Finite homological certificates
Toric BGG Category O
The Toric BGG layer is a finite certificate system: it never claims the model contains a literal abelian category. It attaches small sign-vector posets, standard-filtration masks, sparse differentials, Koszul degree profiles, Gale-dual labels, and derived signatures to training records. Late-phase weights remain small so BPB stays primary.
This page combines the hand-written method explainer with the generated Category O audit. If a metric is listed as unavailable, that does not mean the method is absent; it means the copied public run did not emit the exact finite certificate required to compute that metric. The late gate is intentionally conservative.
Best-Run BGG-Adjacent Losses
| toric_bgg_loss | 0.164997 |
|---|---|
| koszul_persistence_loss | 0.045398 |
| derived_signature_loss | 3.243159 |
| toric_cca_topology_loss | 1.045991 |
d(m⊗ξ)=Σᵢ xᵢm⊗eᵢ∧ξ. The square vanishes because xᵢxⱼ=xⱼxᵢ while eᵢ∧eⱼ=-eⱼ∧eᵢ.
Generated Audit Gates
| exact_certificate_available | 1.0000 |
|---|---|
| exact_finite_chain_provenance | 1.0000 |
| late_gate_required | 1.0000 |
A late gate of 1 means exact certificate metrics are expected only in late-phase or certificate-bearing records, avoiding noisy penalties during BPB-first early training.
How It Affects Training
- Weight up carefully: improves standard-object routing and homological consistency when records carry exact certificates.
- Weight down early: prevents sparse certificate losses from dominating byte likelihood before BPB stabilizes.
- Use as audit: unavailable exact metrics should trigger data/certificate instrumentation review, not a blind loss increase.
Resolution and Standard-Filtration Metrics
| metric | value/status | provenance | meaning |
|---|---|---|---|
| toric_bgg_d2_residual | 0.032614 | exact_finite_chain_complex_boundary_square | Boundary-square residual for sparse finite differentials. Low means the predicted differential behaves like a chain-complex map with d²=0. |
| toric_bgg_standard_leakage | 0.403288 | exact_finite_chain_poset_standard_mask | Attention mass outside the standard-filtration mask. Low means hidden standard-object routing respects the poset. |
| toric_bgg_koszul_linearity_residual | 0.000000 | exact_finite_koszul_degree_profile | Deviation from the expected Koszul degree profile. Low means the finite BGG/Koszul proxy is degree-consistent. |
| toric_bgg_gale_dual_consistency | 0.000000 | finite_gale_dual_signature_pair | Agreement between an arrangement certificate and its Gale-dual signature. High consistency supports dual-curriculum transfer. |
| toric_bgg_signature_smoothness | 0.000000 | finite_bgg_signature_path | Variation of the BGG signature along the reasoning trajectory. Low values indicate stable categorical coordinates. |
| toric_bgg_standard_entropy | 0.982606 | finite_standard_label_distribution | Entropy of standard-label assignments. It detects collapse to one standard object or diffuse unusable labels. |