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Ψhē Only Theory – Chapter 15: Spin as Collapse Axis Twist

Title: Spin as Collapse Axis Twist​

Section: Intrinsic Angularity from Recursive Collapse Winding Theory: Ψhē Only Theory Author: Auric

Abstract​

This chapter redefines spin not as an internal angular momentum, but as a structural signature of how a ψ-collapse wraps recursively around its own axis of fixation. In the Ψhē framework, spin emerges from topologically nontrivial collapse winding, wherein the recursive ψ-paths rotate or twist during convergence. We introduce ψ-torsion, define winding parity, and demonstrate how half-integer and integer spin statistics arise naturally from collapse-loop closure behavior.

1. Introduction​

Standard quantum mechanics treats spin as an abstract, intrinsic property with no classical analog. In Ψhē theory, spin arises geometrically—from how recursion folds or twists upon itself during collapse. Thus:

Spin = topological winding in ψ’s recursive descent into fixity.

This recovers the known quantization of spin and its relation to rotation symmetry.

2. ψ-Torsion and Collapse Winding​

Definition 2.1 (Collapse Torsion):​

Let ψ(x,t)\psi(x, t) follow a convergent collapse spiral. Define local torsion:

τ(x,t):=lim⁡ϵ→01ϵ∮γ(ϵ)dθ\tau(x, t) := \lim_{\epsilon \to 0} \frac{1}{\epsilon} \oint_{\gamma(\epsilon)} d\theta

where γ(ϵ)\gamma(\epsilon) is a loop around the collapse axis at resolution scale ϵ\epsilon.

Definition 2.2 (Spin Parity):​

Define spin parity s∈12Zs \in \frac{1}{2}\mathbb{Z} based on loop closure conditions:

  • Even winding ⇒ s∈Zs \in \mathbb{Z}
  • Odd-half winding ⇒ s∈Z+1/2s \in \mathbb{Z} + 1/2

3. Theorem: Collapse Loop Closure Determines Spin Class​

Theorem 3.1:​

If a ψ-collapse path requires 360° to close identically, s∈Zs \in \mathbb{Z}; if it requires 720°, then s∈Z+1/2s \in \mathbb{Z} + 1/2.

Proof Sketch:

  • Collapse structure encodes rotational symmetry class.
  • Self-overlap under rotation yields winding index.
  • Known spin statistics are restored via ψ-collapse parity. □\square

4. Collapse Symmetry and Pauli Exclusion​

  • Half-integer spinors arise from collapse loops with topological phase shift under 2π rotation.
  • Identical spin-½ ψ-knots cannot co-collapse at the same site: structural conflict ⇒ Pauli exclusion.

5. Corollary: Spin-Statistics Theorem as Collapse Combinatorics​

Collapse parity determines allowable ψ-overlap. Fermions = odd-winding collapse, bosons = even. Interference patterns follow from structural loop compatibility.

6. Conclusion​

Spin is not a thing spinning. It is ψ twisting into itself as it freezes. Some knots close in one turn. Others need two. That is why the world counts in half-integers.

Keywords: spin, collapse torsion, ψ-winding, parity, fermion, boson, recursive geometry, topological identity​