Paper · code · verified
Three layers
Published equations, the authors’ repository, and a construction that satisfies the claimed invariants. Residuals are computed from the same default parameters on every row.
| Layer | Operator | Claimed | Actual | ε_group | ε_man | ε_det | Verdict |
|---|---|---|---|---|---|---|---|
| paper | FHRE as printed R(θ) head-only | Lorentz rotations | diag(1, SO(n)) ⊊ SO⁺ | 0 | 0 | 0 | holds |
| verified | FHRE, consistent interval R(θ), d² = −2 − 2⟨·,·⟩ | Lorentz rotations | diag(1, SO(n)) ⊊ SO⁺ | 0 | 0 | 0 | holds |
| paper | LorentzKG Λ = R B, dual endpoints | O⁺(1, n) | O⁺(1, n) | 0 | 0 | 0 | holds |
| paper | FlorE as printed R D(σ), σ ∈ (−1, 1) | O(1, n) | not a group | 0.7500 | 0.0517 | 0.5000 | fails |
| code | FlorE, dzh597/FlorE Q · flip_sign, flip_sign ∈ ℝ | O(1, n) | singular at init | 1.0000 | 0.0689 | 1.0000 | fails |
| verified | FlorE repaired Q = H_m ⋯ H_1, det = (−1)ᵐ | O(n) ⊂ O⁺(1, n) | O(n) ⊂ O⁺(1, n) | 0 | 0 | 0 | holds |
| verified | Hybrid Λ_{h,t} = Q_{h,t} B_{h,t} | O⁺(1, n) = SO⁺ ⋊ ℤ₂ | O⁺(1, n) | 0 | 0 | 0 | holds |
FlorE, three columns
Published algorithm, public repository, construction that stays in the group.
| Aspect | Published | Code | Verified |
|---|---|---|---|
| Flip scalar | σᵣ ∈ (−1, 1), one per relation | one flip_sign per FLG module, init 0.0 | pᵣ ∈ ℤ₂ ⊂ O⁺(1, n), spatial only |
| Group membership | claims Λᵣ ∈ O(1, k) | det = 0 at init; unconstrained thereafter | Q B ∈ O⁺(1, n); no time reversal |
| Center cᵣ | cᵣ ∈ Lⁿ (hyperboloid) | nn.Embedding, Euclidean init | ManifoldParameter on Lⁿ |
| Tangent projection | d + ⟨d, c⟩_L c at c ∈ ℍⁿ | Euclidean proto-dir; expmap.proju; c not on sheet | c on sheet, then Lorentz proju |
| Score | Eq. 13: interval + γ ⟨ξ̂, log⟩_L | margin − ‖t′ − h′‖²_L only | PT_{o→h'}(u_r); both vectors in T_{h'} |
| Boosts | omitted (ablation: spatial part suffices) | LorentzBoost2 commented out | optional exact boost from LorentzKG |