ONTOLOGY: RING QUANTUM MODEL (XXII)——The Geometric Origin of Condensed Matter Phenomena – Superconducting Pairing, Topological States, and Superfluidity

Published September 9, 2026 | Version v1

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Building upon the conditions for condensed matter established in Paper XXI

(wavefunction overlap + thermal fluctuation suppression), this paper further derives

the geometric origins of superconducting pairing, topological states, and superfluid

helium.

The core findings are:

1. Superconducting pairing symmetries (s-, d-, p-wave) are geomet

ric projections of the relative orientations of ring vortex cones in the

lattice periodic potential. s-wave corresponds to parallel/random projection, d

wave to anti-parallel projection on a tetragonal lattice, and p-wave to 90◦ rotation

projection.

2. The superconducting gap ratio 2∆0/(kBTc) = 3.52/ cos θeff is a direct

reading of the ring geometric projection angle. The ∼ 10.56 of cuprate

high-temperature superconductors corresponds to the maximum projection angle

(cos θeff = 1/3), the ∼ 7 of iron-based superconductors corresponds to the inter

mediate projection angle (cos θeff ≈ 0.5), and the 3.52 of BCS superconductors

corresponds to the isotropic limit (cos θeff = 1).

3. Topological states: the θ = π state (axion insulator) is the necessary

consequence of the ring’s topological charge Q = 1/2 being truncated at

material boundaries. The phase transition critical exponent ν = 1 comes from

ν = 1/(2Q), and the half-quantized Hall conductance σxy = e 2/(2h) of the axion

insulator is an exclusive test.

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