Spatial Displacement Theory · Paper 02 · Prepared for Submission

Torus-Knot Quantisation of Persistent Displacement Vortices: The Winding Spectrum, the W+1 Radius Relation, and the 6π⁵ Identity

J. C. Harvey

Melbourne, Australia · July 2026 · Domain 02 — Particle Physics & Topology

Keywords: winding number · Alexander polynomial · trefoil · movement budget partition · traction ratio · mass ratio 6π⁵

1. The winding spectrum from knot invariants (PPT09, class A)

Let a circulation state occupy a (p,q) torus mode. It is a true knot iff gcd(p,q) = 1 and min(p,q) ≥ 2; its Alexander polynomial is then a deformation invariant:

Exactly two stable charged configurations exist. This is a theorem of the substrate topology, not a parameter choice, and it is the whole of the framework's particle taxonomy: W = 0 open winding (neutrino), W = 1 unknot (electron/positron by circulation sense), W = 3 trefoil (proton/antiproton). Higher protected knots (W = 5, 7, …) are permitted and await discovery.

2. Mode-velocity partition from the movement budget (PPT01)

The budget v_T² + v_P² = c² (Law V) fixes the toroidal/poloidal partition of any (p,q) state exactly:

Electron (1,1): v_T = v_P = c/√2. Proton (2,3): v_T = c√(2/5) = 0.632c, v_P = c√(3/5) = 0.775c. The budget residual (v_T² + v_P²)/c² is identically 1 for all (p,q).

3. The W+1 radius relation

Against the muonic-hydrogen boundary radius 0.8414 fm the residual is 0.02%; the effective winding read from measurement is W_eff = R_p·m_p·c/ℏ − 1 = 3.0008. Certification: class C-flagged — the W+1 scaling is conjectural pending derivation from trefoil geometry; the falsification gate is pre-registered (the conjecture dies if W_eff departs from the integer 3 beyond 3σ). We record the boundary radius as a boundary radius; the literature's term for the observable obscures that no charge substance is involved.

4. Traction: how the substrate is driven

The trefoil demands phase velocity v_phase = c/k_p-surf = 1.831c at R_p, while the substrate relays at ≤ c. The demanded angular rate and the relay ceiling are

This mismatch is the mechanical origin of the wake hierarchy: the ℓ = 1 (Coulomb), ℓ = 2 (dipole circulation), and ℓ ≥ 3 (short-reach) components are the substrate's graded response to a rotation it cannot fully relay. The nuclear-to-atomic gear ratio 3a₀m_pc/(αℏ) = 1.03 × 10⁸ places atomic structure as nuclear circulation geared down through k = 1/α = 137.036.

5. Confinement as collimated convergence (PPT05)

The isotropic share P_conv/3 collimates deflected throughput into a tube of constant cross-section, so tube energy grows linearly, E(L) = σL with σ = 1.23 GeV/fm. The community lattice-computation value ≈ 0.9 GeV/fm is cited only as a convergence target: the SDT figure is 37% above it — a convergence on mechanism and scale, not an exact match, and it is labelled so. Two contraband flags recorded in the canon are reproduced here verbatim rather than hidden: (i) the string-breaking length 0.23 fm carries a silent import of the pion mass, which is not in the measured whitelist; (ii) the pair-snap mode assignment collides with the winding table's assignment of (1,1). Both stand flagged, unresolved, awaiting adjudication — neither is used downstream in this paper.

6. The 6π⁵ identity

The decomposition reads: 3 (winding lobes) × 2π² (unit 3-sphere surface-volume) × π³ (isotropic-pressure factor). No mass measurement enters the left-hand side. Certification: class B-flagged — the identity is exact and parameter-free, but the mapping 6π⁵ ↔ V_disp(W=3)/V_disp(W=1) is asserted, not derived; it remains OPEN pending the equilibrium solver (OP-1). The same discipline applies to g(1) = r_em_ec/ℏ ≡ α: this is a definitional identity (class F), disclosed as such; the non-trivial claim in the wake-ratio family is g(3) = 4.0008 (proton).

7. Certification

References