_{i}$	internal phase of mode i	rad
$A_{i} \equiv \nabla ϕ_{i}$	phase–gradient field	m⁻¹
$B [ψ_{i}]$	bias field generated by $ψ_{i}$	–
$α$	spatial anchoring stiffness	J m⁻¹
$β$	bias-energy coefficient	J m⁻³
$λ$	cross-gradient coupling	J m⁻¹

1 Total anchoring cost

For N overlapping modes

\begin{matrix} (AH-1) & C_{tot} = \sum_{i = 1}^{N} \int ρ_{i}^{2} [α | A_{i} |^{2} + β B^{2} [ψ_{i}]] d^{3} x + \sum_{i < j} \int ρ_{i} ρ_{j} [λ (A_{i} \cdot A_{j}) + β B [ψ_{i}] B [ψ_{j}]] d^{3} x \end{matrix}

Self terms (first sum) penalise each mode’s own curvature and
bias energy.
Cross terms (second sum) reward (or penalise) compatible phase
gradients and bias-field overlap.

If
$A_{i} \cdot A_{j} > 0$
and the bias fields align, the integral is negative → binding.

2 Binding / unbinding criterion

Define

Δ C_{i j} = C_{tot} [ψ_{i} + ψ_{j}] - (C_{self, i} + C_{self, j}) .

A composite forms when

\begin{matrix} (AH-2) & Δ C_{i j} < 0 \end{matrix}

and breaks when environmental changes (temperature, extra modes,
saturation) flip the sign.

3 Examples

System	Why (phase view)	Result
Baryon (3 quark-like modes)	Three 120° phase sheets give net $A = 0$ ; cross terms cancel curvature.	Stable, colour-neutral object.
Hydrogen atom	Electron winding matches nuclear bias; $A_{e} \cdot A_{p} < 0$ lowers cost.	Bound state at Bohr radius.
$H_{2}$ molecule	Two electrons share bias overlap between nuclei; minimises (AH-1).	Bond length emerges from cost minimum.

4 Saturation and breakup

When a composite enters a region with high external bias
or exceeds the local saturation threshold,
the second line in (AH-1) turns positive and
condition (AH-2) fails → the mode ejects a constituent or fissions into
lower-cost pieces.

5 Take-aways

Binding is not a separate force; it is the negative part of the
anchoring cost functional.
Quantitative properties (sizes, energies) follow from the same
constants $α, β, λ$ fixed in the Foundations.
Composite stability is environment-dependent and automatically
explains ionisation, dissociation, and nuclear fission thresholds.

Appendix AG - Emergent Charge | [Index](./Appendix Master) | Appendix AI - Modal Decay