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Research-Stack/4-Infrastructure/shim/rrc_equation_classifier_curriculum.jsonl
Brandon Schneider 38ddec024d Add RRC projection receipts and roadmap mirrors
2026-05-08 14:50:03 -05:00

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{"completion": {"projection_signature": {"shape_distance": 0.193428, "top_axes": [["projection_declared", 1.0], ["shape_closure", 0.63], ["witness_declared", 0.6], ["semantic_entropy", 0.479167]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "hardware_affinity", "history_depth", "negative_control_strength"]}, "receipt_hash": "43b52a1c1a45b0bb8a314112f9a08a739a74e6cc214ed857419b7120b51aab8b", "rrc_shape": "CognitiveLoadField", "status": "CANDIDATE"}, "prompt": {"equation": "L_threshold_hist = L_threshold_eff * exp(-rho_B * B_overflow)", "equation_id": "connectome_protective_cognitive_load_reweighting_receipt:core_equations:bandwidth_adjusted_threshold", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.229561, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "1fcbedc686a13186e3f68c56b3d31d1c2627b8a124864bf1b93dc4328ae71853", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "B_gate_emotional = exp(-gamma_emotional * DeltaE_emotional_hist / kT_emotional_hist)", "equation_id": "connectome_protective_cognitive_load_reweighting_receipt:core_equations:emotional_gate", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.230737, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5b2ba33fcc1aa76e85bfd29c319ddabf2c801e9f058b03f34378942b5567f4af", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "DeltaE_emotional_hist = DeltaE_emotional_eff + chi_B * B_overflow", "equation_id": "connectome_protective_cognitive_load_reweighting_receipt:core_equations:historical_emotional_barrier", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.290188, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.625], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "218df5724324d7c981f1ac73e6efca089fc375ebd3af528e1c93a6a4d9e22455", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "g = cos(\u0394\u03b8\u00b72\u03c0/16) \u00b7 cos(\u0394\u03c6\u00b7\u03c0/8) \u00b7 (1 - 2|\u03c7_i - \u03c7_j|)", "equation_id": "math_model_map:17", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.291006, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.645833], ["geometric_mass", 0.628571], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "da486cc353663b25d97693aae5ba6f9b4aa32341024d4efb8c3f090eaf0a7fca", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "|S_\u03bc| = 8\u00d78\u00d716\u00d716\u00d78\u00d78\u00d72\u00d716\u00d78\u00d74 = 8,589,934,592 \u2248 2^33", "equation_id": "math_model_map:30", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.289427, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.604167], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "978313fd4d1d82817b4987aa29151a0376cf699e71fd934a60439951f607172f", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "|S_total| \u2248 2^{5,900,000}", "equation_id": "math_model_map:32", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.246118, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["semantic_entropy", 0.59375], ["shape_closure", 0.59]], "weak_axes": ["topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2fd8e1819fc82014fff6a2cb052ac4d5b78cebbce50ff3954cf405a044da8974", "rrc_shape": "SignalShapedRouteCompiler", "status": "HOLD"}, "prompt": {"equation": "|S_reachable| \u2248 |S_total| / 10^{29}", "equation_id": "math_model_map:33", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288392, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "af590f387dfb29bda308e53c1d73ba7a3d55e821c88897928d5cab048e5a3a6d", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "P = (\u0394V, \u0394t, \u03c0, \u03c4, \u03c7, C, A)", "equation_id": "math_model_map:34", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288392, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8c87ddfc20e69d60b1bc8fd5a695ae8768306e1e451e6158ec9624fd1ad8107b", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "T_throat = {(i,j) | \u03a6_topo(i,j) >> \u03a6_metric(i,j)}", "equation_id": "math_model_map:35", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288076, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "4e61c9a2c9b8bd7e4fb3b0ddbb9fa9a45a2c6cbd4258fd6fd82d73764aed0bdf", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "w_ij = w_p \u00b7 w_\u03c0 \u00b7 w_\u03c4 \u00b7 w_\u03c7 \u00b7 w_topo \u00b7 w_\u03c3", "equation_id": "math_model_map:36", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288076, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "0e63345785cc5b350073c20c3de1ad34aba6a04370663fc7f53b55f1ba4a2851", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "Hol(\u03b3_loop) = \u222e_\u03b3 T(p) dp", "equation_id": "math_model_map:37", "task": "project_equation_via_rrc"}}
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{"completion": {"projection_signature": {"shape_distance": 0.258417, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["semantic_entropy", 0.59375], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "a2a9eb76072fcec04119b2a7932f0f745b03efef35863aefed18e81c52cab5cd", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "Windows: (3,5), (8,12), (16,20) \u03bcm; transmission=1.0 inside, exp(-dist\u00b70.5) outside", "equation_id": "math_model_map:69", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.257299, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.625], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "286c033a790b4f1f94b40ee8e16d36dbaa492f99aca2c2465a9819527099700b", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "\u03bd = 10\u2074/\u03bb [cm\u207b\u00b9]; DFB: \u00b17.5 cm\u207b\u00b9, EC: \u00b1200 cm\u207b\u00b9; \u03bb_min=10\u2074/\u03bd_max, \u03bb_max=10\u2074/\u03bd_min", "equation_id": "math_model_map:70", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.222839, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["compression_pressure", 0.533333]], "weak_axes": ["geometric_mass", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "4c88a751d5ffea677b72f7228d7250fc705783f346a987cd8b4db760b040b191", "rrc_shape": "SignalShapedRouteCompiler", "status": "HOLD"}, "prompt": {"equation": "MI(x) = baseline_bpb(x) - actual_bpb(x)", "equation_id": "math_model_map:71", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.282379, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.625], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "7b0b137dd804b4e6c11a846eefbeb715aab91f9e42b775afcb0a075d3b58dbc5", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "MI_pred = \u03a3_i (w_i \u00b7 MI_i \u00b7 S_i) / \u03a3_i (w_i \u00b7 S_i); w_i = 1/(d_i + \u03b5)", "equation_id": "math_model_map:72", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288392, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2357eaebdd3fd8cb9079886cf1327def9664f0efabfc2b5613514156be9cf7dd", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "surprise = log(1 + |MI_actual - MI_predicted|)", "equation_id": "math_model_map:73", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.245865, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.583333]], "weak_axes": ["geometric_mass", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "699153c1c31bc79342a9bc0ac1cb47758e9011c9db2cbf9f116517f437e34708", "rrc_shape": "SignalShapedRouteCompiler", "status": "HOLD"}, "prompt": {"equation": "\u03c1(x) = MI(x) / (cost(x) + \u03b5)", "equation_id": "math_model_map:74", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.281874, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.65625], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6bf5176d2cc3c31cb132c13188cf3adc0b3ce653be560a230448c62f743ffc81", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "d(z\u2081,z\u2082) = \u221a\u03a3_i w_i \u00b7 ((z\u2081_i - z\u2082_i) / s_i)\u00b2", "equation_id": "math_model_map:75", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.280052, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "583b13f2e4992fb2fbaacee7b0016a8257d771cbfd7a36432ee0d64520735916", "rrc_shape": "CadForceProbeReceipt", "status": "HOLD"}, "prompt": {"equation": "\u03a3 F_in = \u03a3 F_out at every node", "equation_id": "math_model_map:76", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.294433, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["semantic_entropy", 0.59375], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b1fd0f221b264e7e1730772ac71b661be7fa55385e6b051518d19e74b8e633a8", "rrc_shape": "CadForceProbeReceipt", "status": "HOLD"}, "prompt": {"equation": "\u03c3 = C : \u03b5; \u03b5 = \u00bd(\u2207u + \u2207u^T)", "equation_id": "math_model_map:77", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.2697, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.771429], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c12fda3eaf6f12c00e39e2b211cf32a75bdf148a0f8abc164388f3217685ea9d", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "V(G) = \u2227_{v\u2208V} (\u03a3F_in = \u03a3F_out)", "equation_id": "math_model_map:78", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288723, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5937379a24d00e4ff9984061a66536cb320035aa643f32be350b511545999192", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "cos = x \u00b7 x_ref / (\u2016x\u2016 \u00b7 \u2016x_ref\u2016)", "equation_id": "math_model_map:79", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288392, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2279300b812c7ddddafa063bbb82c459482d5984477f9526b7fbf81f81a24953", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "alignment = \u2207g_i \u00b7 \u2207g_j / (\u2016\u2207g_i\u2016 \u00b7 \u2016\u2207g_j\u2016)", "equation_id": "math_model_map:80", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288076, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f319772c36046078415dd5430a65eb5c2beedabeaa9edb59f6e108e5d9d4493d", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "phase += \u03a3 y \u00b7 dx", "equation_id": "math_model_map:81", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.290188, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.625], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "fb789ee71553ed410ba9705005bd3574b002cc9eeb106fdd0a9c5c2b783506c8", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "g_tt = M\u00b2, g_tp = 0, g_pp = (N\u00b7cos(\u03b8))\u00b2; a=C_eq/(2\u03c0), c=C_mer/(2\u03c0), f=(a-c)/a, e\u00b2=2f-f\u00b2, N=a/\u221a(1-e\u00b2sin\u00b2\u03b8), M=a(1-e\u00b2)/(1-e\u00b2sin\u00b2\u03b8)^(3/2)", "equation_id": "math_model_map:82", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287773, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "50bffeea9ce27d6b59131ae118ffc312de14583708d1f760a1ad8a5c550fd0d7", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "ds\u00b2 = g_tt\u00b7d\u03b8\u00b2 + 2\u00b7g_tp\u00b7d\u03b8\u00b7d\u03c6 + g_pp\u00b7d\u03c6\u00b2", "equation_id": "math_model_map:83", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.289427, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.604167], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "e4b7c532a3e75046db87636459bfa9f761b5ccef72f21d37de67663fcf063abe", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "Gamma^k_ij = \u00bd\u00b7g^kl\u00b7(\u2202_i g_jl + \u2202_j g_il - \u2202_l g_ij)", "equation_id": "math_model_map:84", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.290188, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.625], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "26899341e8d34fa22f740443bf87c050787e485ca6a78daf9b8f56392f01ecf8", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "a^\u03b8 = -(Gamma^\u03b8_tt\u00b7v_\u03b8\u00b2 + 2\u00b7Gamma^\u03b8_tp\u00b7v_\u03b8\u00b7v_\u03c6 + Gamma^\u03b8_pp\u00b7v_\u03c6\u00b2); v' = v + a\u00b7dt; x' = x + v'\u00b7dt", "equation_id": "math_model_map:85", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.290188, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["semantic_entropy", 0.625], ["witness_declared", 0.6]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "e2cc8c9bb833872a3f6f95cda2cd554a0a36d793dd1ee130dd191d26ac5753f9", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "u = 2\u00b7tan(\u03b8/2)\u00b7cos(\u03c6), v = 2\u00b7tan(\u03b8/2)\u00b7sin(\u03c6); \u03b8 = 2\u00b7atan(\u221a(u\u00b2+v\u00b2)/2), \u03c6 = atan2(v,u); select when |cos(\u03b8)| < 0.01", "equation_id": "math_model_map:86", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288392, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6bb555a669bad82ae5d1c668929aa67bb518e14d63246b72e8740ceaa4d922ef", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "D+D\u2192D, L+L\u2192L, D+L\u2192W(COLLAPSE\u2192W); chiralityToTernary: D\u2192Active, L\u2192Active, W\u2192Latent", "equation_id": "math_model_map:87", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.21559, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.697917], ["shape_closure", 0.63], ["witness_declared", 0.6]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength"]}, "receipt_hash": "2e8552e99f652db8237beea4bde835bbefacb9363a83dd0f41ba4ee0ff69579b", "rrc_shape": "CognitiveLoadField", "status": "CANDIDATE"}, "prompt": {"equation": "\u03c6 = (\u03c4 mod T)/T, T=942/1000; ternary: \u03c6<1/3\u2192Q, \u03c6<2/3\u2192A, else L; \u03c6' = (\u03c6+1/4) mod 1 if surprise>threshold", "equation_id": "math_model_map:88", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288076, "top_axes": [["projection_declared", 1.0], ["geometric_mass", 0.628571], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6a10a0d8a553ec61b5a89184e27bf2ea584b32a362750f016b528e7f1be5852a", "rrc_shape": "ProjectableGeometryTopology", "status": "HOLD"}, "prompt": {"equation": "Same acceleration as M85 + Verlet velocity update + chart transition check", "equation_id": "math_model_map:89", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.273853, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.614583], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8d1568106a8f7520ae1863fab42ed3885f3974bd97e38a46eaea1b86eac25dfc", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "risk(s) = (1 + \u03b3\u00b7(1-cos(\u03b8)))/d\u00b2 + \u03b7\u00b7h; \u03b3=3, \u03b7=0.8", "equation_id": "math_model_map:90", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.25882, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.583333]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "d7bb15aec563dc01d7ba7b0fc653af3e8dd71e301d46df4f9f3ab09b863e8836", "rrc_shape": "CognitiveLoadField", "status": "HOLD"}, "prompt": {"equation": "h' = \u03b1\u00b7h + \u03b2\u00b7a; \u03b1=0.95, \u03b2=0.2", "equation_id": "math_model_map:91", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "9434289633435ec35d6da5c7614184653246d2dc20201d1cb7d75c72c81056bd", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "a_\u03bc = (|g| - 2) / 2 = 0.001165920705(114)", "equation_id": "extracted_md:0", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "820adc2fb7a8153855c493adda824f2b534b37e8111cf904caf4555e998efb9b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03c3 = 0.127 ppm (0.000000114)", "equation_id": "extracted_md:1", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.289065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6046a6f88be65f3b5a69c362584ff039c771fede60385e995bf9bba2175ceeb0", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "a_\u03bc^theory = a_\u03bc^experiment within 0.5\u03c3", "equation_id": "extracted_md:2", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286858, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c821d0a37f32d268d8ca7ed07a9872b49314d32cce25306e0413e9a54b52abdf", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "g_\u03bc = -2.00233184122(82)", "equation_id": "extracted_md:3", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286858, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "3e1ff3fb15a91dd72996d429dc66b1efbfe98efc6148b62fc9226142f6d1b250", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03bc = g \u00b7 (e\u210f / 2m) \u00b7 S", "equation_id": "extracted_md:4", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f13e19b6762f0fdfbdde55aae9e40cdbb44703e5a87a0819c1a49f5c4fe46829", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "(D_t^\u03b1 + (-\u2207\u00b2)^\u03b2) \u03a8 = \u03bb |\u03a8|^\u03b3 \u03a8", "equation_id": "extracted_md:5", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "723166936077544dbde68e7c90889c3dc65c555cbc73f3209bf01422b55bf9e8", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "D^\u03b1 f(x) = F^{-1}[ |k|^\u03b1 \u00b7 F[f](k) ]", "equation_id": "extracted_md:6", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287271, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "bf2d4dcdfe099f5a1221eee16669cf33b88301d264844fdb831a95d36ed97e0e", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03a8_observed(x, t) = \u222b K_\u03b8(x - x') \u03a8_unified(x', t) dx'", "equation_id": "extracted_md:7", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "bf83dba12a3149cb7b373cc51f4f8e01c407ecf95044acc166aed78e81eccf5a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "w(\u03b1, \u03b8) = sin(\u03b8)^\u03b1 \u00b7 cos(\u03b8)^{1-\u03b1}", "equation_id": "extracted_md:8", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "87b8d3be5b2863c19d09790a4d25362ce47da8507864a7c50bb778f20e497c5d", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "g_n(\u03b8_obs) = g_0 \u00b7 sin(\u03b8_obs)^{1/n} \u00b7 cos(\u03b8_obs)^{1 - 1/n}", "equation_id": "extracted_md:9", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b7c558563b14a58620e82e1066031fd992796f6fcd1ca1916a5d83ee6822d1cd", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Q_n = (1/2\u03c0) \u222e_C \u2207_n \u03c6 \u00b7 dn = m/n for m \u2208 \u2124", "equation_id": "extracted_md:10", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "fb04a05581f4d0110aec8264a3f76aa693194b995055b6ff5d0f055e24e4767b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "ds\u00b2 = -d\u03b8\u00b2/\u03c9(\u03b8)\u00b2 + a(\u03b8)\u00b2 [dr\u00b2/(1-kr\u00b2) + r\u00b2 d\u03a9\u00b2] + \u2113_P\u00b2 d\u03b8\u00b2 \u0393(\u03b8)", "equation_id": "extracted_md:11", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "d0692883ba5beb46828be902f6de8909fe7d50e7be58053564c9f7e7678e1dd6", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "H_eff = \u03c9(\u03b8)", "equation_id": "extracted_md:12", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b49ad77d116fe1e4d31ba52bcc595bf8975a41c0028265361ce6e5dcfc55ce3a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03c1_DE(\u03b8) = \u03c1_foam \u00b7 (1 - \u03b8/\u03b8_max)\u00b2", "equation_id": "extracted_md:13", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "02077d7cbc6dba209f888ae509ae0ae834865c5a1ec65e7c624b85a0ad02bc72", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "S(R) \u2264 C' \u00b7 R^{D_H} \u00b7 T^{(D_H - 1)}", "equation_id": "extracted_md:14", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "977920ad1364a6ac997511acf26594c4501a362c3f1700654506fcba4dd185a1", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "S \u2264 C' \u00b7 R^{1.44} \u00b7 T^{0.44}", "equation_id": "extracted_md:15", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b894b0b740339fe04f5dfb040b46653e8dcbfa1a607b97fde11115b929c8daff", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "A(r) = 2\u03c0 (cosh(r) - 1) \u2248 \u03c0 \u00b7 exp(r) for r >> 1", "equation_id": "extracted_md:16", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f8b98d829c552f87d22f3a05646984a4e9f03a780dee7c5285e99d14c114a48f", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "L_{n+1} / L_n \u2248 exp(d_inj) \u2248 \u03a6\u00b2 \u2248 2.618", "equation_id": "extracted_md:17", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.28814, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "3d4a0a3264301b89b2b5d1fdc18de6151dab6b8cb19bc4069f9a096da0afaa0c", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "D_s = 2 D_H / (1 + D_H)", "equation_id": "extracted_md:18", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "270b734fc2aba4abb480cce5a841acccf50134f89d3e5c6764cced8a887c8ccd", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "D_s = 4/3 \u2248 1.333", "equation_id": "extracted_md:19", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ccee8a0dfc45270e15b2a346e817d61f890020693d3864544b8e06781480c1ad", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03c1(E) ~ E^{D_s/2 - 1} = E^{-1/3}", "equation_id": "extracted_md:20", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.289549, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8de47c80aca3e577a006919723d7ae34de6487f0b18a00b87cf4cb7c465dd32a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "E_n ~ n\u00b3", "equation_id": "extracted_md:21", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "620384c50857a63e7c4a1c8ae3321dfd74989f48aec7903de34d2889ea472362", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "n_s = 1 - 2/(1 + \u03b8_max/\u03b8_recombination) \u2248 0.965", "equation_id": "extracted_md:22", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287271, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b58bc6bf8ef3dc86a387cd355643c4e91089d051b23ebd63e8e8e388bbd20102", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "E_dissipated \u2265 k_B T \u00b7 ln(2) per bit erased", "equation_id": "extracted_md:23", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287271, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "509e4220562f4989a3d584ced23444b5f851af71ff25ad0eaee5dc6c9e108877", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "H(X) = -\u03a3 p(x) log p(x)", "equation_id": "extracted_md:24", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "d832958d6158607b2804c391c283519d309cd368a4d1b7bab791911d8c39cfe8", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "S \u2264 2\u03c0 R E / (\u210f c ln 2) = A / (4 G \u210f)", "equation_id": "extracted_md:25", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "31e1c43d5181798de3a068ada6b13f9a4d63206652b59b7dff673a1e8304fad0", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "C_V = (12\u03c0\u2074/5) N k_B (T/\u0398_D)\u00b3 \u221d T\u00b3", "equation_id": "extracted_md:26", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288596, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b2828c6913e2eb54e36f296dab61dfc8279ec0c0988a7142deddab8c953e3eb6", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "C_V \u221d T^{D_s}", "equation_id": "extracted_md:27", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288596, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f91d4d36242769d52d4bdb7a5ab1f055c9d097ce7c8df24c67c099b8df9ae3fe", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "C_V \u221d T^{1.18}", "equation_id": "extracted_md:28", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8927580859f5227425295b590601b2d7d1e344e36bda46cb2e74da96c98d90b4", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u27e8exp(-\u03b2 W)\u27e9 = exp(-\u03b2 \u0394F)", "equation_id": "extracted_md:29", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ee58ce129e44dfa94d0adc6465bbcac64dc919e4e632197b4848ad8c429d8ac1", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "(\u0394J)\u00b2 / \u27e8J\u27e9\u00b2 \u00b7 \u03c3 \u2265 2 k_B", "equation_id": "extracted_md:30", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "9fdbb18506fb7adcd3050fe8923e8d5d7a43b1409795dd16eee21c9aa31b3553", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u0394x \u00b7 \u0394p \u2265 \u210f/2", "equation_id": "extracted_md:31", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "0fa5a1523cc0b05a786c865918bb3371f9b0950cdd3cb7373f8e26efa4eec702", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03a8_total = \u03a8_A + \u03a8_B = A \u00b7 exp(i \u03c9_\u03a8 \u03b8) \u00b7 [exp(i k_\u03a8 x_A) + exp(i k_\u03a8 x_B)]", "equation_id": "extracted_md:32", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "733730562f92d3d87ee4fdfb29e9df74f2b0a0adc0a9197a4d5d69831e640536", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "|\u03a8_total|\u00b2 = 2|A|\u00b2 \u00b7 [1 + cos(k_\u03a8 (x_A - x_B))]", "equation_id": "extracted_md:33", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287271, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ed7d300c022f6fbb6190516bed43784c69b01781f47a4d927966d2b3deb410c1", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "E = \u210f \u03c9 = \u03c9_\u03a8 (in natural units \u210f = 1)", "equation_id": "extracted_md:34", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288596, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "e9627d15e5b48fd71b0f2afcd0cade14b268ba1450e9361d17e67e53369d74c9", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "p = \u210f k = d\u03b8_0/dx = k_\u03a8", "equation_id": "extracted_md:35", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.289065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8a9b46a64121eec6763b28790838109c860f5467eecfa807ba95219bdbf64958", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "\u03bb = 2\u03c0 / k_\u03a8 = 2\u03c0 / p", "equation_id": "extracted_md:36", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "186711871c51207179bc7dc122a8a4ce71d52731bf983d3d6a3cb537ef8e989a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Phenotype(x, t) = \u03a8_E [ Genotype(x) \u00d7 Regulatory_State(t) ]", "equation_id": "extracted_md:37", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.232964, "top_axes": [["projection_declared", 1.0], ["shape_closure", 0.64], ["decoder_declared", 0.6], ["witness_declared", 0.6]], "weak_axes": ["geometric_mass", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "cf30a90a927873ec56e26f5dc3d62bf33c8e9c7d6930d205a70a710c49652852", "rrc_shape": "SignalShapedRouteCompiler", "status": "HOLD"}, "prompt": {"equation": "Residual(n) = \u03a8_decode [ Basis, Context(n) ] XOR Byte(n)", "equation_id": "extracted_md:38", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "7b8c3a96925b3cdc9fe6a3349a77e60bc6ee166199bfb6292c3b2a2eb8a4b5f0", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "H_\u03a8(data) = -\u03a3_n p(n) log_2 p_\u03a8(n) \u2264 H_uniform(data) = 8 bits/byte", "equation_id": "extracted_md:39", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ac5d92474fa932d45e99aaf09161cdaa41a9028ac6c6e7b3b4206bdc61030308", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "G = (V, E) represents a quantum state |\u03c8\u27e9 through a\ncollection of local tensors {Tv }v\u2208V , one for each vertex", "equation_id": "eq_21f33f4110064dbd", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ea800e23de7ceddb38f1622e12f8417ea01cf44b28b43ec0b34b2f863fe343ed", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "injec= CD and Hv \u223c\ntive if this map has trivial kernel", "equation_id": "eq_14d74623112c018f", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.265908, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ff0d46411960c99d581c302547de341e46ea6d963bbbbc522b2582b7df3552be", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "d \u2265 D|N (v)| , which generically holds after coarse-graining\nwhen D = O(1)", "equation_id": "eq_fcd40dc2de7c7324", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "31da9991424da41b45b73a144b363be19c56a6053ef9e7ecaff91fceb27928b0", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "min \u2265 \u03b4", "equation_id": "eq_416bbe45bfd12c68", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6e94370b8c903d8d952a4cec40757a5f24e4a7538c29e7b72b30f6c999017420", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9", "equation_id": "eq_290815ff34ff74a2", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "eb82ce56c1700190e6c9e6a55ca6d2267671ce2f7932ee484a0c2d8a26a943b5", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, n) of\nthe graph, the tensor Tv \u22c6 T\u0304v of the norm network defines\na superoperator on the virtual space from any set of legs\nto its complement Eq", "equation_id": "eq_5ea587928ed366f5", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "89343fde1c5afae10e79b25cde1c21d595882c272b073f0be07d24d7299b6ce4", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = ZBP \uf8ed1 +\nZ\u2113 \uf8f8", "equation_id": "eq_e1646b712978905e", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "85b74ce61f591a2d0ec08e41b69b35c7142fca7ac2a5b9819d59544484c37037", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = log ZBP +\n\u03d5(W)ZW ,\n(7)\nconnected W\n\nwhere a cluster is collection of loops with multiplicities,\nW = {(\u21131 , \u03b11 ), (\u21132 , \u03b12 ),", "equation_id": "eq_ed91aac060fde4cf", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.280384, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ffb50cb92661f1282f30cdd89752fbf609429f9141ee12206aa547465f9e5c91", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "i \u2265 1 is its multiplicity in the cluster", "equation_id": "eq_9b66cadab83a4e1e", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2efdb15aff690ae9174b012709da7bf039fcb510a7cbe0eca14d823cba932272", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "ZW = i Z\u2113\u03b1ii and the\nP\nweight of a cluster is |W| := i \u03b1i |\u2113i | where |\u2113| for a loop\n\u2113 \u2208 L denotes the number of edges in \u2113", "equation_id": "eq_6fd0d43c1e1577a1", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "7152eaff8affcc4281af20854b12d2f57c0a8207f5141f81bbda9d69bcc4fa4a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "c > c0 =\nO(log \u2206)", "equation_id": "eq_8605b7c5ccdbcf5f", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283511, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.572917]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8afa195d251266167aae5d398e00c6a8bb5371680e9870c8c93363ec8b8ee8ce", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9 to 1/ poly(N ) multiplicative\nerror\n(ii) local observables \u27e8OA \u27e9 = \u27e8\u03c8|OA |\u03c8\u27e9/\u27e8\u03c8|\u03c8\u27e9 to\n1/ poly(N ) multiplicative error, given \u27e8OA \u27e9 \u0338= 0\n(iii) correlation functions \u27e8OA OB \u27e9 \u2212 \u27e8OA \u27e9 \u27e8OB \u27e9 to\nO\u0303(1/ poly(N )) additive error\nwhere A, B \u2282 V are disjoint local regions", "equation_id": "eq_dbd7ee74be207cc7", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "44effbc7d62a598c897faedd9f562e9c42f3e99767721fed573d3e7751eb069b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, n) with d(v, A) =\nr, we have\n\u0010\n\u0011\n\u2225\u00b5\u2032\u22c6,\u20d7e \u2212 \u00b5\u22c6,\u20d7e \u22251 = O e\u2212r/\u03be\u2217\n(10)\n\n\f5\nWhere 1/\u03be\u2217 = O(log \u03b5\u2217 /\u03b5)", "equation_id": "eq_9138cc21d6e4da27", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "75015934af99cbbd7c7bda525905053c38aeef136fdf483ab1baee199a28de4f", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "W = Wnear \u222a Wfar , where clusters in\nWnear are distance at most a cutoff Rth away from B", "equation_id": "eq_5be87eaf6cffbaae", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "329c6502c0823c5cf5db572511d4d6891240841c846f830a88e1b1bc46b5d0c4", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "t < r and\nstarts increasing thereafter, owing to the strict lightcone\nin the message-passing dynamics [see SM for a proof]", "equation_id": "eq_927277d74e6531cc", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "fcdf797b99db402de51a0195d73595963defba4c5c6b3a614d8b9608d6c89ba1", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "t = r\u22121, leading\nto\n\u0010\n\u0011\n\u2225\u00b5\u22c6,\u20d7e \u2212 \u00b5\u2032\u22c6,e \u22251 = O e\u2212r/\u03be\u2217\n(13)\nestablishing locality of the BP fixed-points", "equation_id": "eq_353043888999f4ac", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "29b3d3cd8f35c5908084976e831d9267669e7a777a0802c1663afb7b8f2ce584", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "d = c \u2212 c0 = O(1)", "equation_id": "eq_532f389442891ed2", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b61b83549338279d4c4adb4c5a93626ae38a827b4b701628a60cb0c2fd2d1968", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "R = \u0398(1)", "equation_id": "eq_fdfae5689dd4a9a5", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2badceabd488a2e2090127bf493dedbd573fa69f92534e2ccaf04a6d3026a860", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "X = 1}", "equation_id": "eq_9138aeb942929155", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283255, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.583333]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "758e17af3e072042f4572c1ce8eee674da8c6892cde49f1199166b26d85d691d", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "i=1\n\nFor p = \u221e we have,\n\u2225x\u2225\u221e := max |xi |\n1\u2264i\u2264m\n\n(S3)\n\nFor an operator A \u2208 B(H), we define the operator Schatten norm in terms of its singular values \u03c3(A) as,\n\u2225A\u2225p := \u2225\u03c3(A)\u2225p\n\n(S4)\n\nfor any p \u2208 [1, \u221e]", "equation_id": "eq_0cd087779e7532a9", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f32748cb72d0238ecbae7d2c3da9c54c169451c3b170d2fbbf174c4b73570574", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "p \u2264 q \u2264 \u221e and any operator A, we have,\n\u2225A\u2225p \u2265 \u2225A\u2225q\n\n(S5)\n\n\u2225A\u22251 \u2265 \u2225A\u22252 \u2265 \u2225A\u2225\u221e\n\n(S6)\n\nIn particular,\n\n\f10\n2", "equation_id": "eq_5593e036fe6f7dd7", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283781, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.5625]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5f172d76bcc85f6f2c8797d6e018b6b797ad57b704828cda6bdf6b4317af3079", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "p \u2264 q \u2264 \u221e and any non-zero operator A,\n1\n\n1\n\n\u2225A\u2225p \u2264 rank(A) p \u2212 q \u2225A\u2225q\n\n(S7)\n\np\nrank(A)\u2225A\u2225\u221e\n\n(S8)\n\n\u2225Ax\u22252\n= \u03c3max (A)\nx\u0338=0 \u2225x\u22252\n\n(S9)\n\nIn particular,\n\u2225A\u22252 \u2264\nWe also note that, the \u221e\u2212norm obeys,\n\u2225A\u2225\u221e = sup\n\nwe will denote \u2225 \u00b7 \u2225\u221e by just \u2225 \u00b7 \u2225 for convenience", "equation_id": "eq_ea9b6d38a8a0c1b0", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "9b1519dfe4e81be5b4e8eb97a7e1360d257a8a42225817492b5e679a046914ef", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "B = Ac , reshape the tensor into a linear map\nT : HA \u2192 HB\nby grouping the indices in A and B into multi-indices a = (ia1 ,", "equation_id": "eq_e0537400fa963370", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287699, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5e2aa9f91b47f5f3a7204c65b75cb2367c1a3b871e3d207c0fc7ad885de3f099", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "b = (ib1 ,", "equation_id": "eq_23fc8ebaf9c8ae4a", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283511, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.572917]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "55f40198b0ef268596ca51e481023b890b488c996a15558791e19b85fefae8f2", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "q = 1 we have,\n\u0001\n| tr A\u2020 B | \u2264 \u2225A\u2225p \u2225B\u2225q\n\n(S12)\n\nGraphs For a graph G = (V, E), we define some useful notation", "equation_id": "eq_8b4895ac1c7ebca0", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "09f8d81bc79c3c37219afdef58392fdb33628e7307096d30bb863b06e49d5695", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = {v, w} \u2208 E, we will refer to it\u2019s directed versions \u20d7e = (v, w) and \u2190\ne = (w, v)", "equation_id": "eq_25d49db810143f41", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c988688aa9ea9d3dbc6d281e7849c0f15eee8659f2b151fd980983d30ff7330c", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = {v, w}, we define, for any u \u2208 V ,\nd(e, u) := min{d(v, u), d(w, u)}\nand similarly, for any A \u2282 V ,\nd(e, A) := min d(e, u)", "equation_id": "eq_5034975bc936419d", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "27ba4b17fe70105fa967b149f865c5dd831b68e2d301ed58a5fea3db8f31dda1", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "li=1 be finite-dimensional Hilbert spaces (the l virtual legs), and\nlet Hphys be the physical Hilbert space", "equation_id": "eq_ab6c35762bc86081", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5389fded63bd6e67c838e74d08fef56b7784b209bbfc79a383f08ea38c6d0e9e", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "i=1\n\nDefinition S1", "equation_id": "eq_60b71803ce04c1ba", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283511, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.572917]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "0d092cba889d317b62c4bc308b8269804f4a750d3e2a4a24988136a6bc86f116", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "i=1 Hi \u2212\u2192 Hphys from\n\nWe perform a singular value decomposition\nT = V \u03a3U \u2020 \u2208 Cnp \u00d7nv ,\n\n(S14)\n\nwhere np \u2265 nv by injectivity, where U \u2208 Cnv \u00d7nv and V :\u2208 Cnp \u00d7nv satisfy U \u2020 U = 1nv (unitarity) and V \u2020 V = 1nv\n(isometry), and \u03a3 = diag(\u03bbe ) \u2208 Cnv \u00d7nv collects the singular values", "equation_id": "eq_7d5beabf91af593e", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "08e3486cfb6bc395126c774bce551d2c0ef4f6bb9b6492d02c6e64d60c20e097", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = \u03b4 \u2208 (0, 1]", "equation_id": "eq_8daf48607920ba7b", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "3626d12afaa3089f4036363e0919b91c7d33e9a28b424907f47c7f082792559d", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = 1,\ne\n\ne\n\n(S15)\n\nWe call \u03b4 the injectivity parameter", "equation_id": "eq_3b521e5a6156cbc1", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "1471d21dd79182101f7fc66a6243a12a8bbe19889caf8896986e3f7a1477dd54", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "T = V U \u2020 itself is an\nisometry", "equation_id": "eq_7cf50458351623ec", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.288596, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b0a6a12afdbbc708227f19d1a6ba7d4f0bf33a2f7b29141c4c507138d8fa630b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "a = 1", "equation_id": "eq_b5bc1ffd90912fb1", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "79f63c48ff596104c605b52c5a2519023f3adfe50fa26585e66dc58c01d2a044", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Lc = dim(HLc )1L ,\nKa,L\u2192Lc Ka,L\u2192L\n(S18)\nc = dim(HL )1Lc ,\na\n\na\n\nwhich follow from the fact that U is a unitary", "equation_id": "eq_51e4f2d85951d30b", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "26c408a3c2d3dc114a311051f6ce4fd9a4947510c1a2c3fdfa221a45b2eb34e0", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Lc =\n[U\u0304 a ](i\u2032L ),(iLc ) [U a ](iL ),(iLc ) =\n\u03b4iL ,i\u2032L = dim(HLc )1L\na\n\na,iLc\n\niLc\n\nMoreover, when T is \u03b4\u2212injective with \u03b4 = 1 (i", "equation_id": "eq_51dad4a7954a91e6", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "3cc3f66aa1da3314dac2f6a29d5dd3c69c7a83f029c7e8253f29be10a678484a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "G = (V, E) with vertex set V and edge set E", "equation_id": "eq_10109834ac3ac602", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283255, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.583333]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6957a59f11a406de94859ae1d671eea0c89a9a773b5aea81ad4630db529effe9", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "G = (V, E) with uniform virtual\ndimension D and physical dimension dp is specified for each v \u2208 V by a tensor\nO\nTv :\nH(n,v) \u2192 Hphys \u223c\n= Cdp ,\nn\u2208N (v)\n\nwhere N (v) denotes the set of neighbors of v, and each virtual space H(n,v) \u223c\n= CD", "equation_id": "eq_6c7adf4f4e980afe", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "32d69a72eb5d1e87c16fa666cc787ab0438eda6d8fbf3eb3de238b667e28626c", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, w), we define a bond-space H\u20d7e \u223c\n= CD", "equation_id": "eq_dde05d0258ec36a5", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "69aadec231bdc41b6ba25290e3d26ecbef94bf3354644fdab10c718e40233fee", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, w) is a positive\noperator \u00b5\u20d7e \u2208 Pos(H\u20d7e ) representing the message from v to w", "equation_id": "eq_36e95d88ecbda2b0", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8cf99e737ab8c4ec3e4af775d465c88944a5a940bf94fbae365aeac02b218a62", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, n) \u2208 \u20d7\nE,\n\uf8f6\n\uf8eb\nO\n\u00b5(m,v) \uf8f8", "equation_id": "eq_25b43848714b6ceb", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5d5fbdcf840b0cef5c46dc49fd7b8bb6b8a04b8d69e96b76009e49928f44ae92", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e > 0 such that\nPEPS if for each directed edge \u20d7e \u2208 E,\nf\u20d7e (\u00b5\u22c6 ) = \u03bb\u20d7e \u00b5\u20d7e", "equation_id": "eq_139255306999d8d8", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c6450f574e62718b7cc23e8829ff21b38d745251ffacef3837d52fdffaab2071", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9\ncan be formally described as a Taylor series in terms of \u2018loops\u2019 on the network as described below", "equation_id": "eq_2a06ffe17253ce3c", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8524fbfbe618913c034e0e1a5a7cb1c60c0f9fa3c19a0c9d9090ca53b7492d8a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = ZBP \uf8ed1 +\nZl \uf8f8\n\n(S21)\n\n\u0393\u2282L\nl\u2208\u0393\n\u0393 finite, compatible\n\nwhere the sum runs over all finite sets \u0393 of mutually compatible loops", "equation_id": "eq_e598d78992b57382", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6acddfb16f6030a79e9205eac08f341597c743f06196925624df985b4d653b2d", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "W = {(\u21131 , \u03b11 ), (\u21132 , \u03b12 ),", "equation_id": "eq_14f71a50573b7fd0", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.287271, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6816c632531998f46388d39b007b8bcb04031b2cbe2c5c49ecf35a1f3b6a7f22", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "nW = i=1 \u03b1i", "equation_id": "eq_44bf0ed184c8c0f8", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "b59f99533ebb319b687a09773d2383cdd40e53a22fdc1569649f92bb5fac2d4f", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "W = {(\u21131 , \u03b11 ),", "equation_id": "eq_4c772de7743acc11", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5a034e3e1b9b047142db36b9c662af8e3af3132a3d72b6c7602829e6374def04", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "ZW =\n\nk\nY\n\nZ\u2113\u03b1ii", "equation_id": "eq_3fe8c2f5d04f57ff", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "78217ae50ced51b7b636d56bb3f9179a169faac49a93d3edbf5836d1d2786fc9", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "i=1\n\nWe call a cluster W connected if the interaction graph GW is connected, meaning there is a path between any two\nvertices in the interaction graph", "equation_id": "eq_1abc29f7b62428f6", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283781, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.5625]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "70cfd4deeaadfab6cc643ad42e3c0a90ecdc0741f16568687be3b92d0d22563b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "GW =\nPk\n(VW , EW ) has |VW | = i=1 \u03b1i vertices, with loop \u2113i corresponding to \u03b1i vertices", "equation_id": "eq_f4bc63354f272a96", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8be5322a1d59e4e2951fc63ff442814748c67512a8e58ace4f4323d28843797b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "W = {(l1 , \u03b11 ), (l2 , \u03b12 ),", "equation_id": "eq_2a47fee92e986d5e", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286858, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "4c62ae5ea9980f4f0d983cfe107fd62bc4dceb0fe9011751e102b82e7910b591", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "ZW =\nZl\u03b1i i", "equation_id": "eq_4a18868e412a28db", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "e03f389ce08e8879221255cddcee474f99e8f0219c9248138b1772000323cbfb", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = log ZBP +\n\u03d5(W)ZW ,\n\n(S25)\n\nconnected W\n\nwhere the sum runs over all connected clusters W", "equation_id": "eq_d14e04ca5ba01499", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "5698e30987c13f13c4a90915a8e0b1a8f0614f90d5cbcfb9af1932382727bff2", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "c > c0 := log(2e\u2206) + 12 such that\n|Zl | \u2264 e\u2212c|l|\n\n(S27)\n\nthen, the series for log Z converges absolutely", "equation_id": "eq_405936ddfb9bc98c", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283014, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["semantic_entropy", 0.59375], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c42c5119cddb3a98270f1976ec0a8128d79a8b7185380b24446545fddf686901", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Fm = log ZBP +\n\nX\n\n\u03d5(W)ZW ,\n\n(S28)\n\nconnected W\n|W|\u2264m\n\nis bounded by\n|log Z \u2212 Fm | \u2264 N e\u2212d(m+1)\nwhere d = c \u2212 c0 , \u2206 is the degree of the graph, and N is the number of vertices", "equation_id": "eq_6a72df8385b3a8fb", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285005, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "6bebf737792b75732cfafd0ce02baa397bf425c8f0c42a36be9d5fecd2117a0a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "c > c0 = log(2e\u2206) + 1/2", "equation_id": "eq_52e8897befaf699d", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "1f11f224ef810f5a7e76e5c4f77ffeee12c7e85063867f1f46b6e466ca813b1c", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "G = (V, E)", "equation_id": "eq_f05cdb2173e0a64b", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284364, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.541667]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "bba3f804941b888be94b46f857b2829fd68b6f74773ba47afdebc806e1338e5d", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "T = \u27e8\u03c8|\u03c8\u27e9 and\nA\nT = \u27e8\u03c8|OA |\u03c8\u27e9 respectively after suitable BP normalization", "equation_id": "eq_f93b08fedbc0a5af", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "37c41cdc8f918cedbd280a36c207a693eaca39146965718514d9158381e1797f", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9 and Z A = \u27e8\u03c8|OA |\u03c8\u27e9\nfor a local observable with \u27e8OA \u27e9 \u0338= 0", "equation_id": "eq_d0869069ce0fee51", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283014, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["semantic_entropy", 0.59375], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "0a7fc9355ad818198f9dd4583422a30f98cd43caf587a589a5d0f1003ff18387", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "m = \u27e8OA \u27e9BP \u00b7 exp\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\nconn W\u2190LA\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8f4\n\uf8fe\n\uf8f3supp(W)\u2229A\u0338=\u2205\n|W|\u2264m\n\nleads to a relative error \u03b4m = | \u27e8OA \u27e9 \u2212 \u27e8OA \u27e9m |/| \u27e8OA \u27e9 | bounded by\n\u0010\n\u0011\n\u03b4m \u2264 O |A|e\u2212(c\u2212c0 )(m+1)\n\n(S33)\n\nwhere d = c \u2212 c0 = O(1)", "equation_id": "eq_e2dad9249b00119f", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283781, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.5625]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "4d86b95fef402e299320e82e95ad79c8215f8505d2dbd64440e9734290556128", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "A = \u2202\u03bb log Z\u03bbA |\u03bb=0 = \u27e8OA \u27e9BP +\n\u03d5W Z W\n\u03b1l\n\u2212 \u27e8OA \u27e9BP", "equation_id": "eq_2b301ef1b6ca651f", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "aa9e138e99cba44fa622f53b37d0ffda3b931b2dcf658989f993151b14a5eee4", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9 and Z A = \u27e8\u03c8|OA |\u03c8\u27e9", "equation_id": "eq_f4aab7b26b55c832", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.28814, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "24ca0a37b5ce4963619f2b06989eaf6467670a26b0ea5d098074c81b8913e841", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "d = c \u2212 c0", "equation_id": "eq_1a6dee6dca2f084b", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286074, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "c35422d2bc3f1322f55a2413cb0b178de920d80af0d018416791cfcdaf49ec91", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "c =\n\nX\n\nA\n\u03d5W \u2202\u03bb ZW,\u03bb\n\nconn", "equation_id": "eq_0698ebc502ab24cc", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286858, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "563d963a08c32cf299cf5bc32ed42c97188735b2b17d7c4b69e2225d98387117", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "A = (A1 ,", "equation_id": "eq_2edfb68a37ab3d1f", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286858, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "d541c7b2ef543d84e2612a7ee172a13a46029a3b93297163d8553db103cd5fb4", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "degree \u2265 2 everywhere except in all regions Ai", "equation_id": "eq_7957e0f2cb86a72c", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284678, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "bd49b7c2a246bab22c25f599a8b22bf7a000276abc5f654796f7fced93990a60", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "c \u2264 O e\u2212d(A,B)/\u03be\nfor a finite correlation length \u03be \u2264 O((1/(c \u2212 c0 ))) and d(A, B) being the graph distance", "equation_id": "eq_10a508b9d741a6f6", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283511, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.572917]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "2f920bb852c44abd591b98d69e1f6c81bcc164922537aa8146ab9c66816a4a5b", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Z = \u27e8\u03c8|\u03c8\u27e9 can be computed to 1/ poly(N ) multiplicative error in poly(N ) time\n(ii) local observables \u27e8OA \u27e9 = \u27e8\u03c8|OA |\u03c8\u27e9/\u27e8\u03c8|\u03c8\u27e9 with \u27e8OA \u27e9 \u0338= 0 can be computed to multiplicative error \u03f5 in poly(1/\u03f5)\ntime\n(iii) 2-point correlation functions can be computed to additive error O\u0303(1/ poly(N )) in poly(N ) time\n\n\f18\nProof", "equation_id": "eq_fb10b15989eae47c", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.283255, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.583333]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "f9e4fbaf07b96a5b5adcb92f57027d2b5909be6e3b415d6da6f7f8aa94c56e6a", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "m \u2264 N e\u2212d(m+1)\n\n(S37)\n\nHence, to ensure log Z \u2212 F\u0303m < O(1/ poly(N )) we require m = \u2126(log N )", "equation_id": "eq_984644a6b0dbab3e", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285347, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "266d34bc34255770efc7792968a9c6897bdd640da81b731e0ae290b711f51768", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "m = \u2126(log N )", "equation_id": "eq_837e24a6c715af63", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "e9d2b1e05e24a44253a6bba021402450f29434878ff7b8556a8290ab86e49590", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "m\n\u2264 O e\u2212d(m+1) \u2264 \u03f5\n\u27e8OB \u27e9\n\n(S38)\n\nwhich can be ensured given m = O(log 1/\u03f5), again computable in poly 1/\u03f5 time", "equation_id": "eq_e54ad45a68cb6a93", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.282576, "top_axes": [["projection_declared", 1.0], ["semantic_entropy", 0.614583], ["witness_declared", 0.6], ["shape_closure", 0.59]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "1e2244bc8cabfa83a106ad2198da92d674fa455de0e5bb353bf91de3ccba2749", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "m = d(A, B) + O(log 1/\u03f5), leading to an error O m2 (|A| + |B|)e\u2212dm , which is\n(note that (|A| + |B|) = O(1),\n!\n\u0012\n\u00132\n\u0012\n\u0013\n1\n1\n2\n\u2212d[d(A,B)+log 1/\u03f5]\nO d(A, B) log\n\u00b7e\n= O log N \u00b7\n= O\u0303(1/ poly(N ))\n(S39)\n\u03f5\npoly(N )\nsince we already have d(A, B) = O(log N ) and we choose \u03f5 = 1/ poly(N )", "equation_id": "eq_21f71c6c28c71003", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "8e1f505732912d568416a7536b6d3f698c2d2ae578cb362336435df436577b1f", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Ka = D1L\u0338=i", "equation_id": "eq_717ace3c9565c03b", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.284065, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["semantic_entropy", 0.552083]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "35d1fb2df523e9386cd67c3e00a7bb4e704ea771eaed61953e2b85205b9eda73", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "a \u2265 \u03b4 2 > 0 and\n\n\u2020\na Ka Ka = D\n\nP\n\n(S43)\n\na\n\n1,\nX\n\n\u03bb2a Ka\u2020 Ka \u2ab0 \u03b4 2\n\nX\n\na\n\nKa\u2020 Ka = \u03b4 2 D 1,\n\na\n\nhence\nTr f (X) \u2265 \u03b4 2 D Tr(X) = \u03b4 2 D", "equation_id": "eq_ccca83f09f701b34", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.286459, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "38aeca630c95999201a73c7063db3bd1845139908a41915bbea262433649b3e5", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "e = (v, n) \u2208 E", "equation_id": "eq_bc5bda52965ac0f8", "task": "project_equation_via_rrc"}}
{"completion": {"projection_signature": {"shape_distance": 0.285704, "top_axes": [["projection_declared", 1.0], ["witness_declared", 0.6], ["shape_closure", 0.59], ["residual_risk", 0.54]], "weak_axes": ["geometric_mass", "compression_pressure", "topology_torsion", "receipt_density", "field_energy", "hardware_affinity", "history_depth", "negative_control_strength", "scale_band_declared"]}, "receipt_hash": "ee83fc87cfebe1d5bbe8ef177b7b502acc40ac5f8c4530287e6f67becdf90123", "rrc_shape": "HoldForUnlawfulOrUnderspecifiedShape", "status": "HOLD"}, "prompt": {"equation": "Ka = dim(HL ) L", "equation_id": "eq_d4a2926485bb8572", "task": "project_equation_via_rrc"}}
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