Appendix M — Derivation 13: Modal Topology and Structure Vectors

Overview

In the Standard Model, particles are assigned internal quantum numbers—charge, spin, colour, flavour—each with associated symmetries and conservation laws.

In modal dynamics, no such labels are fundamental.
Instead, each mode’s identity arises from its internal phase topology and how it anchors into the coherence field.

This appendix defines modal structure vectors such as ${\vec{κ}}_{i}$ , and shows how they encode topological invariants that govern interaction behaviour, coherence class, and exclusion.

1. Coherence Function and Phase Surface

Each mode is described by:

ψ_{i} (x, t) = ρ (x, t) e^{i ϕ (x, t)}

But the observable features of a mode depend not on $ρ$ alone, but on the global winding and internal symmetry of $ϕ (x, t)$ .

These features are captured by a vector of topological descriptors:

{\vec{κ}}_{i} = (w, σ, χ, ξ, λ, \dots)

Where each component describes a specific structural invariant:

2. Components of the Structure Vector

$w$ : Winding number — Total $2 π$ phase rotations over the mode’s closed surface (related to electric charge)
$σ$ : Spinor symmetry — Indicates whether the phase surface is symmetric (bosonic) or antisymmetric (fermionic)
$χ$ : Coherence class — Indicates whether modes can co-anchor (bosonic) or exclude (fermionic) based on saturation (see Appendix H)
$ξ$ : Topological geometry — Classifies internal alignment patterns (e.g. helical, toroidal, spherical), encoding transformation behaviour under symmetry operations
$λ$ : Chirality index — Indicates internal handedness (e.g. left- or right-aligned phase progression), affecting anchoring response

Each mode’s identity is fully defined by ${\vec{κ}}_{i}$ .
There is no need to assign particle types or quantum numbers—these emerge from $ϕ (x)$ ’s structure.

3. Structural Equivalence and Transformation

Two modes are considered structurally equivalent if their ${\vec{κ}}_{i}$ are related by an allowed transformation under anchoring symmetry.

These transformations define modal analogues of:

Gauge rotations (U(1), SU(2), SU(3))
Lorentz boosts
Chirality flips
Charge conjugation

A transformation is valid only if it leaves the total anchoring cost $C [ψ]$ invariant.

Interactions between modes arise from topological compatibility between their $\vec{κ}$ vectors.

We define an interaction operation:

{\vec{κ}}_{i} ⋆ {\vec{κ}}_{j} \to {\vec{κ}}_{i j}

This operation is associative but not necessarily commutative, and encodes:

Whether two modes can bind
Whether the result is stable (anchorable)
Whether the modes interfere destructively

This replaces coupling constants and interaction vertices with structural rules grounded in coherence.

5. Structural Origin of Conservation Laws

Conservation arises from anchoring invariance under evolution:

Winding conservation $\Rightarrow$ charge
Spin topology conservation $\Rightarrow$ angular momentum
Chirality conservation $\Rightarrow$ parity selection
$ξ$ conservation $\Rightarrow$ mode class (e.g. lepton/hadron distinction)

These are not imposed—they are consequences of maintaining structural coherence in a deforming field.

6. Degeneracy and Symmetry Breaking

Degenerate modes (e.g. neutrino flavours) have:

Identical anchoring costs
Different ${\vec{κ}}_{i}$ under symmetry operation

Symmetry breaking occurs when the anchoring landscape $B (x)$ favours one vector over another, causing spontaneous re-anchoring into a lower-cost configuration.

This replaces the Higgs mechanism with a field-dependent structural alignment process.

Conclusion

Modal identity is not labelled—it is embedded in the geometry of internal phase.
The structure vector ${\vec{κ}}_{i}$ captures all interaction-relevant features and determines whether modes attract, bind, exclude, or decay.

Appendix L | [Index](./Appendix Master) | Appendix N