Geometric model verification · Lorentz KGE · August 2026
FHRE, LorentzKG, and FlorE all say they act by Lorentz isometries on a hyperboloid. The printed equations and the public FlorE repository do not agree with that claim. This atlas is a laboratory for those invariants — not a brochure for the MRR tables.
00
19 falsified · 5 holds · 4 blocked · 0 unsupported · 1/5 reproduced families · 62/62 frozen identities. Public code is not the published table until a checkpoint edge closes.
FlorE
failsinconsistent · Constraint mismatch · Equation mismatch
Paper: σᵣ ∈ (−1, 1) and Λᵣ ∈ O(1, k) because Dᵣ is a reflection.
Dᵀ D − I = diag(0,…,0,σ²−1), so ε_group = |σ²−1|. The open interval contains neither ±1. There is no permitted σᵣ in the printed domain for which Dᵣ is an isometry. The useful target group is O⁺(1, n), not all of O(1, n).
FlorE
failsviolates · Parameter mismatch · Default-value mismatch · Constraint mismatch
Paper: a relation-specific σᵣ. Code: self.flip_sign = nn.Parameter(0.0), then ww[:,:,-1] *= flip_sign.
One flip_sign Parameter per FLG module (head_linear and tail_linear), shared across all relations in that module, initialized at 0. Not a relation-indexed σᵣ. At init the last column is zero: det(RD) = 0. The matrix is not in O(k), O(1,k), or GL(k).
FlorE
failsviolates · Constraint mismatch · Algorithm mismatch
Paper: cᵣ ∈ ℍⁿ. Code: nn.Embedding + normal_(std=0.01), then expmap(c, ·).
Not merely init: rel_center is an ordinary Euclidean parameter, not a ManifoldParameter. Nothing forces ⟨c,c⟩_L = −k during training. proju’s tangent identity requires that constraint. Later projx can put the *output* on the sheet; that is not exp_{c_r} for c_r ∈ ℍⁿ.
FlorE
failsviolates · Equation mismatch · Algorithm mismatch · Constraint mismatch
Paper Eq. 8: ξ from a Lorentz tangent projection at cᵣ ∈ ℍⁿ, then exp_{cᵣ}(ξ).
The DO module builds a Euclidean proto-direction (dot + Euclidean norm). Lorentz.expmap then calls proju with the Lorentz inner product — that internal step is real. It does not rescue Eq. 8: rel_center is not structurally on the sheet, so ⟨c, u'⟩_L = ⟨c,v⟩_L (1 + q/k) need not vanish. Post-hoc projx ≠ exp_{c_r}(ξ_r) with c_r ∈ ℍⁿ.
FlorE
failsviolates · Missing operation · Equation mismatch
Paper: −d²_L + γ ⟨ξ̂, log(t)⟩_L + biases. Code: margin − cinner2(t′−h′, t′−h′) + tanh biases.
The directional inner product never appears in _forward. DO mutates embeddings first, so direction can leak in — that is not Equation 13.
FlorE
failsinconsistent · Equation mismatch
Eq. 10: denominator √(⟨qᵣ, qᵣ⟩_L + 1) with ⟨q,q⟩_L = −1.
On the unit hyperboloid the radicand is zero, so the printed log map is singular exactly where it is supposed to operate. That is a transcription/formula failure, independent of σ.
FlorE
blockedunknown · Equation mismatch · Omitted implementation detail
ξ̂ᵣ ∈ T_{cᵣ}ℍⁿ compared with log_{Λh}(t) ∈ T_{Λh}ℍⁿ by a Lorentz inner product.
Unless cᵣ = Λh, ξ̂ᵣ is not in T_{Λh}: ⟨Λh, ξ̂⟩_L is generically nonzero. An ambient Minkowski pairing exists. It is not an intrinsic Riemannian inner product on one tangent space without transport or an explicit identification. Medium-confidence geometric issue; not the same class as σ² ≠ 1.
FlorE
qualifyunknown · Evaluation mismatch
Ablations: 40.2 full / 37.3 −DO / 24.1 −FLG. Inference that O(1,k) caused the gain.
FLG contributes. Lorentz-group membership does not follow. A truthful non-isometric R D baseline is required before attributing 0.018 MRR to isometry. Hypothetical until trained.
FlorE
failsinconsistent · Evaluation mismatch
The same full-model Z-Paradox numbers appear in the main table and the ablation table.
FB15k-237 Z-Paradox: 52.3 in the main relation-pattern table vs 42.3 in the ablation table. WN18RR: 45.8 vs 52.8. No adjacent note of a different checkpoint, seed, split, or evaluator. This is a reporting inconsistency, not a proof that 0.402 is fabricated.
FlorE
failsviolates · Parameter mismatch · Training mismatch
The public best scripts reproduce the paper’s stated FB15k-237 / WN18RR margins.
Paper: FB 1.07, WN 1.01. Public best scripts: FB 1.00, WN 1.08. Other hyperparameters (lr, negatives, batch) line up more closely. Which configuration produced 0.402 is unresolved.
FlorE
qualifyviolates · Default-value mismatch · Omitted implementation detail
The training entrypoint’s flags match the documented configuration.
Boolean argparse uses type=bool and action=store_true with default=True, so --flag False still stores True. CUDA check is shaped like if torch.cuda.is_available: (missing call). Lower severity than Eq. 5; still a runtime-config discrepancy.
FlorE
failsviolates · Evaluation mismatch · Algorithm mismatch
Paper: filtered ranking of the full test set (FB 20,446 / WN 3,134).
Experiment.evaluate at d276d70 sets max_queries=20 and default batch=100, then breaks the main loop when query_counter >= 20. That is at most 2,000 examples. Reverses are appended after originals, so FB never reaches reverse triples. The same evaluate() is used on valid_data for early stopping, so checkpoint selection is truncated too. Does not prove the published 0.402 used this loop.
FlorE
failsviolates · Evaluation mismatch · Missing operation
Tables 4 and 6 are produced by the public evaluate() path.
relation_map = {} is empty. Per-relation collection runs only if rela_id in relation_map.values(), which is never true. If it were populated, hits_level and hit are referenced before assignment. HEAD cannot regenerate the paper’s relation-pattern statistics. Combined with Table 4 ≠ Table 6, the lineage of those numbers is unresolved.
FlorE
failsviolates · Training mismatch · Omitted implementation detail
Public training writes a checkpoint that can be hashed for Level C.
best_model = deepcopy(model.state_dict()); later load_state_dict(best_model); evaluate(test). No torch.save in this path. Level C against current HEAD requires reconstructing the pipeline, not executing it unchanged.
FlorE
failsinconsistent · Equation mismatch · Constraint mismatch
Paper: h=(3,0), t₁=(4,1), t₂=(5,3) lie on the Lorentz manifold, ⟨tᵢ,tᵢ⟩_L = 16.
With signature (−+), ⟨h,h⟩_L = −9, ⟨t₁,t₁⟩_L = −15, ⟨t₂,t₂⟩_L = −16. Unit-sheet preconditions fail, so the identity d² = −2−2⟨x,y⟩_L is not applied. The paper then uses that identity anyway.
FHRE
failsinconsistent · Equation mismatch
FHRE Eq. 6: d²_L = 2/c − 2⟨x,y⟩_L with c = 1 and ⟨x,x⟩_L = −1.
Putting x = y into the printed formula yields 4, not 0. Repairing the sign gives −2 − 2⟨x,y⟩_L = 2(cosh d_ℍ − 1), a nonnegative interval surrogate monotonic in geodesic distance. That is not a proof of the triangle inequality.
FHRE
failsinconsistent · Constraint mismatch · Equation mismatch
FHRE: ⟨x,x⟩_L = −1, x₀ = √(‖x_s‖² − 1/c), α = √(−c)‖z‖_L, and c = 1.
At c = 1 the printed time formula gives ⟨x,x⟩_L = +1, not −1, and √(−c) is imaginary. Reading c as signed curvature −1 makes the time coordinate, the exp-map scale, and Eq. 6 all consistent. The paper states both conventions at once.
LorentzKG
holdssatisfies · Evaluation mismatch
Householder R ∈ O(n) composed with a boost B(v) is a Lorentz transformation by construction.
Random Householder ⊕ boost on L²: metric residual, determinant residual, and manifold residual all sit at machine precision.
01
Fully hyperbolic means the relation acts linearly in ℝⁿ⁺¹. FHRE and LorentzKG keep exp/log off the scoring path. FlorE’s public forward does not: FLG → DO, and DO calls expmap and Möbius addition (log, parallel transport, exp) before the interval is scored.
2024 · ECAI
diag(1, SO(n))
The Givens factor is an isometry. Printed d² and the c=1 time formula disagree with ⟨x,x⟩_L=−1. Reading c as signed curvature −1 repairs all three.
Liang, Wang, Bao & Gao
Inner Mongolia University
2024 · ACL Findings
O⁺(1, n)
A Lorentz transformation from the positive orthochronous group: Householder rotation/reflection composed with a boost, applied separately to head and tail.
Fan, Xu, Chen, Chen, Das & Yang
Visa Research
2026 · AAAI
claimed O(1, n)
Claims the full Lorentz group plus a directional offset. The printed σ domain never reaches an isometry; each FLG module has one flip_sign, not σᵣ.
Duo, Li, Su & Gao
Inner Mongolia University
02
Three layers, not one formula. The printed FHRE score is not the interval LorentzKG uses. Under ⟨x, x⟩_L = −1 they differ by 4 at coincidence.
| Model | Published | Verified / executed | Note |
|---|---|---|---|
| FHRE | +2 − 2⟨R vₕ, vₜ⟩_L + b | −2 − 2⟨R vₕ, vₜ⟩_L + b | Printed identity is not a distance. |
| LorentzKG | −d²_L(Λ_{r,1} h, Λ_{r,2} t) + b + δ | same; d² = −2 − 2⟨·,·⟩_L | Paper and algebra agree. |
| FlorE | −d²_L + γ ⟨ξ̂, log⟩_L + b | code: margin − ‖t′ − h′‖²_L only | Eq. 13 term is not executed. |
FHRE printed d²(x, x)
4.0000
2 − 2⟨x, x⟩_L
Consistent interval d²(x, x)
0
−2 − 2⟨x, x⟩_L
03
diag(1, SO(n)) ⊂ SO⁺ ⊂ O⁺ ⊂ O is real. The architectures are not supersets. FHRE only rotates the spatial block. LorentzKG adds boosts. FlorE also uses dual maps (Eq. 13, head_linear / tail_linear) but no boosts, and D(σ) leaves the group.
FHRE
LorentzKG
FlorE
Claimed group; invariants fail.
04
A head and two symmetric tails on one rapidity circle. Proper rotations reach one cluster. An orientation-reversing isometry reaches the other — when it is actually an isometry. Watch the manifold residual as σ leaves {±1}.
d²(Λh, t₁)
0.0000
d²(Λh, t₂)
1.0059
ε_group
0
det Λ
1.0000
Λ₀₀
1.0000
q(h)
-1.0000
q(Λh)
-1.0000
identity
applicable
05
FlorE reports FB15k-237 MRR 0.402 against LorentzKG 0.384, and large Z-Paradox slice gains. Those figures are supported by the FlorE paper. They cannot be attributed to public HEAD (d276d70). Until a checkpoint edge closes, they are a citation, not evidence about a Lorentz isometry or about the current repository.
Reproduction gates closed: 0/5. Bars are citations. The public repository diverges from the paper; which implementation produced the table is unknown.
Filtered MRR as cited. Not a causal effect of group scope: the architectures are not nested.
Accuracy (%). FlorE vs best baseline on annotated slices.
Self-reported. Protocols differ. The Atlas will not compute FlorE − FHRE on these bars.
Δ blocked. Different splits, hard-negative protocol, and baseline set. Provenance: FHRE self-report vs FlorE self-report, CoDEx-S.
FlorE FB MRR (%). LorentzKG drops when only the boost remains. Not evidence about group scope.
FlorE on FB15k-237
LorentzKG FB MRR
06
The landscape is not a timeline of bigger groups. It is a table of independent degrees of freedom. Reported MRR rises across the three papers; that is not a causal effect of group scope.
| Model | Geometry | Exact isometry | Boosts | Reflection | Dual maps | Offset | Curvature | Z-Par (empirical) |
|---|---|---|---|---|---|---|---|---|
| FHRE | Lorentz | yes | no | no | no | no | fixed −1 | limited |
| LorentzKG | Lorentz | yes | yes | yes | yes | no | fixed −1 | better |
| FlorE claimed | Lorentz | claimed | no | claimed | yes | claimed | fixed −1 | cited; T4≠T6 |
| FlorE code | mixed | no | commented out | per-module scalar | yes | Euclidean proto + Lorentz proju | fixed −1 | unknown |
| FlorE repaired | Lorentz | yes | no | yes | optional | yes, Lorentz | fixed −1 | untested |
| Hybrid | Lorentz | yes | yes | spatial ℤ₂ | yes | yes, Lorentz | configurable | untested |
Novelty is the combination, not any single column. LorentzKG already has boosts, reflections, and dual maps. FlorE’s claimed extras are orientation reversal and a directional offset — both of which fail their own invariants as printed and as coded. Z-Paradox on repaired/hybrid rows is untested — those models have not been trained. Cited numbers are not reproductions.
07
FlorE’s domain is the open interval (−1, 1). The isometry condition is σ² = 1. There is no overlap. The parameterization never reaches the group it names.
One flip_sign per FLG module, initialized at 0, unconstrained. Euclidean proto-direction; expmap.proju is Lorentz but rel_center is off-sheet. Score without Eq. 13. The inspected repository diverges from the published specification. The lineage that produced the reported table is unresolved.
Represent spatial parity as pᵣ ∈ ℤ₂ inside O⁺(1, n). Build Qᵣ as a product of Householders. det(Qᵣ) = (−1)ᵐ. Do not attempt the two time-reversing components of O(1, n).
Dual endpoint maps plus genuine boosts are not a subset of repaired FlorE. The next model should take those, add exact orientation parity, and only then add a Lorentz-correct offset.
08
Target group O⁺(1, n), not O(1, n). Spatial Householders and proper boosts. Dual endpoint maps. Time orientation fixed so embeddings stay on x₀ > 0.
Λ_{r,h} = Q_{r,h} B_{r,h} ∈ O⁺(1, n)
Λ_{r,t} = Q_{r,t} B_{r,t} ∈ O⁺(1, n)
Q = H₁…Hₘ, each Hᵥ = I − 2vvᵀ/vᵀv, det Q = (−1)ᵐ · spatial parity only
B(v) a proper orthochronous boost
h′ = Λ_{r,h} h, t′ = Λ_{r,t} t
Option B — keep the map an isometry. Direction lives in the score:
f = −d²_L(h′, t′) + γ ⟨ξ̂_r, log_{h′}(t′)⟩_L + b
Option A — exp-map displace h′ — is not an isometry. Do not call that relation map Lorentz.
LorentzKG already has Householders, boosts, and dual maps. The narrow addition is explicit spatial parity plus a Lorentz-native directional score that does not move the points. Explore the generators.
09
The public FlorE repository diverges from the published specification. The provenance needed to show which implementation produced the reported table is unavailable. Until that edge closes, MRR 0.402 is a citation — not evidence about O(1, n).