02
◇ part I · foundations

Zero-locus defects of the complex order parameter ψ(x,w,t) are topologically protected

The parent matter field is introduced as a complex scalar ψ:R3×Rw×Rt→C\psi: \mathbb{R}^3 \times \mathbb{R}_w \times \mathbb{R}_t \to \mathbb{C} of finite L2L^2 norm away from background. Zeros of ψ\psi — where both real and imaginary parts vanish — form codimension-2 submanifolds in the spatial slice and carry integer winding that is stable under smooth deformation.

Winding invariance3D slice homotopysector: matter4D · Action & ProjectionsPrior · throat ontology
◇ matter sector · order parameter

ψ is a complex scalar; ρ = |ψ|² is the hydrodynamic density

Following the Madelung decomposition ψ=ρ eiθ\psi = \sqrt{\rho}\, e^{i\theta}, the matter sector resolves into a density ρ=∣ψ∣2\rho = |\psi|^2 and a phase θ\theta whose gradient defines a velocity potential. Regions with ρ>0\rho > 0 are ordinary fluid. Points at which ρ=0\rho = 0 are singular for the phase — there is no phase without a density — and it is precisely at such points that nontrivial topology can be hosted.
◇ matter decomposition · Part I Eq. 2.1
θ\theta is multi-valued in the presence of defects; it lives on the universal cover of the punctured domain.
ψ(x,w,t)  =  ρ(x,w,t)  eiθ(x,w,t),ρ  ≥  0, θ∈R/2πZ\begin{aligned} \psi(x,w,t) &\;=\; \sqrt{\rho(x,w,t)}\; e^{i\theta(x,w,t)}, \\ \rho &\;\geq\; 0,\ \theta \in \mathbb{R}/2\pi\mathbb{Z} \end{aligned}
◇ winding invariant · π₁(S¹)

The winding integer is a homotopy invariant

Let γ\gamma be a small closed loop in a 2D slice transverse to the defect, encircling exactly one zero of ψ\psi and lying entirely in the smooth region ρ>0\rho > 0. The winding number
◇ winding invariant · Part I Eq. 2.2
Integer-valued by virtue of π1(S1)=Z\pi_1(S^1) = \mathbb{Z}. Independent of the representative loop within the defect's homotopy class.
n(γ)  =  12π∮γdθ  ∈  Zn(\gamma) \;=\; \frac{1}{2\pi}\oint_\gamma d\theta \;\in\; \mathbb{Z}
Because n(γ)n(\gamma) takes discrete values, it is a conserved winding of the smooth sector: any continuous deformation of ψ\psi that keeps γ\gamma inside ρ>0\rho > 0 preserves nn. The only way to change nn locally is for a second defect of opposite winding to be pulled through the loop — a non-smooth event in the reduced sense.
In the three-dimensional brane slice Σ3={w=0}\Sigma_3 = \{w=0\}, the zero set of ψ\psi is generically a one-dimensional submanifold — a curve, or a family of curves — whose transverse winding profile defines the local defect structure. Closed curves produce ring-type defects; curves ending on the bulk define the throat-type defects handled in topic 04.
◇ stability

Stability is kinematic, not dynamical

A crucial feature: the persistence of an isolated winding defect is a kinematic statement about the smooth sector. No potential barrier, no binding energy, and no exotic force is needed for the winding integer to be invariant under continuous deformations that keep the loop in ρ>0\rho > 0. Under those regularity assumptions, an isolated winding-1 defect cannot be erased by a local smooth motion.
Annihilation of defect pairs, by contrast, is allowed: a winding-1 defect and a winding-(−1) defect can meet and vanish, with total winding conserved. This gives one allowed channel by which bound defect structures — relevant to the atom sector in topic 09 — can decay, and by which the brane can absorb configurations into smooth field radiation.
Allowed
n ↔ n + 0 (smooth motion); n + (−n) → 0 (pair annihilation); curve reconnections preserving total winding.
Forbidden
n → n + 1 or n → 0 by a single smooth deformation; creation of a single isolated defect from the vacuum.
◇ energetics · controlled reduction

Core energetics are a controlled scaling target

The current source record supports order-of-magnitude energetics for controlled defect and throat reductions, not a unique generic core profile theorem. In the prior brane-bulk throat ontology, the cylindrical throat modes have the scaling
◇ throat-mode scaling · prior ontology
At fixed mode amplitude, energy scales with transverse and bulk wave numbers; at fixed mode normalization, lower wave-number modes are energetically preferred.
Emn  ∝  Amn2(kr2+kw2)a2L,Qmn  ∝  Amn2a2L\begin{aligned} \mathcal E_{mn} &\;\propto\; A_{mn}^{2}(k_r^2+k_w^2)a^2L, \\ \mathcal Q_{mn} &\;\propto\; A_{mn}^{2}a^2L \end{aligned}
The exponent n=5n = 5 in the EoS is inherited from the parent weak-field optical matching discussed in topic 01; it is not chosen to set a defect size. Profile uniqueness and the full moving-wall realization remain open issues rather than settled consequences of the stiff EoS alone.
Mode scalingProfile uniquenessconsumed by: topic 05 (inflow), topic 09 (atoms)
◇ forward reference

What this does not yet include

Two essential refinements remain. First, defects acquire structure along a direction that is not part of the brane's 3D slice — the hidden ww axis of topic 03. Second, a distinguished class of defects — throats — has support on the brane but extends into the bulk with a non-trivial profile; these are the proper geometric setting for particle-like configurations, and are handled in topic 04.
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