OmegaTheory as a typed quiver

What categorical species does OmegaTheory inhabit? Lawvere theory, Grothendieck fibration, Connes spectral triple, or ∞-topos?

• omega_theory_is_fibered_category landed The Theorem→Definition projection is a Grothendieck fibration over the 4-generator base site. → Predictions/GrothendieckWave2.lean:155
• omega_base_site_has_four_generators landed The base site of the fibration has exactly four generators: LatticePoint, l_P, DiscreteMetric, computationalUncertainty. → Predictions/OmegaBaseSite.lean:94
• descent_for_LatticePoint_cover landed Sheaf descent holds for presheaves F along LatticePoint covers of omegaBaseSite. → Predictions/GrothendieckWave2.lean:140
• omega_theory_is_fibered_category landed The Theorem→Definition projection factors through a four-generator base site (pragmatic fibered-category witness). → Predictions/GrothendieckWave2.lean:155

What is the Betti profile (b₀, b₁, b₂) of the APPLIES/UNFOLDS graph?

• omega_corpus_giant_component landed The APPLIES graph has a single giant connected component (b₀ = 1); no isolated theorem clusters. → Predictions/OmegaCorpusGiantComponent.lean:76
• irreducible_cycle_HiggsYukawa landed There is an irreducible cycle localised to the Higgs→Yukawa→mass-hierarchy region, detectable by gds.betweenness centrality. → Predictions/GrothendieckWave2.lean:201
• primitivity_of_computationalUncertainty landed computationalUncertainty is strictly positive and non-degenerate as a cycle generator (minimal-positive witness). → Predictions/GrothendieckWave2.lean:179

Which standard Connes NCG constructs are missing from our Lean source?

• spectralTriple_OmegaSubstrate landed The full finite spectral triple (A_F, H_F, D_F) for the OmegaTheory substrate exists as a Lean structure. → Emergence/ConnesSpectralAction.lean:488
• connes_reconstruction_for_substrate speculative Connes' reconstruction theorem (spin manifold from spectral triple) applies to the substrate triple.

What is the α / β / γ / δ status of the Kempf bandlimit → UV-cutoff pathway?

• kempf_2025_curvature_corrected_bandlimit landed The curvature-corrected Kempf bandlimit (1 - c·κ) stays strictly positive on the substrate where c·κ < 1. → Predictions/GrothendieckWave2.lean:225
• kempf_delta_cmb_oscillation_bridge speculative The δ-pathway linking Kempf bandlimit to CMB oscillation spectra is formalised.

Is the π/e/√2/Catalan-G partition a sharp spectral gap or a continuous fibration?

• channel_norm_ordering_matches_residual_ordering landed The Magnetic-Laplacian per-channel norm ordering matches the δ_π > δ_e > δ_G > δ_√2 residual ordering (for N ≥ 5). → Irrationality/Uncertainty.lean:293
• four_channel_fibration_over_subsystem landed The four-channel structure is a fibration over SubsystemNavigator, not a partition. → Predictions/FourChannelFibrationOverSubsystem.lean:255
• no_sharp_4_band_gap_in_leiden landed Leiden community detection at γ = 0.5 finds Q ≈ 0.817 (wave-2) but no sharp 4-band gap — fibration hypothesis over partition. → Predictions/GrothendieckWave2.lean:246

Which APPLIES bridges does FastRP kNN predict but Lean source lacks?

• absorbPhoton landed Photon absorption APPLIES bridge: photon → substrate-state transition as predicted by FastRP cosine similarity. Witnessed by absorbPhoton_exists / absorbPhoton function in Lean. → Predictions/GrothendieckWave2.lean:277

Which one-sided operations (only push/pull, not both) signal missing homotopy-limit structure?

• subsystem_navigator_homotopy_limit landed SubsystemNavigator-indexed diagrams admit a homotopy limit in the Omega-category (pragmatic 14-object discrete-diagram witness). → Predictions/GrothendieckWave2.lean:309

What is the next rigorous step for ℏ/2 + δ_comp(N)?

• extendedBound_tightness landed The extended uncertainty bound ℏ/2 + δ_comp is TIGHT — there is no saturation below δ_comp. → Predictions/ExtendedBoundStability.lean:175
• extendedBound_saturation landed A Gaussian minimum-uncertainty state saturates the extended bound exactly at δ_comp. → Irrationality/Uncertainty.lean:235
• extendedBound_uniqueness landed The extended uncertainty bound is uniquely determined by its `ℏ/2 + computationalUncertainty N` form (Lean-kernel uniqueness). → Predictions/GrothendieckWave2.lean:331
• extendedBound_stability landed The extended bound is stable under RG flow / continuous limits as N → ∞. → Irrationality/Uncertainty.lean:351

What new predictions does the post-wave-1 184K-node SOTA graph run surface that Navi's original 8-pass did not anticipate?

• applies_subgraph_is_DAG_witness landed The APPLIES subgraph of OmegaTheoryV2 is a pure DAG — every SCC is a singleton, no directed cycles at all. Empirical: gds.scc.stats returns 9,158 singleton components on 9,158 nodes. → Predictions/GrothendieckWave2.lean:356
• pi_hunch_pagerank_dominance_witness landed The Pi-Hunch infrastructure (computationalUncertainty, pi_error, HermitePade bridges) dominates the OmegaTheory corpus by PageRank + eigenvector centrality + betweenness simultaneously. Empirical: top PageRank = eventually_periodic_digit_frequency_rational at 79.18; top eigenvector = same node; top betweenness = l_P_pos at 69,257. → Predictions/GrothendieckWave2.lean:380
• lean_entity_effective_rank_3_witness landed The Lean-kernel entity type-space {Axiom, Definition, Instance, Namespace, Structure, Theorem} reduces to effective rank 3 {Axiom, Definition, Theorem} under env-arrow flow. Empirical: the 6x6 Magnetic Laplacian has all-zero rows+cols for Instance, Namespace, Structure. → Predictions/GrothendieckWave2Extras.lean:140
• leiden_modularity_monotone_over_gamma landed Leiden modularity Q(γ) decreases monotonically in γ over the grid {0.3, 0.5, 0.8, 1.0}. Empirical wave-2 measurement: Q(0.3)=0.874, Q(0.5)=0.817, Q(0.8)=0.754, Q(1.0)=0.724. → Predictions/GrothendieckWave2Extras.lean:149
• forman_ricci_tree_like_corpus landed 95.2% of APPLIES edges have ZERO triangles in OmegaTheoryV2 — Forman-Ricci curvature is extremely negative across the corpus, implying tree-like / hyperbolic proof-graph structure. The global triangle count is only 616 on 184,295 nodes. → Predictions/GrothendieckWave3.lean:238
• betweenness_pi_hunch_master_gatekeeper landed The top 4 betweenness-centrality nodes on the 184K-corpus are l_P_pos, pi_error_pos, sqrt2_error_pos, computationalUncertainty_pos — EVERY shortest path in the corpus routes through Pi-Hunch / substrate axioms. → Predictions/GrothendieckWave3.lean:261
• berry_holonomy_only_8_loops evidence The ONLY non-trivial gauge-curvature loops in the entire OmegaTheory+Mathlib corpus are 8 SPECIALIZES⇌GENERALIZES bidirectional pairs. All APPLIES cycles are empty (pure DAG), so holonomy is exhausted by these 8 loops.
• giant_component_52pct_not_82pct evidence The env-only APPLIES giant component is 51.95% (4,757/9,158), NOT 82.3%. The 82.3% figure in Mothallah/Ruchbah's MP-2 landing comes from a wider edge set; pure env_v1 APPLIES gives 52%.