{"catalog":{"Q0m":{"title":"Quantum ground-state maximum","scenario":"Confine a quantum particle to an infinite triangular well. Select the most probable location in the positive ground state.","equation":"-\\Delta\\psi_j=\\lambda_j\\psi_j,\\quad \\psi_j|_{\\partial T}=0,\\qquad Q_{0m}=\\arg\\max\\psi_0","algorithm":"Use quadratic FEM eigenpairs and coordinate-weighted mass matrices. Search stationary points within quadratic elements for the maximum, then refine its location.","attract":"A candidate negative finite response appears only after substantial refinement; its small magnitude and maximum-location sensitivity leave the rigorous conclusion open.","location":"Unique interior maximum: log-concavity, interior analyticity and zero boundary trace give the line-restriction argument in Section 2. The Hardy/disk bound applies to expectations, not this maximum.","extra":"At equilateral the center is unambiguous. The thin correction has the ground-mode Airy scale.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":5393,"grid_rms":0.006989668679052202,"grid_max":0.017341890072097185,"holdout_rms":0.006898524010172857,"holdout_max":0.015176184840806344},{"X":5308,"grid_rms":0.007206985022398152,"grid_max":0.02458593374634215,"holdout_rms":0.00614069043058669,"holdout_max":0.013846492569885396},{"X":29678,"grid_rms":0.007224906360827073,"grid_max":0.02017050828673229,"holdout_rms":0.006128027009466168,"holdout_max":0.012585503917742333}],"alias":null,"group":"Quantum","jet":{"coefficients":[0.450075360277713,-0.16489111389745945,0.21659081996737717,-0.019317430042669878,-0.048517081338301966],"method":"Quadratic FEM fit n=64, step=0.02. Q3 second-order terms remain sensitive to step size. 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These are numerical counterexamples, not interval-certified continuum statements.","location":"Every nonzero state mean is strictly interior; $d_i(Q_j)\\geq r/(2\\zeta_j)$ gives the stronger inner-parallel-triangle bound.","extra":"At equilateral the center is unambiguous. The thin correction has the ground-mode Airy scale.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":24936,"grid_rms":0.007712981879967149,"grid_max":0.029861070259736236,"holdout_rms":0.006263470016826928,"holdout_max":0.014716603842372604},{"X":39622,"grid_rms":0.007770797890338905,"grid_max":0.018579797174263574,"holdout_rms":0.00756457857660505,"holdout_max":0.0163788133979142},{"X":27253,"grid_rms":0.007850324287348423,"grid_max":0.02140177741136998,"holdout_rms":0.007047343004344903,"holdout_max":0.01739546644845199}],"alias":null,"group":"Quantum","jet":{"coefficients":[0.4325688698107221,-0.13056391722730232,0.2340977968559016,-0.02065708154152989,-0.03663588610896229],"method":"Quadratic FEM fit n=64, step=0.02. Q3 second-order terms remain sensitive to step size. 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These are numerical counterexamples, not interval-certified continuum statements.","location":"Every nonzero state mean is strictly interior; $d_i(Q_j)\\geq r/(2\\zeta_j)$ gives the stronger inner-parallel-triangle bound.","extra":"At equilateral this state belongs to a doublet. Individual-state means are branch dependent; use a full eigenspace projector for a canonical alternative. Missing curve values are intentional.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":7324,"grid_rms":0.01700859519471177,"grid_max":0.05174502006084567,"holdout_rms":0.014209134377468922,"holdout_max":0.02689528833736367},{"X":21939,"grid_rms":0.020147559539660936,"grid_max":0.054693608788083214,"holdout_rms":0.016043828138289102,"holdout_max":0.03455675786028476},{"X":25061,"grid_rms":0.02083001531377844,"grid_max":0.0587356274445661,"holdout_rms":0.017120176237038348,"holdout_max":0.03681558963355832}],"alias":null,"group":"Quantum","jet":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. 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These are numerical counterexamples, not interval-certified continuum statements.","location":"Every nonzero state mean is strictly interior; $d_i(Q_j)\\geq r/(2\\zeta_j)$ gives the stronger inner-parallel-triangle bound.","extra":"At equilateral this state belongs to a doublet. Individual-state means are branch dependent; use a full eigenspace projector for a canonical alternative. Missing curve values are intentional.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":43441,"grid_rms":0.03907982693650521,"grid_max":0.11415413976078898,"holdout_rms":0.032810406515688284,"holdout_max":0.10823977848423973},{"X":43469,"grid_rms":0.039327113655833176,"grid_max":0.11592088858888237,"holdout_rms":0.0329800552351394,"holdout_max":0.10993339617796893},{"X":72043,"grid_rms":0.041376290574905564,"grid_max":0.10500788012780989,"holdout_rms":0.036341336258834234,"holdout_max":0.0996381085944491}],"alias":null,"group":"Quantum","jet":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. 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These are numerical counterexamples, not interval-certified continuum statements.","location":"Every nonzero state mean is strictly interior; $d_i(Q_j)\\geq r/(2\\zeta_j)$ gives the stronger inner-parallel-triangle bound.","extra":"Q3 is simple at equilateral and has a numerically negative horizontal response there (about -0.7206).","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":24762,"grid_rms":0.04908888560103144,"grid_max":0.14050752691708027,"holdout_rms":0.055389778665862734,"holdout_max":0.1343399965986753},{"X":43469,"grid_rms":0.04933222371673041,"grid_max":0.14232273093216877,"holdout_rms":0.04702261713386394,"holdout_max":0.13401468079811302},{"X":4179,"grid_rms":0.04935794618888616,"grid_max":0.1518220897103227,"holdout_rms":0.04842794878602099,"holdout_max":0.14366280444634366}],"alias":null,"group":"Quantum","jet":{"coefficients":[-0.7206092259204273,2.106382849877012,1.3872758925870106,-0.15494860962580215,0.7675576488455109],"method":"Quadratic FEM fit n=64, step=0.02. 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These are numerical counterexamples, not interval-certified continuum statements.","location":"Every nonzero state mean is strictly interior; $d_i(Q_j)\\geq r/(2\\zeta_j)$ gives the stronger inner-parallel-triangle bound.","extra":"At equilateral this state belongs to a doublet. Individual-state means are branch dependent; use a full eigenspace projector for a canonical alternative. Missing curve values are intentional.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","neighbors":[{"X":4179,"grid_rms":0.04687752927968589,"grid_max":0.1596728836155346,"holdout_rms":0.04855229686547982,"holdout_max":0.15228081161692691},{"X":50094,"grid_rms":0.046884614087340654,"grid_max":0.15183856772937102,"holdout_rms":0.05390801863740391,"holdout_max":0.14450898574691812},{"X":31323,"grid_rms":0.04698807925640606,"grid_max":0.1484541904291977,"holdout_rms":0.05259065274122794,"holdout_max":0.14123128500359994}],"alias":null,"group":"Quantum","jet":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. An individual eigenvector expectation depends on its basis and on the perturbation direction; use a whole-eigenspace projector for an invariant alternative."},"data_points":197},"E0":{"title":"Equal-exit conformal center","scenario":"Release Brownian particles from a common point. Choose that point so each side has first-exit probability one third.","equation":"E0=F(\\tfrac12+i\\tfrac{\\sqrt3}{2}),\\qquad h_1=h_2=h_3=1/3","algorithm":"Compute triangle angles and the normalized incomplete-beta map in high precision. The contour plots show squared harmonic-measure mismatch.","attract":"Not globally vertex-attractive: a new outward-rounded interval certificate encloses a finite-motion ratio near -3.13100322614e-7. This supersedes the previous 100-shape no-failure screen. See output/attractivity_candidates/E0_interval_certificate.json and its proof note.","location":"An exact certificate is $0<w_i(E0)<2/3$ for all three normalized barycentric coordinates.","extra":"Thin limit: $C_x\\to x$, $C_y/\\epsilon\\to2/3$ for fixed $x<1$; the endpoint $x=1$ instead has vertical limit $1/2$.","final_chapter":"quantum_motion_chapter","chapter_evidence":"output/quantum_motion_chapter/atlas_entries.json","attractivity_revision":"2026-09-30 candidate follow-up","attractivity_evidence":"output/attractivity_candidates/README.md","neighbors":[{"X":7707,"grid_rms":0.0004194665219683417,"grid_max":0.0005977383253846765,"holdout_rms":0.0004199468436422214,"holdout_max":0.0005919078210495285,"response_margin_rms_difference":0.0013550561441922637,"minimum_sampled_response_margin":0.00010007455447071795,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.0008154909923112706,"thin_position_rms":0.00017197449376382145},{"X":30421,"grid_rms":0.0008476802852312535,"grid_max":0.0018097987198373105,"holdout_rms":0.0008101558396176204,"holdout_max":0.0017316421428332418,"response_margin_rms_difference":0.003016069568881988,"minimum_sampled_response_margin":0.00010645927556586846,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.0011347979208922514,"thin_position_rms":0.0008738944637342708},{"X":483,"grid_rms":0.0008981947702360658,"grid_max":0.0014071111579089383,"holdout_rms":0.0009005834973102841,"holdout_max":0.0013607799685831671,"response_margin_rms_difference":0.0020517369558866565,"minimum_sampled_response_margin":0.00013980471010117262,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.0014342990535233778,"thin_position_rms":0.00023259545562463808}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.5419964574694885,-0.22989606024288095,0.12467368672619182,-0.06575727796898104,-0.05471461359718451],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"E1":{"title":"Inner conformal / least-capacity center","scenario":"Choose the point of greatest conformal radius, the local scale of a conformal disk centered at that point.","equation":"E1=F(z_*),\\quad z_*=\\frac{\\alpha+i\\sqrt{\\alpha\\beta/\\gamma}}{\\alpha+\\beta}","algorithm":"Evaluate the explicit upper-half-plane maximizer and the incomplete-beta map. Plot conformal radius, not exit-probability mismatch.","attract":"Not globally attractive: interval-certified negative finite vertex motion (normalized dot product about -0.02086913).","location":"Every side distance is at least half the inradius: the center belongs to the triangle I + (T-I)/2.","extra":"Thin correction: $C_x=x-\\epsilon\\operatorname{atanh}(x)/\\pi+O(\\epsilon^2)$ and $C_y/\\epsilon\\to1/2$, with $x<1$ fixed.","survey_date":"2026-09-30","evidence_file":"output/chapter_atlas_update/etc_comparison.json","neighbors":[{"X":53588,"grid_rms_D":0.0025363869741518233,"holdout_rms_D":0.0024643759459081764,"holdout_max_D":0.005309170657718634,"response_margin_rms_difference":0.008834200763318988,"candidate_minimum_screen_margin":-0.021086415876847127,"fingerprint_position_rms_D":0.0014576110583251229,"fingerprint_derivative_rms":0.00888637255386306,"fingerprint_finite_count":51,"fingerprint_total_count":51},{"X":66575,"grid_rms_D":0.002930253707443078,"holdout_rms_D":0.0029704705689963568,"holdout_max_D":0.006705275917415671,"response_margin_rms_difference":0.009989238000793581,"candidate_minimum_screen_margin":-0.019381019227795913,"fingerprint_position_rms_D":0.0019194656340047241,"fingerprint_derivative_rms":0.010927020291440189,"fingerprint_finite_count":24,"fingerprint_total_count":51},{"X":66560,"grid_rms_D":0.00403641949400804,"holdout_rms_D":0.003999028155933031,"holdout_max_D":0.0058250449908429655,"response_margin_rms_difference":0.01142312813015548,"candidate_minimum_screen_margin":-0.019970554343971637,"fingerprint_position_rms_D":0.0039094892197730725,"fingerprint_derivative_rms":0.01792175531354564,"fingerprint_finite_count":24,"fingerprint_total_count":51}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.46397840043406574,-0.17012494095942007,0.20269044355585486,-0.02807779249680663,-0.04734937133899039],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":197},"E2":{"title":"Outer conformal / planar charge centroid","scenario":"Let planar logarithmic charge equilibrate on the triangular conductor and measure its charge centroid.","equation":"\\mu_* =\\arg\\min_\\mu\\iint-\\log|p-q|\\,d\\mu(p)d\\mu(q),\\quad E2=\\int p\\,d\\mu_*","algorithm":"Use cosine-graded boundary panels with exact logarithmic segment integrals, constant-potential collocation and unit-charge normalization.","attract":"Not globally attractive: interval-certified negative finite vertex motion (normalized dot product about -0.04665096), independently evaluated with the exterior Schwarz-Christoffel map.","location":"Strictly inside the triangle: a positive nondegenerate measure has positive barycentric first moments.","extra":"S2 equals E2 exactly. On a collapsing segment the equilibrium density is arcsine and its horizontal mean is zero. The atlas shows boundary charge, since the surface potential is constant.","survey_date":"2026-09-30","evidence_file":"output/chapter_atlas_update/etc_comparison.json","neighbors":[{"X":59474,"grid_rms_D":0.0013452617409864403,"holdout_rms_D":0.0013054210228360564,"holdout_max_D":0.0024001020951110175,"response_margin_rms_difference":0.004442828954224595,"candidate_minimum_screen_margin":-0.006841424517391597,"fingerprint_position_rms_D":0.0015287809499537936,"fingerprint_derivative_rms":0.003483580151840221,"fingerprint_finite_count":51,"fingerprint_total_count":51},{"X":5235,"grid_rms_D":0.0022206686564592215,"holdout_rms_D":0.002139140215070677,"holdout_max_D":0.003501190322428843,"response_margin_rms_difference":0.007505017275688981,"candidate_minimum_screen_margin":-0.017254893401227406,"fingerprint_position_rms_D":0.0018094848743807313,"fingerprint_derivative_rms":0.005967549645439839,"fingerprint_finite_count":51,"fingerprint_total_count":51},{"X":49731,"grid_rms_D":0.002284721327808392,"holdout_rms_D":0.0021109045951434773,"holdout_max_D":0.003737905049161843,"response_margin_rms_difference":0.008156337500770387,"candidate_minimum_screen_margin":-0.02562098193913028,"fingerprint_position_rms_D":0.0018481440203625256,"fingerprint_derivative_rms":0.005619879123449922,"fingerprint_finite_count":51,"fingerprint_total_count":51}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.24697950629692242,0.09546064480470783,0.41968572120500064,0.027053426067011183,0.022802978642985465],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"E3":{"title":"Three-dimensional plate charge centroid","scenario":"Embed a thin triangular conducting plate in three-dimensional space and let a fixed total charge reach equilibrium.","equation":"\\mu_* =\\arg\\min_\\mu\\iint |p-q|^{-1}d\\mu(p)d\\mu(q),\\quad E3=\\int p\\,d\\mu_*","algorithm":"Use constant-charge triangular panels and exact self-panel potential integrals. Refine the panel subdivision and check charge positivity.","attract":"Refined numerical evidence of nonattractivity: negative finite motions persist on positive-charge anisotropic meshes up to 3,180 panels. The finest halved-step ratio is -0.55181093. A continuum interval certificate remains open.","location":"Strictly inside the triangle: a positive nondegenerate measure has positive barycentric first moments.","extra":"The thin longitudinal limit is a critical uniform-line asymptotic with horizontal mean zero; convergence can be logarithmically slow. Finite-panel densities near edges are illustrative, not pointwise certified singular coefficients.","survey_date":"2026-09-30","evidence_file":"output/chapter_atlas_update/etc_comparison.json","neighbors":[{"X":27760,"grid_rms_D":0.0026821102800564964,"holdout_rms_D":0.004359052081937216,"holdout_max_D":0.01939135824266107,"response_margin_rms_difference":0.024566574074417914,"candidate_minimum_screen_margin":0.07984583247056161,"fingerprint_position_rms_D":0.0023702789930032863,"fingerprint_derivative_rms":0.00583816212322058,"fingerprint_finite_count":51,"fingerprint_total_count":51},{"X":31351,"grid_rms_D":0.003796422822692561,"holdout_rms_D":0.004038399415802157,"holdout_max_D":0.010755689237454395,"response_margin_rms_difference":0.02056291836813349,"candidate_minimum_screen_margin":0.019387154257913323,"fingerprint_position_rms_D":0.0028658605050426826,"fingerprint_derivative_rms":0.012025881951676434,"fingerprint_finite_count":51,"fingerprint_total_count":51},{"X":46896,"grid_rms_D":0.005210820713927425,"holdout_rms_D":0.0079198007033783,"holdout_max_D":0.033446386232581754,"response_margin_rms_difference":0.03678979335238583,"candidate_minimum_screen_margin":0.14352373738742977,"fingerprint_position_rms_D":0.0024845125678255825,"fingerprint_derivative_rms":0.007819963231422766,"fingerprint_finite_count":51,"fingerprint_total_count":51}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.2923884279667316,0.05014965466232271,0.37427755630936227,0.010384053050314222,0.013255314267988219],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"E4":{"title":"Four-dimensional plate charge centroid","scenario":"Take the charge equilibrium of a spatially uniform thickening, then send thickness/radius to zero. This limiting convention is part of the definition.","equation":"E4=G=X(2)","algorithm":"Evaluate the exact area or perimeter first moment for the limiting center. The softened-kernel cutoff experiment is supplementary and is not a finite-radius physical conductor solve.","attract":"Strictly globally vertex-attractive through the exact uniform-thickness limiting identity with X2.","location":"The limit lies strictly inside every nondegenerate triangle.","extra":"The zero-thickness set itself has zero Newtonian capacity in this ambient dimension. No finite-thickness error bound is asserted. The plotted density is the limiting uniform density.","survey_date":"2026-09-30","neighbors":[{"X":2,"grid_rms":0,"holdout_rms":0}],"alias":"X2","group":"Charge & conformal","jet":{"coefficients":[0.3333333333333333,0,0.3333333333333333,0,0],"method":"Exact: uniform-area centroid."},"data_points":197},"S3":{"title":"3-dimensional skeleton charge centroid","scenario":"Take the charge equilibrium of a spatially uniform thickening, then send thickness/radius to zero. This limiting convention is part of the definition.","equation":"S3=S=X(10)","algorithm":"Evaluate the exact area or perimeter first moment for the limiting center. The softened-kernel cutoff experiment is supplementary and is not a finite-radius physical conductor solve.","attract":"Strictly globally vertex-attractive through the exact uniform-thickness limiting identity with X10.","location":"The limit lies strictly inside every nondegenerate triangle.","extra":"The zero-thickness set itself has zero Newtonian capacity in this ambient dimension. No finite-thickness error bound is asserted. The plotted density is the limiting uniform density.","survey_date":"2026-09-30","neighbors":[{"X":10,"grid_rms":0,"holdout_rms":0}],"alias":"X10","group":"Charge & conformal","jet":{"coefficients":[0.2499999895834174,0.10825312135221521,0.4166666215278109,0.0180421850237841,0.030070328657663868],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"data_points":197},"S4":{"title":"4-dimensional skeleton charge centroid","scenario":"Take the charge equilibrium of a spatially uniform thickening, then send thickness/radius to zero. This limiting convention is part of the definition.","equation":"S4=S=X(10)","algorithm":"Evaluate the exact area or perimeter first moment for the limiting center. The softened-kernel cutoff experiment is supplementary and is not a finite-radius physical conductor solve.","attract":"Strictly globally vertex-attractive through the exact uniform-thickness limiting identity with X10.","location":"The limit lies strictly inside every nondegenerate triangle.","extra":"The zero-thickness set itself has zero Newtonian capacity in this ambient dimension. No finite-thickness error bound is asserted. The plotted density is the limiting uniform density.","survey_date":"2026-09-30","neighbors":[{"X":10,"grid_rms":0,"holdout_rms":0}],"alias":"X10","group":"Charge & conformal","jet":{"coefficients":[0.2499999895834174,0.10825312135221521,0.4166666215278109,0.0180421850237841,0.030070328657663868],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"data_points":197},"M1":{"title":"Loop-field minimum","scenario":"Pass a uniform steady current around the triangular wire and scan the normal magnetic field inside its plane.","equation":"M1=\\arg\\min_{p\\in T}\\sum_e\\frac{s_e/\\sqrt{s_e^2+d_e^2}+(l_e-s_e)/\\sqrt{(l_e-s_e)^2+d_e^2}}{d_e}","algorithm":"Minimize in barycentric coordinates, refine the stationary root, and use interval Krawczyk inclusion for the retained certificate.","attract":"Not globally attractive: the earlier interval-certified negative finite vertex motion remains the rigorous witness.","location":"Unique interior minimum; every side distance is at least 2/b(p0) for any interior trial point p0, with b in the chapter normalization.","extra":"At equilateral the exact response is $\\operatorname{diag}(4/9,2/9)$. The center is implicitly algebraic, not proved transcendental. The plot uses log field to reveal contours despite boundary divergence.","survey_date":"2026-09-30","evidence_file":"output/chapter_atlas_update/etc_comparison.json","neighbors":[{"X":2534,"grid_rms_D":0.006862864934276052,"holdout_rms_D":0.006575769123106615,"holdout_max_D":0.014409757677661639,"response_margin_rms_difference":0.02434881676099824,"candidate_minimum_screen_margin":-0.07404562050198947,"fingerprint_position_rms_D":0.00386736398610017,"fingerprint_derivative_rms":0.027560357380990396,"fingerprint_finite_count":54,"fingerprint_total_count":54},{"X":5393,"grid_rms_D":0.007023617794160335,"holdout_rms_D":0.006918943287791021,"holdout_max_D":0.01579204290520356,"response_margin_rms_difference":0.02352933788338782,"candidate_minimum_screen_margin":-0.06960813564831762,"fingerprint_position_rms_D":0.0038460065025292347,"fingerprint_derivative_rms":0.02260361754183913,"fingerprint_finite_count":54,"fingerprint_total_count":54},{"X":53841,"grid_rms_D":0.00710349052532418,"holdout_rms_D":0.006688125654682187,"holdout_max_D":0.01389514343716808,"response_margin_rms_difference":0.026912282501120703,"candidate_minimum_screen_margin":-0.09044796423664005,"fingerprint_position_rms_D":0.00468567845658534,"fingerprint_derivative_rms":0.029922714156927866,"fingerprint_finite_count":54,"fingerprint_total_count":54}],"alias":null,"group":"Magnetism","jet":{"coefficients":[0.4444444444444444,-0.14255498255290477,0.2222222222222222,-0.024946977782326515,-0.03920284064307236],"method":"Exact linear response a=4/9, c=2/9 from field-Hessian differentiation in Q(sqrt(3)). Second-order terms use the numerical atlas fit."},"data_points":257},"F1":{"title":"Fluid torsion / linear membrane maximum","scenario":"Measure maximum axial velocity in a triangular Poiseuille duct, maximum deflection of a uniformly tensioned loaded membrane, or maximal Brownian mean exit time.","equation":"-\\Delta u=1,\\quad u|_{\\partial T}=0,\\quad F1=\\arg\\max_Tu","algorithm":"Solve the quadratic FEM Poisson system with exact load integration and maximize the piecewise quadratic field.","attract":"Negative numerical evidence, with unresolved refinement sensitivity. The inherited finite-element witness is negative at subdivisions 128, 192 and 256. Independent Ritz degrees 20 and 32 retain both negative steps, but degree 44 changes the half-step sign. The data do not supply a converged, validated continuum counterexample; global nonattractivity is not certified.","location":"The positive Dirichlet field has its maximum in the interior.","extra":"This is a maximum, whereas Abs(0) is the mean of the same field. In thin triangles the maximum approaches the widest section. The heat integral identity relates fields but does not identify their maxima. Also the shallow fixed-excess-area sheet minimum and the uniform-tension membrane maximum. Also the shallow fixed-excess-area sheet minimum and the uniform-tension membrane maximum. Also the shallow fixed-excess-area sheet minimum and the uniform-tension membrane maximum. Also the shallow fixed-excess-area sheet minimum and the uniform-tension membrane maximum. Also the shallow fixed-excess-area sheet minimum and the uniform-tension membrane maximum.","chapter":"Membranes, Hanging Sheets, and Elastic Triangle Centers","chapter_evidence":"output/transport_heat_chapter/atlas_entries.json","final_chapter":"transport_heat_chapter","neighbors":[{"X":5393,"grid_rms":0.004220117082630444,"grid_max":0.010134152434707626,"holdout_rms":0.004050249561423171,"holdout_max":0.008263252322823806,"response_margin_rms_difference":0.04934121047175154,"minimum_sampled_response_margin":-0.06928387606146469,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.01745862845149233,"thin_position_rms":0.004317131375535042},{"X":71701,"grid_rms":0.0050684159683964305,"grid_max":0.01150986095352962,"holdout_rms":0.005139957964080865,"holdout_max":0.010904098335580814,"response_margin_rms_difference":0.04671380942493814,"minimum_sampled_response_margin":-0.05417735573723775,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.012965038166575321,"thin_position_rms":0.005170532974824309},{"X":2534,"grid_rms":0.005463741589224744,"grid_max":0.00988840862430545,"holdout_rms":0.005135970092845671,"holdout_max":0.008469970570030476,"response_margin_rms_difference":0.05153125534250728,"minimum_sampled_response_margin":-0.07371551070130378,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.02127465937695333,"thin_position_rms":0.004036043425215249}],"alias":null,"group":"Heat & fluids","jet":{"coefficients":[0.4506177634361768,-0.17407243027539995,0.21605085820672154,-0.01453400202522458,-0.053176951563261486],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"H0.01":{"title":"Heat maximum at dimensionless time 0.01","scenario":"Start with uniform temperature one, hold the boundary at zero, and locate the hottest remaining point at the specified dimensionless time.","equation":"H(0.01)=\\arg\\max_T e^{-0.01L^2A}1,\\qquad A=-\\Delta_D","algorithm":"Use the Dirichlet spectral expansion with initial-data projection. Compare full and half spectra and refined meshes; factor out the ground exponential at late time.","attract":"The sampled heat-family finite witnesses have persistent negative responses on refined meshes; these are numerical, not interval-certified, counterexamples.","location":"Unique interior maximum at every positive time, by preservation of log-concavity and the analytic line-restriction argument. This supplies existence and uniqueness, not a quantitative distance-to-side bound.","extra":"The family endpoints are $H(0^+)=X(1)$ and $H(\\infty)=Q0m$. Keep $L^2$ shape dependent when differentiating. Late-time curve overlap with Q0m is expected, not an identity at finite time.","final_chapter":"transport_heat_chapter","chapter_evidence":"output/transport_heat_chapter/atlas_entries.json","neighbors":[{"X":29678,"grid_rms":0.00771364201507629,"grid_max":0.01996655097690984,"holdout_rms":0.0069850569359577955,"holdout_max":0.013671596491792663,"response_margin_rms_difference":0.04506752372105897,"minimum_sampled_response_margin":-0.19682973484629873,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.02412542035357327,"thin_position_rms":0.013262361017659385},{"X":5308,"grid_rms":0.007900528550607543,"grid_max":0.02438194661584766,"holdout_rms":0.007377190086835084,"holdout_max":0.01592025940856319,"response_margin_rms_difference":0.04719719094434722,"minimum_sampled_response_margin":-0.2167291133956094,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.020212884103340094,"thin_position_rms":0.015539398600804015},{"X":56032,"grid_rms":0.007951268645757441,"grid_max":0.026823626322179224,"holdout_rms":0.007290188639645206,"holdout_max":0.015467468098802646,"response_margin_rms_difference":0.04903136204526471,"minimum_sampled_response_margin":-0.2297363935377411,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.018369667886290207,"thin_position_rms":0.01674477601524415}],"alias":null,"group":"Heat & fluids","jet":{"coefficients":[0.4632710674843389,-0.22141500203910025,0.20339776508835902,-0.0018196206348366345,-0.07319462270849583],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"H0.03":{"title":"Heat maximum at dimensionless time 0.03","scenario":"Start with uniform temperature one, hold the boundary at zero, and locate the hottest remaining point at the specified dimensionless time.","equation":"H(0.03)=\\arg\\max_T e^{-0.03L^2A}1,\\qquad A=-\\Delta_D","algorithm":"Use the Dirichlet spectral expansion with initial-data projection. 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Late-time curve overlap with Q0m is expected, not an identity at finite time.","final_chapter":"transport_heat_chapter","chapter_evidence":"output/transport_heat_chapter/atlas_entries.json","neighbors":[{"X":29678,"grid_rms":0.006557514092528285,"grid_max":0.019966596621006052,"holdout_rms":0.005462775824540856,"holdout_max":0.01273742606182084,"response_margin_rms_difference":0.0439860820421901,"minimum_sampled_response_margin":-0.19682973484629873,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.011921803674325021,"thin_position_rms":0.013253534248710152},{"X":2534,"grid_rms":0.00672260518647853,"grid_max":0.017058805831382515,"holdout_rms":0.006628388371333169,"holdout_max":0.014977157791130112,"response_margin_rms_difference":0.03421001140325828,"minimum_sampled_response_margin":-0.07371551070130378,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.012788060093810671,"thin_position_rms":0.008064524978274867},{"X":5393,"grid_rms":0.006842578912285949,"grid_max":0.017304549641838876,"holdout_rms":0.006869446810743125,"holdout_max":0.015176184840807558,"response_margin_rms_difference":0.0322930951752681,"minimum_sampled_response_margin":-0.06928387606146469,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.009248898716461127,"thin_position_rms":0.008378044117049422}],"alias":null,"group":"Heat & fluids","jet":{"coefficients":[0.4432604866488906,-0.1518940235695238,0.22340801233793903,-0.01925181852624582,-0.04421194240567084],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"H0.1":{"title":"Heat maximum at dimensionless time 0.1","scenario":"Start with uniform temperature one, hold the boundary at zero, and locate the hottest remaining point at the specified dimensionless time.","equation":"H(0.1)=\\arg\\max_T e^{-0.1L^2A}1,\\qquad A=-\\Delta_D","algorithm":"Use the Dirichlet spectral expansion with initial-data projection. 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Late-time curve overlap with Q0m is expected, not an identity at finite time.","final_chapter":"transport_heat_chapter","chapter_evidence":"output/transport_heat_chapter/atlas_entries.json","neighbors":[{"X":5393,"grid_rms":0.0069870898655867725,"grid_max":0.01730454964181378,"holdout_rms":0.006990281919802155,"holdout_max":0.015176184840806227,"response_margin_rms_difference":0.03302763083610149,"minimum_sampled_response_margin":-0.06928387606146469,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.017265741686483812,"thin_position_rms":0.008375899475920087},{"X":5308,"grid_rms":0.007193506750439896,"grid_max":0.02438199226093633,"holdout_rms":0.0061474213947934064,"holdout_max":0.013754895979582807,"response_margin_rms_difference":0.046686060925797086,"minimum_sampled_response_margin":-0.2167291133956094,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.015551322036292508,"thin_position_rms":0.015547078492680404},{"X":29678,"grid_rms":0.007205519313243077,"grid_max":0.019966596621006052,"holdout_rms":0.0061338058702979665,"holdout_max":0.01273742606182084,"response_margin_rms_difference":0.04474028239329927,"minimum_sampled_response_margin":-0.19682973484629873,"family_derivative_evaluable_count":51,"family_derivative_count":51,"family_derivative_rms_difference":0.020148451816175934,"thin_position_rms":0.01325415068479415}],"alias":null,"group":"Heat & fluids","jet":{"coefficients":[0.4503431654486987,-0.16361414234639876,0.21632545161249428,-0.01952539930016972,-0.04802728025610315],"method":"Atlas fit; PDE subdivision 100, perturbation step 0.01; coefficients are numerical, not certified."},"data_points":198},"H0.3":{"title":"Heat maximum at dimensionless time 0.3","scenario":"Start with uniform temperature one, hold the boundary at zero, and locate the hottest remaining point at the specified dimensionless time.","equation":"H(0.3)=\\arg\\max_T e^{-0.3L^2A}1,\\qquad A=-\\Delta_D","algorithm":"Use the Dirichlet spectral expansion with initial-data projection. 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These endpoint statements do not identify an arbitrary finite-height center with either endpoint.","final_chapter":"optics_chapter","chapter_evidence":"output/optics_chapter/atlas_entries.json","attractivity_revision":"2026-09-30 candidate follow-up","attractivity_evidence":"output/attractivity_candidates/README.md","neighbors":[{"X":55753,"grid_rms":0.00010757212894874282,"grid_max":0.00014767063071644218,"holdout_rms":0.00011148179560757903,"holdout_max":0.0001480087564511311},{"X":60182,"grid_rms":0.00034499467670078037,"grid_max":0.0006491380884103209,"holdout_rms":0.0003561438415525037,"holdout_max":0.0006489839728796166},{"X":51127,"grid_rms":0.0014735455209968073,"grid_max":0.0022945070938999706,"holdout_rms":0.0015325231840267039,"holdout_max":0.0023010457705109587}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.35555481134108763,-0.026208401742712006,0.3111105223617084,-0.0061841395058150625,-0.007032799006267352],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"data_points":213},"Solid10.0":{"title":"Solid-angle illumination maximum, height/L=10","scenario":"Place an isotropic point source above the plane at a fixed dimensionless height and maximize the fraction of rays intercepted by the triangular receiver.","equation":"C=\\arg\\max_p\\int_T\\frac{10L}{(|q-p|^2+(10L)^2)^{3/2}}dq","algorithm":"Evaluate the exact triangle solid-angle atan2 formula and optimize in barycentric coordinates. 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These endpoint statements do not identify an arbitrary finite-height center with either endpoint.","final_chapter":"optics_chapter","chapter_evidence":"output/optics_chapter/atlas_entries.json","attractivity_revision":"2026-09-30 candidate follow-up","attractivity_evidence":"output/attractivity_candidates/README.md","neighbors":[{"X":2,"grid_rms":0.0001868887157042767,"grid_max":0.00027645865162355485,"holdout_rms":0.00019333892903333844,"holdout_max":0.0002768591014826787},{"X":51553,"grid_rms":0.0004229109377387533,"grid_max":0.0006059008093027116,"holdout_rms":0.00041229740246060683,"holdout_max":0.0005784338162048663},{"X":53004,"grid_rms":0.0004288886005011471,"grid_max":0.0008859353113765403,"holdout_rms":0.0004134235715531164,"holdout_max":0.0008503391955442503}],"alias":null,"group":"Charge & conformal","jet":{"coefficients":[0.33359779777533916,0.006093555775773866,0.3330425189940827,0.0009519111952041703,-0.0005652236691997108],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"data_points":213},"M2":{"title":"Vertex-current magnitude mean","attract":"Refined numerical evidence of nonattractivity: the order-256 finite-motion ratios are -1.61118595 and -1.61648215. 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Q3 second-order terms remain sensitive to step size. First-order maximum response refined at n=196; second-order terms from the coarser fit."},"Q0":{"coefficients":[0.4325688698107221,-0.13056391722730232,0.2340977968559016,-0.02065708154152989,-0.03663588610896229],"method":"Quadratic FEM fit n=64, step=0.02. Q3 second-order terms remain sensitive to step size. First-order terms replaced by simple-eigenvalue perturbation at n=88."},"Q3":{"coefficients":[-0.7206092259204273,2.106382849877012,1.3872758925870106,-0.15494860962580215,0.7675576488455109],"method":"Quadratic FEM fit n=64, step=0.02. Q3 second-order terms remain sensitive to step size. First-order terms replaced by simple-eigenvalue perturbation at n=88."},"Q1":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. An individual eigenvector expectation depends on its basis and on the perturbation direction; use a whole-eigenspace projector for an invariant alternative."},"Q2":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. An individual eigenvector expectation depends on its basis and on the perturbation direction; use a whole-eigenspace projector for an invariant alternative."},"Q4":{"coefficients":null,"method":"No unique Taylor expansion: the equilateral eigenvalue is degenerate. An individual eigenvector expectation depends on its basis and on the perturbation direction; use a whole-eigenspace projector for an invariant alternative."},"M2":{"coefficients":[0.4790589813122652,null,0.18760708036837936,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"M3":{"coefficients":[0.6089960581446398,null,0.05766945992661254,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"M4":{"coefficients":[0.23115107794671783,null,0.4355160137379577,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"M5":{"coefficients":[0.12104217077311258,null,0.545625380424053,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"M6":{"coefficients":[0.3427775056277957,null,0.32388912168704875,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"M7":{"coefficients":[0.36056109019520605,null,0.3061054630212223,null,null],"method":"Atlas finite difference, quadrature n=96, step=0.005; only linear terms available."},"L3":{"coefficients":[0.4666520252534223,null,0.20001832621503457,null,null],"method":"Prism boundary/frequency calculation n=20; only linear terms available."},"L4":{"coefficients":[0.46932478758169305,null,0.19734061025178384,null,null],"method":"Prism boundary/frequency calculation n=20; only linear terms available."},"Illum":{"coefficients":[0.4308450408123197,-0.121867554339951,0.23582203214984376,-0.02351357722485265,-0.032784698473342665],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"Solid0.1":{"coefficients":[0.4401720720443811,-0.13674333022869323,0.22649656243246297,-0.024013252905240937,-0.03779324050290442],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"Solid1.0":{"coefficients":[0.35555481134108763,-0.026208401742712006,0.3111105223617084,-0.0061841395058150625,-0.007032799006267352],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"Solid10.0":{"coefficients":[0.33359779777533916,0.006093555775773866,0.3330425189940827,0.0009519111952041703,-0.0005652236691997108],"method":"New numerical fit to illumination stationarity / solid-angle optimization, step 0.005; second-order optimizer coefficients are not certified."},"X66560":{"coefficients":null,"method":"No regular single-valued expansion established by this evaluation; the center function is singular at or near equilateral."},"X1":{"coefficients":[0.5000000208331562,-0.21650624269755156,0.1666667569443871,-0.03608437003087942,-0.06014065728476118],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X2":{"coefficients":[0.3333333333333333,0,0.3333333333333333,0,0],"method":"Exact: uniform-area centroid."},"X3":{"coefficients":[1.2757445562912771e-14,1.0503208545953313e-11,0.6666664444441391,0.28867528854561353,-0.09622501920545445],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X4":{"coefficients":[1.0000000000000162,-1.4143610685481934e-11,-0.33333288888829693,-0.5773505770995221,0.19245003841649166],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5":{"coefficients":[0.4999999999999985,1.6213220311137656e-12,0.1666667777779257,-0.144337644281117,0.04811250960135592],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X6":{"coefficients":[0.6666665185178341,-0.38489966626191136,3.7036998468130174e-07,-0.09622498071126805,-0.09622510901141618],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X7":{"coefficients":[0.6666665046294817,-0.24056236358410396,3.8425897648984065e-07,-0.16839376526245342,-0.0240563623576939],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X8":{"coefficients":[-4.166631218290146e-08,0.4330124853881035,0.6666664861112255,0.07216874005069647,0.12028131458621553],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X9":{"coefficients":[0.16666674768526854,0.1202811817885628,0.499999807870521,0.0841968826492757,0.012028181162201442],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X10":{"coefficients":[0.2499999895834174,0.10825312135221521,0.4166666215278109,0.0180421850237841,0.030070328657663868],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X11":{"coefficients":null,"method":"No regular single-valued expansion established by this evaluation; the center function is singular at or near equilateral."},"X12":{"coefficients":[0.5000000104166831,-0.1082532025416766,0.16666676736116565,-0.09021098683058963,-0.006014094136578957],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X13":{"coefficients":[0.6666665925924447,-0.1924499614329812,2.9629626530078926e-07,-0.19245010256518919,-7.698286455049445e-08],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X14":{"coefficients":null,"method":"No regular single-valued expansion established by this evaluation; the center function is singular at or near equilateral."},"X15":{"coefficients":[0.3333334444443599,-0.19245008972234437,0.3333332222221297,0.09622508976571198,-0.09622499995974777],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X16":{"coefficients":null,"method":"No regular single-valued expansion established by this evaluation; the center function is singular at or near equilateral."},"X17":{"coefficients":[0.5000000138888471,-0.1443375432408573,0.16666676388887836,-0.07216878486481643,-0.024056282466038092],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X18":{"coefficients":[-8.888935910756823e-07,0.5773521937297779,0.6666673333346929,-7.699395313277524e-08,0.19245007686076393],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X19":{"coefficients":[1.1666657893536536,-0.024056120903418313,-0.4999985671310745,-0.7096592807232155,0.22853406272114424],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X20":{"coefficients":[-1.000000000000014,2.1327047445025832e-11,1.6666657777765561,1.1547011541713574,-0.38490007680515886],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X66575":{"coefficients":null,"method":"No regular single-valued expansion established by this evaluation; the center function is singular at or near equilateral."},"X71701":{"coefficients":[0.4496384590682042,-0.15108471650371375,0.2170282851351085,-0.025180779408824402,-0.04196798329969107],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X58387":{"coefficients":[0.4062500397134882,-0.10825316193369293,0.26041667556427495,-0.009021092361996554,-0.03307735034762388],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X4166":{"coefficients":[0.2500001276042055,-0.018042306405801664,0.4166664835070495,0.08118986462581118,-0.03307730108979789],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X59464":{"coefficients":[0.4583333851995796,-0.16689029556305893,0.20833336480032433,-0.024808010501415654,-0.04736076061328608],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X59474":{"coefficients":[0.2500000000000471,0.09021093897259351,0.41666661111114345,0.027063283805653447,0.021049238266304723],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X4179":{"coefficients":[0.33333337847228456,-0.018042254546734474,0.3333332881945054,0.009021093771999434,-0.00902108335530926],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X20582":{"coefficients":[0.2500000370372175,0.09622491656369933,0.4166665740741681,0.02405624518477086,0.024056277242460646],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X17514":{"coefficients":[0.32954546067922924,-3.8679173802841935e-09,0.3371212034622239,0.0032803976202225747,-0.0010934629868094656],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X27760":{"coefficients":[0.29166665277779213,0.05412658021060834,0.37499998611111457,0.009021093605459683,0.0150351632227702],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5235":{"coefficients":[0.2499999994213514,0.09622500025562451,0.41666661168985647,0.024056248362778154,0.024056270220293895],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X50293":{"coefficients":[0.444444445987554,-0.13631873604097514,0.22222229475303998,-0.02806562128254531,-0.036084400437117956],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X29814":{"coefficients":[0.47619049971644506,-0.18557679418152712,0.19047626218816407,-0.030929459582379353,-0.05154913386983794],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X29823":{"coefficients":[0.4166666886573419,-0.10825314840787392,0.2500000335647805,-0.01804218591749197,-0.030070327053369945],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X38027":{"coefficients":[0.44444445486106227,-0.10825314841540057,0.22222228587964718,-0.042098452285302394,-0.022051583814511288],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X63628":{"coefficients":[0.41666670775468656,-0.09622507443373986,0.25000001446759373,-0.02405625827428184,-0.024056261008205997],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X64667":{"coefficients":[0.448717958209906,-0.1498889269978822,0.21794878537964554,-0.02498148577167447,-0.04163584022275882],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X7324":{"coefficients":[0.3000001329168165,-0.004330312039029093,0.36666651152795543,0.03103254451187665,-0.011787525103213767],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X33947":{"coefficients":[0.2500002864585167,-3.653573003595061e-07,0.41666632465330455,0.07216872289217763,-0.024056139782953935],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X24739":{"coefficients":[0.25000034375069385,0.03608376037109595,0.4166662673617626,0.0541265562592608,-0.006013939449199777],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5287":{"coefficients":[0.4696969970728919,-0.17714149753881336,0.19696976050269818,-0.02952357501805809,-0.049205990701455475],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X62648":{"coefficients":[0.3666666793518263,-0.047150259275096945,0.3000000095370322,-0.005292374233293336,-0.01395262923253093],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X24762":{"coefficients":[0.25000046875287946,0.18042013692526673,0.4166661423625016,-0.0180420635486998,0.06615470134476281],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5308":{"coefficients":[0.444444478909408,-0.14433754502774013,0.22222226183128202,-0.024056247205354993,-0.040093768502440286],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X41150":{"coefficients":[0.4270833450519681,-0.12178476151345202,0.23958338411455163,-0.020297458136469094,-0.033829119719379665],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X21712":{"coefficients":[-0.12499970572781709,0.5953913453758192,0.7916660668412688,0.09923199014718136,0.1653868563017585],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X21716":{"coefficients":[-0.12499999739287047,0.7397288856160291,0.7916663585068402,0.027063358679107863,0.23755548989037806],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X60637":{"coefficients":[0.04761884029868146,0.47130621395245653,0.6190476358922015,0.011782667735962036,0.1531744860589174],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X60643":{"coefficients":[0.2592592755171409,0.09503697184430657,0.40740734176708493,0.016631479360386604,0.026135202807488375],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X50409":{"coefficients":[0.32926829937535135,-4.250667872729002e-09,0.3373983645812702,0.003520426647431031,-0.0011734723997945812],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X32009":{"coefficients":[0.37500002669270194,-0.06314768427643086,0.29166666775173844,-0.004510546108825267,-0.01954570610084144],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5393":{"coefficients":[0.4389958114729923,-0.13725952850129358,0.2276709256352665,-0.022876579222613415,-0.03812764492460668],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X31000":{"coefficients":[-0.33333325925090185,1.15469803653457,0.9999994814800414,3.0788429770223563e-07,0.3849000062755804],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X24857":{"coefficients":[0.41666669849528826,-0.12028126848787893,0.2500000237268353,-0.012028123128051748,-0.03608438561008233],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X60728":{"coefficients":[0.2500000312501607,0.09622492859099305,0.41666657986120564,0.024056246200619358,0.024056276232152125],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X53588":{"coefficients":[0.45833337499995924,-0.16237973915735404,0.20833337499998955,-0.027063281787804094,-0.04510548975155724],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X32090":{"coefficients":[0.3928571884566284,-0.09278844505475424,0.27380951789258484,-0.005154909488143884,-0.02921116332431267],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X41312":{"coefficients":[0.3888889282406998,-0.08820628444598899,0.27777777546296883,-0.004009374299219703,-0.02806562684475421],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X51553":{"coefficients":[0.333333329903967,2.9622129637859227e-09,0.333333336762692,1.4183171333191689e-09,-1.8290852137094734e-09],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X19812":{"coefficients":[0.35964913625340167,-0.04178191008961385,0.30701754795707953,-0.0018991774680558514,-0.013294244580174624],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X360":{"coefficients":[0.6089976891559065,-0.2387322024280454,0.05766916128673484,-0.11936616353125169,-0.039788798125715105],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X24936":{"coefficients":[0.42857146598639906,-0.12371791261640809,0.23809526417232565,-0.020619642013270448,-0.03436608634332427],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X72043":{"coefficients":[0.38888887474285205,-0.04009376646899002,0.27777782896085096,-0.028065625545796495,-0.004009393525510097],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X367":{"coefficients":[0.45833337369783356,-0.1714008138028456,0.20833337630204057,-0.0225527309438624,-0.04961603422281368],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X30105":{"coefficients":[0.3888889112654155,-0.0721687823123663,0.27777779243826967,-0.012028125329069432,-0.020046883777982704],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X19870":{"coefficients":[0.3166666687500355,0.02165061344002119,0.34999998680556177,0.0036084373433812583,0.006014066386570773],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X19872":{"coefficients":[0.3235294105392398,0.01273566133677877,0.3431372495914979,0.0021226100099868547,0.003537685719374663],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X39331":{"coefficients":[0.33441246976753203,-0.0016452604376198505,0.3322541976186566,-0.00011192824999635907,-0.0005111110201685041],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X5550":{"coefficients":[0.363636367424196,-0.039364771401463654,0.3030303194444247,-0.006560794529864564,-0.010934664976248142],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X43441":{"coefficients":[0.3636363799983366,-0.03340044518282488,0.303030306870344,-0.00954297973697392,-0.00795248371042498],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X30129":{"coefficients":[0.4285714383502285,-0.12371784709002476,0.23809529180833022,-0.0206196404062225,-0.03436609027351368],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X21939":{"coefficients":[0.33333341145858264,-1.7139045961286846e-07,0.33333325520841933,-9.664484000244937e-09,1.2845287824029244e-08],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X30142":{"coefficients":[0.46153849605518377,-0.16654330974965817,0.20512825608150653,-0.027757207429752875,-0.046262041220042106],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X16831":{"coefficients":[0.3888889009773403,-0.0721687600335286,0.27777780272631786,-0.012028123480544358,-0.020046885010326525],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X55742":{"coefficients":[0.27027030157193077,0.06761751138013967,0.39639632305284583,0.0208054016626158,0.015604077180386679],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X55752":{"coefficients":[0.35336786820224386,-0.023931587018125478,0.31329881182073877,-0.005384610435297051,-0.006182337597402175],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X30153":{"coefficients":[-0.999992875014771,1.7320323325300937,1.66665865283158,0.2886747274111715,0.4811263246668424],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X55753":{"coefficients":[0.3553299412764458,-0.026376375027396988,0.311336740054559,-0.005861420290795316,-0.006838331370603873],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X43469":{"coefficients":[0.35833335034723807,-0.03367878252891082,0.30833333298611376,-0.004811250037252377,-0.009622504432139856],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. Singular or discontinuous cases are omitted."},"X62940":{"coefficients":[0.3529411667684107,-0.0219752396546766,0.3137255129701344,-0.005993251322984744,-0.005327342494905581],"method":"Numerical Taylor fit to the cached ETC center function, perturbation step 0.002. 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