s stringlengths 1 39 | o stringclasses 996
values | p stringlengths 7 77 | confidence float64 0.4 1 | predicate_class stringclasses 2
values | community int64 0 14 ⌀ | equation_name stringlengths 14 73 ⌀ | equation_latex stringlengths 5 217 ⌀ | equation_sympy stringclasses 617
values | equation_form stringclasses 472
values | variables stringlengths 2 85 ⌀ | domain stringclasses 119
values | scm_name stringclasses 165
values | source stringlengths 9 50 ⌀ | extraction_method stringclasses 8
values |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
query | theta | urn:hailstone:math:structural_equation[9f962bd58da8] | 1 | structural_equation | 4 | adr0033.operation1.gor_hash | \theta(query) = (2\pi \times \sum_i ord(query_i) \times p_i) \bmod 2\pi | \theta(query) = (2\pi \times \sum_i ord(query_i) \times p_i) \bmod 2\pi | closed_form | ["theta", "query", "ord", "p_i"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
community_k | arc_assignment | urn:hailstone:math:structural_equation[6fdd2ebddc2b] | 1 | structural_equation | 4 | adr0033.operation1.community_mapping | k = \lfloor \theta \times 15 / 2\pi \rfloor | k = \lfloor \theta \times 15 / 2\pi \rfloor | closed_form | ["k", "theta"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
fractional_chromatic_number | kneser_graph | urn:hailstone:math:structural_equation[de4546a01c00] | 1 | structural_equation | 4 | adr0033.operation2.lovasz_chromatic | \chi_f(KG(n,k)) = n - 2k + 2 | \chi_f(KG(n,k)) = n - 2k + 2 | closed_form | ["chi_f", "n", "k"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
lovasz_bound_kg15_4 | fractal_depth | urn:hailstone:math:structural_equation[7033894b21fe] | 1 | structural_equation | 4 | adr0033.operation2.kg_15_4 | \chi_f(KG(15,4)) = 15 - 8 + 2 = 9 | \chi_f(KG(15,4)) = 15 - 8 + 2 = 9 | closed_form | ["chi_f"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
W_model | W_graph | urn:hailstone:math:structural_equation[ffe83fa4daea] | 1 | structural_equation | 4 | adr0033.operation3.procrustes_optimization | R^* = \arg\min_R ||W_{model} - R \times W_{graph}||_F | R^* = \arg\min_R ||W_{model} - R \times W_{graph}||_F | closed_form | ["R", "W_model", "W_graph"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
SVD_decomposition | optimal_rotation | urn:hailstone:math:structural_equation[fa5e607189f3] | 1 | structural_equation | 4 | adr0033.operation3.procrustes_svd_solution | W_{model}^T @ W_{graph} = U \times \Sigma \times V^T \Rightarrow R^* = U \times V^T | W_{model}^T @ W_{graph} = U \times \Sigma \times V^T \Rightarrow R^* = U \times V^T | closed_form | ["U", "Sigma", "V", "R"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
alignment_quality | frobenius_residual | urn:hailstone:math:structural_equation[13eeecf479db] | 1 | structural_equation | 4 | adr0033.operation3.alignment_metric | alignment\_quality = ||W_{model} - R^* \times W_{graph}||_F / ||W_{graph}||_F | alignment\_quality = ||W_{model} - R^* \times W_{graph}||_F / ||W_{graph}||_F | closed_form | ["W_model", "W_graph", "R"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
community_k | n_gateways | urn:hailstone:math:structural_equation[97221c9f021e] | 1 | structural_equation | 4 | adr0033.operation4.spectral_gap_gateways | n\_gateways(k) = |{i : \lambda_i(L_k) - \lambda_{i-1}(L_k) > \Delta\_threshold}| | n\_gateways(k) = |{i : \lambda_i(L_k) - \lambda_{i-1}(L_k) > \Delta\_threshold}| | closed_form | ["lambda_i", "L_k", "Delta_threshold"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
cheeger_constant | spectral_gap | urn:hailstone:math:structural_equation[707fce258a66] | 1 | structural_equation | 4 | adr0033.operation4.cheeger_inequality | h(G)^2/2 \le \lambda_2(L) \le 2h(G) | h(G)^2/2 \le \lambda_2(L) \le 2h(G) | closed_form | ["h", "lambda_2", "L"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
W_mesh_gradient | loss_attribution | urn:hailstone:math:structural_equation[87d6b7d98bab] | 1 | structural_equation | 4 | adr0033.operation5.mesh_gradient | \partial L / \partial W\_mesh[i,j] = \partial L / \partial output \times output_j | \partial L / \partial W\_mesh[i,j] = \partial L / \partial output \times output_j | closed_form | ["L", "W_mesh", "output"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
output | weighted_sum | urn:hailstone:math:structural_equation[bf87ca065ec6] | 1 | structural_equation | 4 | adr0033.operation5.mesh_output | output = \sum_k W\_mesh[query\_community, k] \times output_k | output = \sum_k W\_mesh[query\_community, k] \times output_k | closed_form | ["W_mesh", "output_k"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
claim | hck_score | urn:hailstone:math:structural_equation[596c7165d207] | 1 | structural_equation | 4 | adr0033.operation7.hck_identifiability | HCK(C, G) = P(C | do(X=x), G) / P(C | G) | HCK(C, G) = P(C | do(X=x), G) / P(C | G) | closed_form | ["HCK", "C", "G", "X"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
backdoor_criterion | causal_effect | urn:hailstone:math:structural_equation[82b19b07c3ff] | 1 | structural_equation | 4 | adr0033.operation7.backdoor_formula | P(C | do(X)) = \sum_Z P(C | X, Z) \times P(Z) | P(C | do(X)) = \sum_Z P(C | X, Z) \times P(Z) | closed_form | ["P", "C", "X", "Z"] | adr0033 | adr0033 | hailstone:adr0033 | hailstone_adr_extraction |
loss_total | causal_consistency | urn:hailstone:math:structural_equation[686885f6b372] | 1 | structural_equation | 4 | adr0035.causal_calculus.math_loss | L_{math} = L_{LM} + L_{HCK} + L_{consistency} | L_{math} = L_{LM} + L_{HCK} + L_{consistency} | loss_function | ["L_LM", "L_HCK", "L_consistency"] | adr0035 | adr0035 | hailstone:adr0035 | hailstone_adr_extraction |
derivative | algebraic_rule | urn:hailstone:math:structural_equation[7bfd01b06603] | 1 | structural_equation | 4 | adr0035.causal_calculus.differentiation_rule | d/dx[x^n] = n*x^{n-1} | d/dx[x^n] = n*x^{n-1} | closed_form | ["n", "x"] | adr0035 | adr0035 | hailstone:adr0035 | hailstone_adr_extraction |
product_rule | differentiation | urn:hailstone:math:structural_equation[3922fa7f0bdc] | 1 | structural_equation | 4 | adr0035.causal_calculus.product_rule | d/dx[fg] = f*(dg/dx) + g*(df/dx) | d/dx[fg] = f*(dg/dx) + g*(df/dx) | closed_form | ["f", "g", "x"] | adr0035 | adr0035 | hailstone:adr0035 | hailstone_adr_extraction |
chain_rule | composite_function | urn:hailstone:math:structural_equation[5a210ded173c] | 1 | structural_equation | 4 | adr0035.causal_calculus.chain_rule | d/dx[f(g(x))] = f'(g(x))*g'(x) | d/dx[f(g(x))] = f'(g(x))*g'(x) | closed_form | ["f", "g", "x"] | adr0035 | adr0035 | hailstone:adr0035 | hailstone_adr_extraction |
do_calculus_integral | causal_reasoning | urn:hailstone:math:structural_equation[cd00045367fe] | 1 | structural_equation | 4 | adr0035.causal_calculus.continuous_do_calculus | P(Y|do(X=x)) = \int P(Y|X=x, Z=z) P(Z=z) dz | P(Y|do(X=x)) = \int P(Y|X=x, Z=z) P(Z=z) dz | closed_form | ["P", "Y", "X", "Z"] | adr0035 | adr0035 | hailstone:adr0035 | hailstone_adr_extraction |
initial_state | rotated_eigenvectors | urn:hailstone:math:structural_equation[5f047dfe538d] | 1 | structural_equation | 4 | adr0036.gor_init.procrustes_initialized_state | h_0^k = R^* \times V_k | h_0^k = R^* \times V_k | closed_form | ["h_0", "R", "V_k"] | adr0036 | adr0036 | hailstone:adr0036 | hailstone_adr_extraction |
total_compression | triples_to_state | urn:hailstone:math:structural_equation[706e54ada261] | 1 | structural_equation | 4 | adr0036.gor_init.total_compression_ratio | 10B * 20 bytes / 64KB ≈ 3,000,000× compression | 10B * 20 bytes / 64KB ≈ 3,000,000× compression | closed_form | [] | adr0036 | adr0036 | hailstone:adr0036 | hailstone_adr_extraction |
diffusion_score | causal_guidance | urn:hailstone:math:structural_equation[9c4b5b665c98] | 1 | structural_equation | 4 | adr0037.causal_diffusion.causal_score | s_θ(x_t,t,G) = ∇log p(x_t|G) + λ × ∇log P_HCK(x_t|G) | s_θ(x_t,t,G) = ∇log p(x_t|G) + λ × ∇log P_HCK(x_t|G) | closed_form | ["s_theta", "x_t", "G", "lambda"] | adr0037 | adr0037 | hailstone:adr0037 | hailstone_adr_extraction |
confidence_decay | transitive_closure | urn:hailstone:math:structural_equation[f28a1f85ee5a] | 1 | structural_equation | 4 | adr0037.causal_diffusion.confidence_decay | conf(A→C) = conf(A→B) × conf(B→C) × decay(predicate_classes) | conf(A→C) = conf(A→B) × conf(B→C) × decay(predicate_classes) | closed_form | ["conf", "decay"] | adr0037 | adr0037 | hailstone:adr0037 | hailstone_adr_extraction |
attention_output | compressed_storage | urn:hailstone:math:structural_equation[2eb55e941c5c] | 1 | structural_equation | 4 | adr0038.attention_storage.attention_as_storage | O = A × V; h = HailStorm.compress(O, k) | O = A × V; h = HailStorm.compress(O, k) | closed_form | ["O", "A", "V", "h"] | adr0038 | adr0038 | hailstone:adr0038 | hailstone_adr_extraction |
recurrent_state | state_delta_storage | urn:hailstone:math:structural_equation[a2a5722bd9cb] | 1 | structural_equation | 4 | adr0038.attention_storage.mamba_as_storage | Δh = h_{t+1} - h_t; if |Δh|² > τ: store to graph | Δh = h_{t+1} - h_t; if |Δh|² > τ: store to graph | closed_form | ["h_t", "Delta_h", "tau"] | adr0038 | adr0038 | hailstone:adr0038 | hailstone_adr_extraction |
retrieval_reward | graph_growth | urn:hailstone:math:structural_equation[55c7e18b3090] | 1 | structural_equation | 4 | adr0038.attention_storage.retrieval_reward | retrieval_reward(h) = mean(HCK_score) × retrieval_frequency(h) | retrieval_reward(h) = mean(HCK_score) × retrieval_frequency(h) | closed_form | ["retrieval_reward", "HCK_score"] | adr0038 | adr0038 | hailstone:adr0038 | hailstone_adr_extraction |
two_communities | geodesic_distance | urn:hailstone:math:structural_equation[b0fd304a81e5] | 1 | structural_equation | 4 | adr0039.geodesic_selection.geodesic_distance | d_geodesic(k,j) = min(|θ_k - θ_j|, 2π - |θ_k - θ_j|) mod 2π | d_geodesic(k,j) = min(|θ_k - θ_j|, 2π - |θ_k - θ_j|) mod 2π | closed_form | ["d_geodesic", "theta_k", "theta_j"] | adr0039 | adr0039 | hailstone:adr0039 | hailstone_adr_extraction |
minimum_parameters | community_budget | urn:hailstone:math:structural_equation[2c53447e5c7a] | 1 | structural_equation | 4 | adr0039.geodesic_selection.minimum_params_formula | params_min(k) = base_params × (1 + α × geodesic_isolation(k)/π) | params_min(k) = base_params × (1 + α × geodesic_isolation(k)/π) | closed_form | ["params_min", "geodesic_isolation", "alpha"] | adr0039 | adr0039 | hailstone:adr0039 | hailstone_adr_extraction |
total_parameters | manifold_integral | urn:hailstone:math:structural_equation[e279a4b3ec00] | 1 | structural_equation | 4 | adr0039.geodesic_selection.total_parameter_budget | P_total = Σ_{k=0}^{14} Σ_{r=0}^{r_max(k)} params(k,r) × 15^r | P_total = Σ_{k=0}^{14} Σ_{r=0}^{r_max(k)} params(k,r) × 15^r | closed_form | ["P_total", "params", "r_max"] | adr0039 | adr0039 | hailstone:adr0039 | hailstone_adr_extraction |
model_weights | parameter_loss | urn:hailstone:math:structural_equation[356ed5f35a03] | 1 | structural_equation | 4 | adr0040.mocm.hck_trained_gradient | ∂L_total/∂θ = ∂L_LM/∂θ + β × ∂L_HCK/∂θ | ∂L_total/∂θ = ∂L_LM/∂θ + β × ∂L_HCK/∂θ | loss_function | ["L_total", "L_LM", "L_HCK", "beta"] | adr0040 | adr0040 | hailstone:adr0040 | hailstone_adr_extraction |
query_evolution | hamiltonian_dynamics | urn:hailstone:math:structural_equation[f7938da24a1b] | 1 | structural_equation | 4 | adr0040.mocm.hamiltonian_evolution | d|ψ(t)⟩/dt = -iH|ψ(t)⟩ where H = W_mesh | d|ψ(t)⟩/dt = -iH|ψ(t)⟩ where H = W_mesh | differential | ["psi", "t", "H", "W_mesh"] | adr0040 | adr0040 | hailstone:adr0040 | hailstone_adr_extraction |
steady_state | query_distribution | urn:hailstone:math:structural_equation[51bd42585557] | 1 | structural_equation | 4 | adr0040.mocm.unitary_evolution | |ψ(t)⟩ = exp(-iH×t)|ψ(0)⟩ | |ψ(t)⟩ = exp(-iH×t)|ψ(0)⟩ | closed_form | ["psi", "t", "H"] | adr0040 | adr0040 | hailstone:adr0040 | hailstone_adr_extraction |
gor_manifold | schrodinger_equation | urn:hailstone:math:structural_equation[35c20d53c6fa] | 1 | structural_equation | 4 | adr0040.mocm.schrodinger_equation | -∇²ψ + V(x)ψ = Eψ [Schrödinger on GOR manifold] | -∇²ψ + V(x)ψ = Eψ [Schrödinger on GOR manifold] | differential | ["nabla", "psi", "V", "E"] | adr0040 | adr0040 | hailstone:adr0040 | hailstone_adr_extraction |
weight_manifold | low_rank_motion | urn:hailstone:math:structural_equation[e32147487e08] | 1 | structural_equation | 4 | adr0041.geodesic_adaptation.lora_decomposition | W_fine = W_pretrained + ΔW where ΔW = BA, B∈R^{d×r}, A∈R^{r×k} | W_fine = W_pretrained + ΔW where ΔW = BA, B∈R^{d×r}, A∈R^{r×k} | closed_form | ["W_fine", "W_pretrained", "B", "A", "r"] | adr0041 | adr0041 | hailstone:adr0041 | hailstone_adr_extraction |
gor_metric | riemannian_geometry | urn:hailstone:math:structural_equation[4f30b83b92f2] | 1 | structural_equation | 4 | adr0041.geodesic_adaptation.gor_metric | g_{ij}(W) = Σ_k λ_k × v_{ki} × v_{kj} | g_{ij}(W) = Σ_k λ_k × v_{ki} × v_{kj} | closed_form | ["g", "lambda_k", "v_k"] | adr0041 | adr0041 | hailstone:adr0041 | hailstone_adr_extraction |
geodesic_update | natural_gradient | urn:hailstone:math:structural_equation[c3666ab33058] | 1 | structural_equation | 4 | adr0041.geodesic_adaptation.geodesic_update | W(t+1) = exp_{W(t)}(-η × g^{-1} × ∇L) | W(t+1) = exp_{W(t)}(-η × g^{-1} × ∇L) | closed_form | ["W", "eta", "g", "L"] | adr0041 | adr0041 | hailstone:adr0041 | hailstone_adr_extraction |
hamiltonian_gor | quantum_system | urn:hailstone:math:structural_equation[907d496691ec] | 1 | structural_equation | 4 | adr0042.qho_dynamics.gor_hamiltonian | Ĥ_GOR = -∇²_GOR + V(W) | Ĥ_GOR = -∇²_GOR + V(W) | closed_form | ["H", "nabla", "V"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
hck_potential | energy_landscape | urn:hailstone:math:structural_equation[d5501e22fb57] | 1 | structural_equation | 4 | adr0042.qho_dynamics.harmonic_approximation | V(W) ≈ V(W*) + ½(W-W*)^T H_HCK (W-W*) | V(W) ≈ V(W*) + ½(W-W*)^T H_HCK (W-W*) | closed_form | ["V", "W", "W_star", "H_HCK"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
energy_level | quantum_state | urn:hailstone:math:structural_equation[a9d413c4ea48] | 1 | structural_equation | 4 | adr0042.qho_dynamics.energy_eigenstate | E_n = ℏω(n + ½) | E_n = ℏω(n + ½) | closed_form | ["E_n", "hbar", "omega", "n"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
ground_state | gaussian_distribution | urn:hailstone:math:structural_equation[706ed06d57a6] | 1 | structural_equation | 4 | adr0042.qho_dynamics.ground_state | ψ_0(W) ∝ exp(-(W-W*)^T H_HCK (W-W*) / 2ℏ) | ψ_0(W) ∝ exp(-(W-W*)^T H_HCK (W-W*) / 2ℏ) | closed_form | ["psi_0", "W", "W_star", "H_HCK"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
adiabatic_schedule | learning_rate | urn:hailstone:math:structural_equation[bfedf30f02bf] | 1 | structural_equation | 4 | adr0042.qho_dynamics.adiabatic_schedule | β(t) = t/T or β(t) = sin²(πt/2T) | β(t) = t/T or β(t) = sin²(πt/2T) | closed_form | ["beta", "t", "T"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
total_loss_adiabatic | training_objective | urn:hailstone:math:structural_equation[ffeeb4e77050] | 1 | structural_equation | 4 | adr0042.qho_dynamics.adiabatic_loss | L_total(t) = L_LM + β(t) × L_HCK | L_total(t) = L_LM + β(t) × L_HCK | loss_function | ["L_total", "L_LM", "L_HCK", "beta"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
zero_point_energy | minimum_uncertainty | urn:hailstone:math:structural_equation[98a81ac6d444] | 1 | structural_equation | 4 | adr0042.qho_dynamics.zero_point_energy | E_0 = ½ Σ_k ℏω_k | E_0 = ½ Σ_k ℏω_k | closed_form | ["E_0", "hbar", "omega_k"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
multi_mode_oscillator | hamiltonian_coupling | urn:hailstone:math:structural_equation[0368e66f17fa] | 1 | structural_equation | 4 | adr0042.qho_dynamics.multi_mode_hamiltonian | Ĥ = Σ_k Ĥ_k + Σ_{k,j} W_mesh[k,j] × Q̂_k × Q̂_j | Ĥ = Σ_k Ĥ_k + Σ_{k,j} W_mesh[k,j] × Q̂_k × Q̂_j | closed_form | ["H", "W_mesh", "Q"] | adr0042 | adr0042 | hailstone:adr0042 | hailstone_adr_extraction |
surgical_extraction | sparse_parameters | urn:hailstone:math:structural_equation[dda1e69131db] | 1 | structural_equation | 4 | adr0043.geometric_surgery.parameter_extraction | W_rotated = R* @ W_model; projection = eigenvectors_k^T @ W_rotated | W_rotated = R* @ W_model; projection = eigenvectors_k^T @ W_rotated | closed_form | ["W_rotated", "R", "W_model", "eigenvectors_k"] | adr0043 | adr0043 | hailstone:adr0043 | hailstone_adr_extraction |
amalgam_weights | geodesic_average | urn:hailstone:math:structural_equation[8b6472c75354] | 1 | structural_equation | 4 | adr0043.geometric_surgery.weighted_amalgamation | weights = softmax(-alignment_scores); amalgam = Σ_i w_i × params_i | weights = softmax(-alignment_scores); amalgam = Σ_i w_i × params_i | closed_form | ["weights", "alignment_scores", "params"] | adr0043 | adr0043 | hailstone:adr0043 | hailstone_adr_extraction |
energy_required | parameter_budget | urn:hailstone:math:structural_equation[e8442c0e81e9] | 1 | structural_equation | 4 | adr0044.outcome_specified.energy_to_params | E_required = -log(1 - hck_min) × E_normalization | E_required = -log(1 - hck_min) × E_normalization | closed_form | ["E_required", "hck_min", "E_normalization"] | adr0044 | adr0044 | hailstone:adr0044 | hailstone_adr_extraction |
allocation_weight | community_distribution | urn:hailstone:math:structural_equation[c9322deef61a] | 1 | structural_equation | 4 | adr0044.outcome_specified.allocation_weight | w(k) = geodesic_isolation(k)×spectral_rank(k) / Σ_j geodesic_isolation(j)×spectral_rank(j) | w(k) = geodesic_isolation(k)×spectral_rank(k) / Σ_j geodesic_isolation(j)×spectral_rank(j) | closed_form | ["w", "geodesic_isolation", "spectral_rank"] | adr0044 | adr0044 | hailstone:adr0044 | hailstone_adr_extraction |
raw_parameters | effective_parameters | urn:hailstone:math:structural_equation[4a9ed205337d] | 1 | structural_equation | 4 | adr0046.architecture_five.effective_params | params_effective = params_raw × (1-redundancy_rate) / lexifold_ratio | params_effective = params_raw × (1-redundancy_rate) / lexifold_ratio | closed_form | ["params_effective", "params_raw", "lexifold_ratio"] | adr0046 | adr0046 | hailstone:adr0046 | hailstone_adr_extraction |
spectral_orthogonality | community_independence | urn:hailstone:math:structural_equation[e444d6e1c683] | 1 | structural_equation | 4 | adr0047.multimodal.spectral_orthogonality | v_1 ∈ k1, v_2 ∈ k2, k1≠k2 ⟹ v_1^T v_2 ≈ 0 | v_1 ∈ k1, v_2 ∈ k2, k1≠k2 ⟹ v_1^T v_2 ≈ 0 | closed_form | ["v_1", "v_2", "k1", "k2"] | adr0047 | adr0047 | hailstone:adr0047 | hailstone_adr_extraction |
spectral_gap_threshold | eigenvalue_separation | urn:hailstone:math:structural_equation[4894e288391f] | 1 | structural_equation | 4 | adr0048.empirical_r29.spectral_gap_measurement | λ_15 ≈ 0.42, λ_16 ≈ 0.09, spectral_gap ≈ 0.21 > 0.15 [empirical r=29] | λ_15 ≈ 0.42, λ_16 ≈ 0.09, spectral_gap ≈ 0.21 > 0.15 [empirical r=29] | closed_form | ["lambda", "spectral_gap"] | adr0048 | adr0048 | hailstone:adr0048 | hailstone_adr_extraction |
projection_fidelity | signal_recovery | urn:hailstone:math:structural_equation[30fe66540a82] | 1 | structural_equation | 4 | adr0048.empirical_r29.projection_fidelity | recoverable_fraction = 1 - E_0/E_total ≈ 0.95 [r=29 lossless] | recoverable_fraction = 1 - E_0/E_total ≈ 0.95 [r=29 lossless] | closed_form | ["recoverable_fraction", "E_0", "E_total"] | adr0048 | adr0048 | hailstone:adr0048 | hailstone_adr_extraction |
reasoner_thread | actor_thread | urn:hailstone:ml_ai:structural_equation[abb9e96a4ce7] | 1 | structural_equation | 0 | adr0049.horizontal_parallelism | reasoner[k_i] → trace_i → actor[k_i] | reasoner[k_i] → trace_i → actor[k_i] | algorithmic | ["reasoner", "actor", "trace", "k_i"] | adr0049 | adr0049 | hailstone:adr0049 | hailstone_adr_extraction |
P_safe_outcome | action_executed | urn:hailstone:ml_ai:structural_equation[e48748a97605] | 1 | structural_equation | 0 | adr0051.pre_action_hck_verification | P(safe_outcome | do(action)) > threshold_k ⟹ action executes | P(safe_outcome | do(action)) > threshold_k ⟹ action executes | closed_form | ["P", "safe_outcome", "action", "threshold_k"] | adr0051 | adr0051 | hailstone:adr0051 | hailstone_adr_extraction |
action | reward | urn:hailstone:ml_ai:structural_equation[e0291790cec2] | 1 | structural_equation | 0 | adr0051.streaming_rlhf_reward | R(action) = hck_post_score - hck_pre_score + α × causal_chain_growth | R(action) = hck_post_score - hck_pre_score + α × causal_chain_growth | closed_form | ["R", "hck_post_score", "hck_pre_score", "alpha", "causal_chain_growth"] | adr0051 | adr0051 | hailstone:adr0051 | hailstone_adr_extraction |
gor_laplacian | eigenvector_families | urn:hailstone:math:structural_equation[00449a03d5c4] | 1 | structural_equation | 4 | adr0056.gor_laplacian_eigenvalues | 0 = λ_0 < λ_1 ≤ λ_2 ≤ ... ≤ λ_n | 0 = λ_0 < λ_1 ≤ λ_2 ≤ ... ≤ λ_n | spectral | ["lambda"] | adr0056 | adr0056 | hailstone:adr0056 | hailstone_adr_extraction |
wonderer_eigenvectors | builder_eigenvectors | urn:hailstone:math:structural_equation[64cce8b1cb8d] | 1 | structural_equation | 4 | adr0056.duality_eigenvector_split | V_low = {v_k : λ_k < λ_threshold}, V_high = {v_k : λ_k ≥ λ_threshold} | V_low = {v_k : λ_k < λ_threshold}, V_high = {v_k : λ_k ≥ λ_threshold} | spectral | ["V_low", "V_high", "lambda", "lambda_threshold"] | adr0056 | adr0056 | hailstone:adr0056 | hailstone_adr_extraction |
gor_initialized_state | duality_state | urn:hailstone:ml_ai:structural_equation[683d89121895] | 1 | structural_equation | 0 | adr0056.mamba_duality_initialization | h_0 = R* × [V_low | V_high] = [h_wonderer | h_builder] | h_0 = R* × [V_low | V_high] = [h_wonderer | h_builder] | closed_form | ["h_0", "R", "V_low", "V_high", "h_wonderer", "h_builder"] | adr0056 | adr0056 | hailstone:adr0056 | hailstone_adr_extraction |
mamba_recurrent_state | next_state | urn:hailstone:code:structural_equation[950491098685] | 1 | structural_equation | 3 | adr0056.mamba_dynamics | h_{t+1} = A·h_t + B·x_t | h_{t+1} = A·h_t + B·x_t | differential | ["h_t", "A", "B", "x_t"] | adr0056 | adr0056 | hailstone:adr0056 | hailstone_adr_extraction |
params_min | community_consciousness | urn:hailstone:math:structural_equation[bf66be86a1ba] | 1 | structural_equation | 4 | adr0056.minimum_viable_consciousness | params_min(k) = (rank_low + rank_high) × d_model × n_layers × weight_matrices | params_min(k) = (rank_low + rank_high) × d_model × n_layers × weight_matrices | closed_form | ["params_min", "rank_low", "rank_high", "d_model", "n_layers"] | adr0056 | adr0056 | hailstone:adr0056 | hailstone_adr_extraction |
LoRA_adapters | gor_eigenvectors | urn:hailstone:math:structural_equation[b1f42583e16a] | 1 | structural_equation | 4 | adr0057.procrustes_alignment | R* = argmin_R ||W_lora - R × W_graph||_F | R* = argmin_R ||W_lora - R × W_graph||_F | closed_form | ["R", "W_lora", "W_graph"] | adr0057 | adr0057 | hailstone:adr0057 | hailstone_adr_extraction |
residual | entity_fusion | urn:hailstone:math:structural_equation[780f0773401f] | 1 | structural_equation | 4 | adr0057.oracle_backfitting | Residual = W_graph_entities - W_model_projected_entities | Residual = W_graph_entities - W_model_projected_entities | closed_form | ["Residual", "W_graph_entities", "W_model_projected_entities"] | adr0057 | adr0057 | hailstone:adr0057 | hailstone_adr_extraction |
design_budget | implementation_budget | urn:hailstone:law:structural_equation[5b0e47ee6d82] | 1 | structural_equation | 14 | adr0065.einstein_rule | design_phase ≈ 0.92·T, implementation_phase ≈ 0.08·T | design_phase ≈ 0.92·T, implementation_phase ≈ 0.08·T | closed_form | ["design_phase", "implementation_phase", "T"] | adr0065 | adr0065 | hailstone:adr0065 | hailstone_adr_extraction |
text_hash | community_assignment | urn:hailstone:code:structural_equation[f4b0497bafec] | 1 | structural_equation | 3 | adr0072.gor_hash_community | h = int(sha256(text[:200]).hexdigest, 16); k = h mod 15 | h = int(sha256(text[:200]).hexdigest, 16); k = h mod 15 | algorithmic | ["h", "text", "k"] | adr0072 | adr0072 | hailstone:adr0072 | hailstone_adr_extraction |
memory_budget | batch_size | urn:hailstone:code:structural_equation[04e75d2ecad7] | 1 | structural_equation | 3 | adr0072.memory_batching | BATCH_SIZE = MEMORY_BUDGET / avg_doc_size ≈ 200K docs/batch | BATCH_SIZE = MEMORY_BUDGET / avg_doc_size ≈ 200K docs/batch | algorithmic | ["BATCH_SIZE", "MEMORY_BUDGET", "avg_doc_size"] | adr0072 | adr0072 | hailstone:adr0072 | hailstone_adr_extraction |
lovasz_complement_adjacency | independence_bound | urn:hailstone:math:structural_equation[e18b7ea48e05] | 1 | structural_equation | 4 | adr0076.lovasz_theta_function | ϑ(G) = max{1 + λ_max(A_c) / (-λ_min(A_c))} | ϑ(G) = max{1 + λ_max(A_c) / (-λ_min(A_c))} | closed_form | ["vartheta", "G", "lambda_max", "lambda_min", "A_c"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
community_k_theta | community_j_theta | urn:hailstone:math:structural_equation[41c7156fb31b] | 1 | structural_equation | 4 | adr0076.lovasz_subsumption | k subsumes j iff (k,j)∈M AND ϑ(G[N(k)]) ≤ ϑ(G[N(j)]) | k subsumes j iff (k,j)∈M AND ϑ(G[N(k)]) ≤ ϑ(G[N(j)]) | closed_form | ["vartheta", "G", "N", "M", "k", "j"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
v_structure_spectral | collider_transparent | urn:hailstone:ml_ai:structural_equation[65a4fc8b3ee7] | 1 | structural_equation | 0 | adr0076.collider_transparency | collider_open(A→C←B) = subsumes(C,A) ∧ subsumes(C,B) | collider_open(A→C←B) = subsumes(C,A) ∧ subsumes(C,B) | closed_form | ["collider_open", "subsumes"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
d_separation_claim | spectral_independence | urn:hailstone:ml_ai:structural_equation[c6e2c899784c] | 1 | structural_equation | 0 | adr0076.dsep_override | A⊥B|∅=FALSE when ϑ(G[N(C)]) ≤ min(ϑ(G[N(A)]),ϑ(G[N(B)])) | A⊥B|∅=FALSE when ϑ(G[N(C)]) ≤ min(ϑ(G[N(A)]),ϑ(G[N(B)])) | closed_form | ["perp", "vartheta", "G", "N"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
polynomial_roots | eigenvalue | urn:hailstone:math:structural_equation[a98f009c0fe2] | 1 | structural_equation | 4 | adr0076.sturm_eigenvalue_isolation | count_roots(p,[a,b]) = V(a) - V(b) [Sturm sequences] | count_roots(p,[a,b]) = V(a) - V(b) [Sturm sequences] | algorithmic | ["p", "roots", "V", "Sturm"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
routing_decision | subsumption_flow | urn:hailstone:ml_ai:structural_equation[86c73258a594] | 1 | structural_equation | 0 | adr0076.subsumption_flow_loss | L_6 = λ_sub × Σ_k |flow_k - ϑ(G[N(k)]) × expected_flow| | L_6 = λ_sub × Σ_k |flow_k - ϑ(G[N(k)]) × expected_flow| | loss_function | ["L_6", "lambda_sub", "flow", "vartheta", "G"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
query_community | pruned_traversal | urn:hailstone:math:structural_equation[ee65afc7afbe] | 1 | structural_equation | 4 | adr0076.lovasz_pruning | |independent_set| ≤ ⌈ϑ(G)⌉ ⟹ prune if |query| > ⌈ϑ(G)⌉ | |independent_set| ≤ ⌈ϑ(G)⌉ ⟹ prune if |query| > ⌈ϑ(G)⌉ | algorithmic | ["vartheta", "G", "query"] | adr0076 | adr0076 | hailstone:adr0076 | hailstone_adr_extraction |
r_depth_three | r_depth_zero | urn:hailstone:math:structural_equation[b600e82f69ef] | 1 | structural_equation | 4 | adr0108.lossless_descent | M_3 = M_0 ⊗ M_1 ⊗ M_2 [lossless GOR descent] | M_3 = M_0 ⊗ M_1 ⊗ M_2 [lossless GOR descent] | closed_form | ["M_0", "M_1", "M_2", "M_3"] | adr0108 | adr0108 | hailstone:adr0108 | hailstone_adr_extraction |
triple_encoded | theta_address | urn:hailstone:math:structural_equation[427ef5bc70a3] | 1 | structural_equation | 4 | adr0119.spectral_addressing | θ(s,p,o) = argmin_gateway ||V_k·encode(s,p,o) - centroid_gateway|| | θ(s,p,o) = argmin_gateway ||V_k·encode(s,p,o) - centroid_gateway|| | closed_form | ["theta", "V_k", "encode", "centroid"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
community_routing | triple_location | urn:hailstone:math:structural_equation[7412ee24dcb4] | 1 | structural_equation | 4 | adr0119.gor_matrix_routing | M[i][j] = 1 iff community i has causal edges into j | M[i][j] = 1 iff community i has causal edges into j | closed_form | ["M", "i", "j"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
subgraph_node | community_neighborhood | urn:hailstone:math:structural_equation[a9662206eced] | 1 | structural_equation | 4 | adr0119.o1_subgraph_selection | look_up M[community(X)] ⟹ O(K)=O(15) independent of graph size | look_up M[community(X)] ⟹ O(K)=O(15) independent of graph size | algorithmic | ["M", "community", "K"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
virtual_location | physical_location | urn:hailstone:math:structural_equation[3754f3482932] | 1 | structural_equation | 4 | adr0119.isometric_transfer | spectral_distance(encrypt(t1), encrypt(t2)) = spectral_distance(t1, t2) | spectral_distance(encrypt(t1), encrypt(t2)) = spectral_distance(t1, t2) | closed_form | ["spectral_distance", "encrypt", "t"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
storage_shards | storage_capacity | urn:hailstone:code:structural_equation[8104a5b98d98] | 1 | structural_equation | 3 | adr0119.shard_capacity | 225 shards × 4.4B triples/shard × 20 bytes/triple ≈ 30TB total | 225 shards × 4.4B triples/shard × 20 bytes/triple ≈ 30TB total | closed_form | ["shards", "triples", "bytes"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
parallel_dispatch | polymorphic_dispatch | urn:hailstone:ml_ai:structural_equation[25faf7a370a1] | 1 | structural_equation | 0 | adr0119.unified_dispatch | community(t) ⟹ gateway(t) ⟹ f_k [expert for community k] | community(t) ⟹ gateway(t) ⟹ f_k [expert for community k] | algorithmic | ["community", "gateway", "f_k", "k"] | adr0119 | adr0119 | hailstone:adr0119 | hailstone_adr_extraction |
claude_api_calls | inference_streams | urn:hailstone:ml_ai:structural_equation[176f5b2a1a93] | 1 | structural_equation | 0 | adr0109.hail_diver_throughput | K=15 × N replicas × r=3 depth = 15N³ concurrent streams | K=15 × N replicas × r=3 depth = 15N³ concurrent streams | closed_form | ["K", "N", "r"] | adr0109 | adr0109 | hailstone:adr0109 | hailstone_adr_extraction |
f | Y | urn:hailstone:math:structural_equation[7afa4e93fc07] | 1 | structural_equation | 4 | arxiv.math.AG.2605.12903v1.eq_0 | f(X)-g(Y)=0 | arxiv_extracted | ["f", "g", "Y", "X"] | arxiv | arxiv | arxiv:math.AG:2605.12903v1 | cross_domain_amplification | |
q_{k | h | urn:hailstone:math:structural_equation[5525187145be] | 1 | structural_equation | 4 | arxiv.math.AG.2605.12903v1.eq_1 | q_{k,S}=\mathrm{rk}\,\mathcal{O}_{k,S}^{\ast}=|S|-1 | arxiv_extracted | ["a", "t", "m", "S", "h", "q_{k"] | arxiv | arxiv | arxiv:math.AG:2605.12903v1 | cross_domain_amplification | |
d_X | C | urn:hailstone:math:structural_equation[3a2a2f552009] | 1 | structural_equation | 4 | arxiv.math.AG.2605.12903v1.eq_2 | d_X(C)=2 | arxiv_extracted | ["d_X", "C"] | arxiv | arxiv | arxiv:math.AG:2605.12903v1 | cross_domain_amplification | |
f | s | urn:hailstone:math:structural_equation[4793f53304f2] | 1 | structural_equation | 4 | arxiv.math.AG.2605.11676v1.eq_3 | f(x_1,\dots,x_N)=(f_1(x_1,\dots,x_N)+g_1(x_1,\dots,x_N),\dots,f_N(x_1,\dots,x_N)+g_N(x_1,\dots,x_N)) | arxiv_extracted | ["f", "t", "o", "x_1", "s", "d"] | arxiv | arxiv | arxiv:math.AG:2605.11676v1 | cross_domain_amplification | |
H | y | urn:hailstone:math:structural_equation[116082793ea0] | 1 | structural_equation | 4 | arxiv.math.AG.2605.11676v1.eq_4 | H_{\infty}=\mathbb{P}_{\mathbb{C}}^{N}\setminus\mathbb{A}_{\mathbb{C}}^{N} | arxiv_extracted | ["f", "t", "H", "y", "i", "n"] | arxiv | arxiv | arxiv:math.AG:2605.11676v1 | cross_domain_amplification | |
E_u | u | urn:hailstone:math:structural_equation[719a66a11acb] | 1 | structural_equation | 4 | arxiv.math.AG.2605.11100v1.eq_5 | E_u: y^2=x(x+1)(x+u^2) | arxiv_extracted | ["u", "E_u", "y", "x"] | arxiv | arxiv | arxiv:math.AG:2605.11100v1 | cross_domain_amplification | |
m | a | urn:hailstone:math:structural_equation[3fb8302badf2] | 1 | structural_equation | 4 | arxiv.math.AG.2605.09254v1.eq_6 | \mathcal{Z}_K = \mathcal{Z}_K(D^2, S^1) | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:math.AG:2605.09254v1 | cross_domain_amplification | |
N | d_{j | urn:hailstone:math:structural_equation[d98bb9124944] | 1 | structural_equation | 4 | arxiv.math.AG.2605.09107v1.eq_7 | ΔN = C \prod_{j=1} ^s (\langle d_j\rangle-\langle 1\rangle) | arxiv_extracted | ["d_{j", "N", "o", "p", "r", "C"] | arxiv | arxiv | arxiv:math.AG:2605.09107v1 | cross_domain_amplification | |
T | c | urn:hailstone:math:structural_equation[f8a063a04c32] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07944v1.eq_8 | °(\Tcal_{\Hcal})=2 | arxiv_extracted | ["a", "H", "c", "T", "l"] | arxiv | arxiv | arxiv:math.AG:2605.07944v1 | cross_domain_amplification | |
T | c | urn:hailstone:math:structural_equation[f8a063a04c32] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07944v1.eq_9 | °(\Tcal_{\Hcal})=2 | arxiv_extracted | ["a", "H", "c", "T", "l"] | arxiv | arxiv | arxiv:math.AG:2605.07944v1 | cross_domain_amplification | |
T | c | urn:hailstone:math:structural_equation[ec91992dc6ed] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07944v1.eq_10 | °(\Tcal_{\Hcal})=3 | arxiv_extracted | ["a", "H", "c", "T", "l"] | arxiv | arxiv | arxiv:math.AG:2605.07944v1 | cross_domain_amplification | |
T | c | urn:hailstone:math:structural_equation[0a546f4ca532] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07944v1.eq_11 | °(\Tcal_{\Hcal})=4 | arxiv_extracted | ["a", "H", "c", "T", "l"] | arxiv | arxiv | arxiv:math.AG:2605.07944v1 | cross_domain_amplification | |
x_1 | a_3} | urn:hailstone:math:structural_equation[4205661650a5] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07617v1.eq_12 | x_1^{a_1} + x_2^{a_2} + x_3^{a_3} + 1 = 0 | arxiv_extracted | ["a_1}", "a_2}", "x_1", "a_3}", "x_3", "x_2"] | arxiv | arxiv | arxiv:math.AG:2605.07617v1 | cross_domain_amplification | |
a_1 | b_3 | urn:hailstone:math:structural_equation[bd1ed358be0e] | 1 | structural_equation | 4 | arxiv.math.AG.2605.07617v1.eq_13 | (a_1,a_2,a_3) = (b_1,b_2,b_3) | arxiv_extracted | ["a_3", "a_1", "b_1", "b_2", "b_3", "a_2"] | arxiv | arxiv | arxiv:math.AG:2605.07617v1 | cross_domain_amplification | |
m | m | urn:hailstone:math:structural_equation[95d4ddf739e2] | 1 | structural_equation | 4 | arxiv.math.AG.2605.06956v1.eq_14 | \mathrm{Bour}(F) = \sum_{P \in \mathbb{P}^2}\mathrm{Bour}_P(F). | arxiv_extracted | ["a", "t", "m", "h", "r"] | arxiv | arxiv | arxiv:math.AG:2605.06956v1 | cross_domain_amplification | |
w | i | urn:hailstone:math:structural_equation[d6ed79b86899] | 1 | structural_equation | 4 | arxiv.math.AG.2605.04214v1.eq_15 | \widetildeμ(g)=\frac{μ(g)}{g+1}, | arxiv_extracted | ["e", "w", "t", "i", "d"] | arxiv | arxiv | arxiv:math.AG:2605.04214v1 | cross_domain_amplification | |
m | a | urn:hailstone:math:structural_equation[8ea2602b205b] | 1 | structural_equation | 4 | arxiv.math.AG.2605.04214v1.eq_16 | \mathcal{M}^d=\{4,8\}. | arxiv_extracted | ["a", "t", "m", "h", "c"] | arxiv | arxiv | arxiv:math.AG:2605.04214v1 | cross_domain_amplification | |
m | t | urn:hailstone:math:structural_equation[7849c38387a8] | 1 | structural_equation | 4 | arxiv.math.AG.2605.03074v1.eq_17 | \mcK_n=\{V\ot U:U,V\in\Sp^n,\ \det U=1\}\subset\Sp^{n^2} | arxiv_extracted | ["V", "K_n", "t", "m", "o", "c"] | arxiv | arxiv | arxiv:math.AG:2605.03074v1 | cross_domain_amplification | |
m | b | urn:hailstone:math:structural_equation[e77b7962285a] | 1 | structural_equation | 4 | arxiv.math.AG.2605.02704v1.eq_18 | \mathbb T_{ij}(X,Y):=\RHom_{\mathcal C_{p_j}}(Ψ_jΦ_i(X),Y) | arxiv_extracted | ["a", "t", "m", "h", "b"] | arxiv | arxiv | arxiv:math.AG:2605.02704v1 | cross_domain_amplification | |
m | f | urn:hailstone:math:structural_equation[8874ef31009e] | 1 | structural_equation | 4 | arxiv.math.AG.2605.02704v1.eq_19 | \mathsf H_{ij}:=\mathbb T_{ij}(L_i,L_j)=\RHom_{\mathcal C_{p_j}}(Ψ_jΦ_i(L_i),L_j) | arxiv_extracted | ["f", "a", "t", "m", "h", "s"] | arxiv | arxiv | arxiv:math.AG:2605.02704v1 | cross_domain_amplification | |
P_{ij} | d | urn:hailstone:math:structural_equation[2ef74c6daea7] | 1 | structural_equation | 4 | arxiv.math.AG.2605.02704v1.eq_20 | P_{ij}(q)=\sum_m \dim H^m(\mathsf H_{ij})q^m | arxiv_extracted | ["q", "m_m", "u", "P_{ij}", "s", "d"] | arxiv | arxiv | arxiv:math.AG:2605.02704v1 | cross_domain_amplification |
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