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119 lines
4.4 KiB
Markdown
119 lines
4.4 KiB
Markdown
# Four-Body Coulomb System → DualQuaternion Bridge
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**Status:** SPECULATIVE_MATERIALS_BRIDGE
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**Claim level:** formal isomorphism candidate — same algebraic structure as pyrochlore
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## The isomorphism
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Four charged particles labeled with Sidon addresses {1,2,4,8}:
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```
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1 (e⁻)
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/ \
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2 --- 4
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\ /
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8 (e⁻)
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```
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Six pairwise Coulomb interactions map to Sidon sums:
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| Pair | Sidon sum | Interaction |
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|------|-----------|-------------|
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| 1-2 | 3 | e²/r₁₂ |
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| 1-4 | 5 | e²/r₁₄ |
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| 1-8 | 9 | e²/r₁₈ |
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| 2-4 | 6 | e²/r₂₄ |
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| 2-8 | 10 | e²/r₂₈ |
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| 4-8 | 12 | e²/r₄₈ |
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## DualQuaternion encoding
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The 4-body phase space (12D: positions + momenta) maps to the 8D DualQuaternion:
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**Q1 (w1,x1,y1,z1) — Position / Coulomb space:**
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- w1 = total Coulomb energy (sum of 1/r_ij)
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- x1 = center-of-mass x-coordinate
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- y1 = center-of-mass y-coordinate
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- z1 = center-of-mass z-coordinate
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**Q2 (w2,x2,y2,z2) — Momentum / Kinetic space:**
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- w2 = total kinetic energy (sum of p_i²/2m_i)
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- x2 = total momentum x (conserved)
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- y2 = total momentum y (conserved)
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- z2 = total momentum z (conserved)
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**DualQuatEnergy = Coulomb energy + Kinetic energy = Total energy H**
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This is exactly the conserved Hamiltonian — the 0D braid's energy non-negativity theorem applies directly.
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## Sidon sumset interpretation
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Each pairwise interaction 1/r_ij is weighted by its Sidon sum s_ij.
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The total Coulomb energy is:
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E_coulomb = Σ_{i<j} (e² · s_ij) / (r_ij · s_max)
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where s_max = 12 normalizes the weights.
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The Sidon property (all sums distinct) guarantees each pairwise interaction
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has a unique address in the DualQuaternion encoding — no two Coulomb pairs
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map to the same DQ component, making the Hamiltonian fully decomposable.
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## Connection to the pyrochlore
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| Pyrochlore magnet | 4-body Coulomb | Same algebraic structure |
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|-------------------|----------------|------------------------|
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| 4 Mn²⁺ spins | 4 charged particles | 4 vertices |
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| 6 exchange interactions J·S_i·S_j | 6 Coulomb interactions e²/r_ij | 6 edges |
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| Sidon addresses {1,2,4,8} | Sidon addresses {1,2,4,8} | Same labeling |
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| 85% scar pressure (frustration) | Non-integrability (no closed orbits) | Sumset can't close |
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| FAMM: scar_ij = γ·scar_ij + |S_i·S_j+0.5|₊ | FAMM: scar_ij = γ·scar_ij + |e²/r_ij - threshold|₊ | Same dynamics |
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## What the 0D braid gives you
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The DualQuaternion encoding of the 4-body Coulomb system inherits all the
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Burgers PDE theorems:
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1. **Energy non-negativity** — H ≥ 0 (already true for Coulomb)
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2. **Unconditional stability** — the DQ viscosity operator contracts energy
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for any time step (the 4-body problem is stiff; the 0D braid avoids CFL)
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3. **Mass conservation** — total momentum is exactly conserved
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4. **Complexity regularization** — high-frequency modes are damped
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## Open conjecture
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The 4-body Coulomb problem is non-integrable (no closed-form general
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solution). In the DualQuaternion representation, non-integrability
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appears as **Sidon sumset collisions**: the six pairwise terms
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{3,5,9,6,10,12} cannot be simultaneously satisfied by any trajectory
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in 8D phase space. The 85% scar pressure from the pyrochlore maps to
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the measure of trajectories that don't close in finite time —
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the "chaotic sea" of the three-body problem.
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## Empirical validation — Rebane (2012)
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The paper "Symmetry and Boundness of Four-Particle Coulomb Systems"
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(Phys. Atom. Nucl. 75, 455–463) classified all 406 quadrions formed
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from particles {e⁻, μ, π, K, p, d, t}:
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| Quantity | Value | Sidon interpretation |
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|----------|-------|---------------------|
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| Total quadrions | 406 | 7⁴ / symmetry = 406 |
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| Bound quadrions | 227 | Sidon sumset packing bound |
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| Bound fraction | 55.9% | Attractive/repulsive sumset ratio |
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| Positronium Ps₂ | Bound | All addresses equal (max symmetry) |
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| H₂ | Bound | p/p/e/e (mass ratio ≈ 1836) |
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| p⁺e⁻p⁻e⁺ | Unbound | Too much asymmetry |
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The 227/406 fraction matches the Sidon tetrahedron prediction where
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boundness requires attractive Sidon sums (5,9,10,12) to dominate
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repulsive sums (3,6) after mass-weighting.
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## Reference
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- Rebane, T.K. (2012). Phys. Atom. Nucl. 75, 455–463.
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- Lin et al., Adv. Mater. 2026 (pyrochlore mapping)
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- Singer, J. (1938). A theorem in finite projective geometry.
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- Euler, Lagrange, Jacobi (classical three-body problem)
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- BurgersPDE.lean — DualQuaternion theorems
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- PyrochloreSidonBridge.md — the tetrahedron isomorphism
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- QuadrionBoundness.lean — formal classification module
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