The Sign-Dual Skeleton of Matter–Antimatter Pairing

Mike Lewis — 19 Jan 2026 · Revision with curvature/spin correspondence and clarified caveats

Abstract

Matter/antimatter pairing exhibits a single binary (±) tag that surfaces simultaneously as curvature sign, Lagrangian coupling sign, Maxwell source sign, and spin projection. We trace that tag through six stacked layers: (1) conservation geometry (antipodal momenta), (2) projective seam with elliptic–hyperbolic split, (3) a universal mixed-term curvature correction of ±1⁄3, (4) sign-flipped electrostatic Lagrangian, (5) charge-conjugation invariance of Maxwell’s equations, (6) spin–magnetic coupling that reveals charge via Zeeman splitting. Curvature duality is strictly an auxiliary-metric analogy in coordinate space, while the physical duality resides in the orientation of the U(1) gauge fibre. The ± 1⁄3 coefficient and ± ½ ħ spin projection are shown to be complementary eigen-spectra of the same ℤ₂ centre.

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1. Kinematic backbone: antipodes on S²

Pair production in the centre-of-mass frame enforces p⃗+ = −p⃗−; dividing by magnitude maps each momentum to antipodal points on the unit sphere. This axis (great circle) is the event’s geometric spine.

p⃗_- + p⃗₊ = 0   ⇒   n̂₊ = − n̂_-,   n̂ ≡ p⃗ / |p⃗|.

2. Projective seam and curvature duality

Using homogeneous coordinates [x:y:z], fix the absolute conic x² + y² − z² = 0. A projective line intersecting this conic divides the plane into an interior hyperbolic domain (K = −1/R²) and an exterior elliptic domain (K = +1/R²). The line itself is the shared geodesic seam.

Interior: x² + y² < z² → K = −1/R²
Exterior: x² + y² > z² → K = +1/R²

Disclaimer A. This curvature split lives in an auxiliary metric space; electrons and positrons inhabit the same flat Minkowski spacetime.

3. Universal mixed-term correction (± 1⁄3)

For a right geodesic triangle with legs a,b ≪ R and hypotenuse c:

Spherical: cos(c/R) = cos(a/R) cos(b/R)
Hyperbolic: cosh(c/R) = cosh(a/R) cosh(b/R)

Series matching through O(a²b²) gives

c² = a² + b² − (K/3) a²b² + O((a/R)⁶),   K = ±1/R².

Thus the coefficient is precisely ±1⁄3 with no intermediate values. The sign flips when the geodesic crosses the seam — an Ising-like jump.

4. Classical dynamics: sign-flipped electrostatics

Non-relativistic Lagrangian in an external potential φ(r):

L = ½ m v² − q φ(r),   F = −q ∇φ.

Charge conjugation q → −q reverses the force. In a conductor-bounded setup:

q = −e	attracted towards ground
q = +e	repelled toward infinity

Disclaimer B. “Ground” is a boundary idealisation; in vacuum both charges simply feel Coulomb forces with no privileged reference point.

5. Maxwell layer: charge-conjugation symmetry

∇·E = ρ/ε₀ ∇·B = 0
∇×E = −∂B/∂t ∇×B = μ₀J + μ₀ε₀∂E/∂t

Charge conjugation flips sources (ρ,J)→−(ρ,J) and fields (E,B)→−(E,B), leaving the equations invariant. The Lorentz force m dv/dt = q(E + v×B) accordingly flips sign.

6. Spin–curvature correspondence

Projecting the Dirac Lagrangian to Pauli two-spinors yields the spin–magnetic coupling

H_mag = − (g q / 2m) S·B, S = (ħ/2) σ.

Eigenvalues are E± = ∓ (g|q|Bħ) / 4m; measuring the Zeeman split thus discloses q’s sign. The binary spectrum ms=±½ħ mirrors the binary curvature coefficient ±1⁄3:

• Charge conjugation q→−q sends (α,ms) → (−α, −ms).
• Both arise from the ℤ₂ centre (orientation flip) of their respective symmetry groups: U(1) fibre for charge, SU(2) for spin.

7. Quantum-field operator layer

In QED the operator C acts on the spinor field via ψ → C ȳψᵗ, sending e ȳψγμψAμ → −e ȳψγμψAμ. Vertex factors and propagators inherit the same sign flip, maintaining the duality at the amplitude level.

8. Stack summary

Layer	Fixed quantity	Sign carrier	Observable flip
Kinematics	p− + p+=0	—	antipodes on S²
Seam geometry	absolute conic	±K	± 1⁄3 mix term
Lagrangian	½ m v²	±q	inward vs outward drift
Fields	tensor form	± (ρ,J)	E,B reversed
Spin	ħ/2 σ	±q	Zeeman split sign
QFT	kinetic ψ̄i∂ψ	±e	vertex sign

9. Discussion

1. The curvature seam is an interpretive lens; the true physical curvature is the electromagnetic field strength F on the U(1) gauge fibre.
2. Spin couples to F; its binary projection acts as the experimental “pointer” disclosing the sign of charge.
3. Higher-order curvature terms retain alternating signs, ensuring the sign-duality persists beyond leading order.

10. Conclusion

Track one ℤ₂ label through six layers and the entire architecture of matter–antimatter pairing falls into place: conservation supplies the axis, sign flips select branches, and spin reveals the choice in the lab. No additional structure — curved spacetime, exotic matter, or otherwise — is required once the sign’s journey is followed consistently.

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References. Jackson (1998); Ratcliffe (2006); Weinberg (1995); Cayley (1859); Dirac (1928).  •  End of document.