Datasets:

PureOne
/

Chisael-Nexythra

{
  "01_event_divisor": {
    "title": "Compiled helicity event scaffold",
    "generation_kind": "analytical_samples",
    "description": "Samples of an exponential helical-shear scaffold with a double initial zero, a simple zero at 0.18, and a double zero at 0.33, together with its energy.",
    "parameter_context": {
      "viscosity": 1,
      "carrier_wavenumbers": [
        1,
        2,
        3,
        4,
        5,
        6
      ],
      "event_times": [
        0,
        0.18,
        0.33
      ],
      "event_multiplicities": [
        2,
        1,
        2
      ],
      "anchor_time": 0.5,
      "anchor_helicity": 1,
      "overall_velocity_scale_delta": 1,
      "compiler_decimal_precision": 100,
      "sample_time_range": [
        0,
        0.8
      ],
      "sample_count": 901,
      "formula_helicity": "sum_k c_k exp(-2 nu k^2 t)",
      "formula_energy": "sum_k |c_k| exp(-2 nu k^2 t) / (2 k)"
    },
    "limitations": "Unscaled heat/shear scaffold: nonlinear advection vanishes, and initial energy is approximately 2612.2564. It is not a small-data nonlinear realization, shock simulation, or experiment. Arbitrary-precision values are exported as floating-point samples; small residuals can remain at nominal zeros.",
    "fields": {
      "time": "Nondimensional operational time t.",
      "helicity": "Signed normalized total helicity H(t), with H(0.5)=1 for the compiled scaffold.",
      "energy": "Nonnegative kinetic energy E(t)=one half of the normalized spatial average of |u|^2, at overall velocity scale delta=1."
    }
  },
  "02_scissor_carriers": {
    "title": "Signed coefficients of the six-carrier scaffold",
    "generation_kind": "analytical_samples",
    "description": "The six alternating signed helicity coefficients used in the event scaffold in 01_event_divisor.",
    "parameter_context": {
      "linked_table": "01_event_divisor",
      "viscosity": 1,
      "carrier_wavenumbers": [
        1,
        2,
        3,
        4,
        5,
        6
      ],
      "event_times": [
        0,
        0.18,
        0.33
      ],
      "event_multiplicities": [
        2,
        1,
        2
      ],
      "anchor_time": 0.5,
      "anchor_helicity": 1,
      "compiler_decimal_precision": 100,
      "coefficient_count": 6
    },
    "limitations": "Coefficients belong to the unscaled one-coordinate helical-shear scaffold, whose nonlinear advection is zero. Signed helicity coefficients are not velocity amplitudes; amplitudes are sqrt(|c_k|/k) before an overall scale. Floating-point CSV values approximate the arbitrary-precision solve.",
    "fields": {
      "wavenumber": "Positive integer Fourier carrier wavenumber k.",
      "coefficient": "Signed helicity coefficient c_k multiplying exp(-2 nu k^2 t); its sign selects carrier chirality."
    }
  },
  "03_triad_kernel": {
    "title": "Exact third mixed-amplitude triad response kernel",
    "generation_kind": "analytical_samples",
    "description": "Samples of K_nu(t)=exp(-6 nu t)[exp(4 nu t)-1-4 nu t]/(8 nu) for three viscosities. For the specified standing-wave triad, the third mixed amplitude derivative of helicity at zero amplitudes is K_nu(t) cos(Phi).",
    "parameter_context": {
      "viscosities": [
        0.2,
        1,
        3
      ],
      "sample_time_range": [
        0,
        0.8
      ],
      "sample_count": 901,
      "phase_factor": "cos(Phi) omitted; columns store K_nu itself",
      "source_formula": "src/make_figures.py and chapters/03_triad_clock.tex"
    },
    "limitations": "An exact response derivative evaluated numerically, not the entire helicity trajectory at finite amplitudes. Kernel positivity alone does not establish positivity of a full finite-amplitude trajectory. No physical experiment is represented.",
    "fields": {
      "time": "Nondimensional time t.",
      "nu_0p2": "K_nu(t) at viscosity nu=0.2.",
      "nu_1": "K_nu(t) at viscosity nu=1.",
      "nu_3": "K_nu(t) at viscosity nu=3."
    }
  },
  "04_phase_gate": {
    "title": "Triad phase dependence of initial helicity acceleration",
    "generation_kind": "analytical_samples",
    "description": "Samples of cos(Phi), the normalized initial acceleration H second derivative at zero divided by 2 nu a b c for the specified standing-wave triad.",
    "parameter_context": {
      "phase_range_radians": [
        "-pi",
        "pi"
      ],
      "sample_count": 501,
      "formula": "H second derivative at zero = 2 nu a b c cos(Phi)",
      "phase_definition": "Phi = phi_1 + phi_2 + phi_3, the closed-loop phase"
    },
    "limitations": "All viscosity and amplitude factors have been normalized away. The plotted cosine is not a finite-amplitude helicity time history; the stronger all-time quadrature extinction uses the symmetry argument in the manuscript.",
    "fields": {
      "phase": "Closed-loop interaction phase Phi in radians.",
      "normalized_acceleration": "cos(Phi), equal to H second derivative at zero / (2 nu a b c) when the denominator is nonzero."
    }
  },
  "05_quadratic_retuning": {
    "title": "Galerkin carrier-amplitude correction versus perturbation",
    "generation_kind": "galerkin_simulation",
    "description": "Three saved nonlinear root solves showing the Euclidean norm of the correction to the axial carrier amplitudes as the transverse perturbation parameter changes.",
    "parameter_context": {
      "viscosity": 1.0,
      "fourier_cutoff_K": 3,
      "quadratic_product_grid": [
        12,
        12,
        12
      ],
      "integrator": "classical fourth-order Runge-Kutta",
      "time_step": 0.001,
      "overall_velocity_scale_delta": 0.1,
      "carrier_wavenumbers": [
        1,
        2,
        3
      ],
      "event_times": [
        0.0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity_target": 0.01,
      "domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
      "provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
      "epsilon_values": [
        0.1,
        0.2,
        0.4
      ],
      "correction_definition": "Euclidean norm of a-a0 before multiplying velocities by delta; a_j=a0_j exp(theta_j)"
    },
    "limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. Three values support an illustrative quadratic scaling; they are not an independent proof of an asymptotic order.",
    "fields": {
      "epsilon": "Transverse parameter epsilon inside the velocity ansatz delta times [carrier field + epsilon transverse field]; actual transverse velocity coefficient is delta epsilon.",
      "amplitude_correction_norm": "Euclidean norm of corrected minus base carrier-amplitude vector before common velocity scaling by delta."
    }
  },
  "06_interaction_orders": {
    "title": "Initial nonlinear diagnostics and carrier retuning",
    "generation_kind": "galerkin_simulation",
    "description": "Raw initial norms of pressure-projected nonlinear interaction and vortex stretching, alongside carrier-amplitude retuning for the same three Galerkin root solves.",
    "parameter_context": {
      "viscosity": 1.0,
      "fourier_cutoff_K": 3,
      "quadratic_product_grid": [
        12,
        12,
        12
      ],
      "integrator": "classical fourth-order Runge-Kutta",
      "time_step": 0.001,
      "overall_velocity_scale_delta": 0.1,
      "carrier_wavenumbers": [
        1,
        2,
        3
      ],
      "event_times": [
        0.0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity_target": 0.01,
      "domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
      "provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
      "epsilon_values": [
        0.1,
        0.2,
        0.4
      ],
      "measurement_time": 0,
      "csv_values": "Raw diagnostics; figure alone divides some curves by their first values"
    },
    "limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. Initial norm trends illustrate first-order interaction and second-order retuning over three epsilon values; the analytical argument supplies the asymptotic claims.",
    "fields": {
      "epsilon": "Transverse parameter epsilon; common velocity scale delta=0.1 is applied in the solver.",
      "advection": "Normalized L2 norm of the Leray-projected nonlinear term P(u cross curl u), equal in norm to projected advective acceleration.",
      "stretching": "Normalized L2 norm of (curl u dot gradient)u at the initial state.",
      "retuning": "Euclidean norm of corrected minus base carrier amplitudes before common delta scaling."
    }
  },
  "07_nonlinear_calendar": {
    "title": "Retuned Galerkin helicity trajectory",
    "generation_kind": "galerkin_simulation",
    "description": "Helicity samples for the final saved epsilon=0.4 Galerkin run, retuned at two event times and one later anchor.",
    "parameter_context": {
      "viscosity": 1.0,
      "fourier_cutoff_K": 3,
      "quadratic_product_grid": [
        12,
        12,
        12
      ],
      "integrator": "classical fourth-order Runge-Kutta",
      "time_step": 0.001,
      "overall_velocity_scale_delta": 0.1,
      "carrier_wavenumbers": [
        1,
        2,
        3
      ],
      "event_times": [
        0.0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity_target": 0.01,
      "domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
      "provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
      "epsilon": 0.4,
      "sample_time_range": [
        0,
        0.5
      ],
      "sample_count": 101
    },
    "limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. The intermediate helicity curve is not prescribed. The analytic shear reference visible in the associated figure is not included as a CSV column.",
    "fields": {
      "time": "Nondimensional evolution time t; 101 increasing samples from 0 to 0.5.",
      "helicity": "Numerical signed total helicity, the normalized spatial average of u dot curl u."
    }
  },
  "09_energy_clock": {
    "title": "Energy-helicity curve from the retuned Galerkin run",
    "generation_kind": "galerkin_simulation",
    "description": "The same epsilon=0.4 Galerkin trajectory as 07_nonlinear_calendar, expressed as energy-helicity pairs without a time column.",
    "parameter_context": {
      "viscosity": 1.0,
      "fourier_cutoff_K": 3,
      "quadratic_product_grid": [
        12,
        12,
        12
      ],
      "integrator": "classical fourth-order Runge-Kutta",
      "time_step": 0.001,
      "overall_velocity_scale_delta": 0.1,
      "carrier_wavenumbers": [
        1,
        2,
        3
      ],
      "event_times": [
        0.0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity_target": 0.01,
      "domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
      "provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
      "epsilon": 0.4,
      "underlying_time_range": [
        0,
        0.5
      ],
      "sample_count": 101,
      "row_order": "Increasing simulation time; decreasing energy"
    },
    "limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. This table removes timestamps but is still derived from one finite-dimensional numerical trajectory. Clock invariance requires energy and helicity to share a common reparameterization.",
    "fields": {
      "energy": "Numerical kinetic energy E=one half of the normalized spatial average of |u|^2.",
      "helicity": "Numerical signed total helicity H=the normalized spatial average of u dot curl u, paired with the energy at the same time."
    }
  },
  "10_common_clock": {
    "title": "Externally prescribed common operational-clock pauses",
    "generation_kind": "analytical_samples",
    "description": "Samples of the unscaled three-carrier shear scaffold composed with an externally imposed operational clock T, with the same T applied to both energy and helicity.",
    "parameter_context": {
      "viscosity": 1,
      "carrier_wavenumbers": [
        1,
        2,
        3
      ],
      "event_times": [
        0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity": 1,
      "overall_velocity_scale_delta": 1,
      "time_units": "nondimensional",
      "laboratory_time_range": [
        0,
        1.1
      ],
      "sample_count": 1101,
      "clock_formula": "T(t)=t-clip(t-0.15,0,0.18)-clip(t-0.65,0,0.20)",
      "pause_intervals": [
        [
          0.15,
          0.33
        ],
        [
          0.65,
          0.85
        ]
      ],
      "operational_time_range": [
        0,
        0.72
      ]
    },
    "limitations": "An imposed operational reparameterization of an unscaled heat/shear scaffold; it is not ordinary fixed-viscosity dynamics in laboratory time and does not freeze physical time. Both observations must share the same clock. Normalization delta=1 is not a certified small-data nonlinear instance.",
    "fields": {
      "laboratory_time": "Nondimensional external observation-time parameter t.",
      "operational_time": "Nondimensional process time T(t), held constant on the two imposed pause intervals.",
      "energy": "Unscaled scaffold kinetic energy E(T(t)); initial value approximately 4.8589048.",
      "helicity": "Unscaled scaffold helicity H(T(t)); the operational anchor H(0.25)=1."
    }
  },
  "12_energy_optimizer": {
    "title": "Finite-library minimum-energy helicity coefficients",
    "generation_kind": "floating_point_optimization",
    "description": "Signed helicity coefficients returned by a weighted L1 linear program over a finite eight-carrier library for two simple zero constraints and a unit anchor.",
    "parameter_context": {
      "viscosity": 1,
      "carrier_wavenumbers": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8
      ],
      "event_times": [
        0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity": 1,
      "overall_velocity_scale_delta": 1,
      "time_units": "nondimensional",
      "objective": "sum_k |c_k| / (2 k)",
      "solver": "scipy.optimize.linprog, method highs",
      "nonzero_carriers": [
        1,
        2,
        8
      ],
      "saved_objective_value": 3.2862485434002364,
      "source_json": "data/optimal_score.json"
    },
    "limitations": "Floating-point optimum within this finite library only; not a global optimum over all spectra or all nonlinear Navier-Stokes states. Unit-anchor coefficients are unscaled. The certificate has numerical residuals and is not interval-certified.",
    "fields": {
      "wavenumber": "Positive integer carrier wavenumber k in the finite library.",
      "coefficient": "Optimized signed helicity coefficient c_k; exact-looking zero entries are solver output, not a symbolic certificate."
    }
  },
  "13_dual_certificate": {
    "title": "Numerical dual evaluations for the finite-library energy optimum",
    "generation_kind": "floating_point_optimization",
    "description": "Absolute dual evaluation |(A transpose z)_k| compared with the energy weight 1/(2k), using the saved numerical optimizer dual vector.",
    "parameter_context": {
      "viscosity": 1,
      "carrier_wavenumbers": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8
      ],
      "event_times": [
        0,
        0.12
      ],
      "event_multiplicities": [
        1,
        1
      ],
      "anchor_time": 0.25,
      "anchor_helicity": 1,
      "overall_velocity_scale_delta": 1,
      "time_units": "nondimensional",
      "observation_rows": [
        "H(0)=0",
        "H(0.12)=0",
        "H(0.25)=1"
      ],
      "energy_weights": "1/(2 k)",
      "source_json": "data/optimal_score.json",
      "saved_max_dual_violation": 4.163336342344337e-14
    },
    "limitations": "Numerical primal-dual evidence for a finite-library linear program, not an exact symbolic or interval certificate. Dual evaluations can exceed the weight by roundoff-scale residuals.",
    "fields": {
      "wavenumber": "Positive integer carrier wavenumber k.",
      "dual_absolute": "Absolute value of the k-th entry of A transpose z for the saved dual vector z.",
      "weight": "Allowed dual envelope and weighted-L1 objective weight 1/(2 k)."
    }
  },
  "14_conditioning": {
    "title": "Conditioning of row-scaled confluent observation systems",
    "generation_kind": "analytical_samples",
    "description": "Numerical condition numbers of square observation matrices for progressively higher-order initial zeros with a fixed later anchor.",
    "parameter_context": {
      "viscosity": 1,
      "vanishing_orders": [
        1,
        2,
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10
      ],
      "initial_event_time": 0,
      "anchor_time": 0.2,
      "carrier_wavenumbers": "For order N, k=1 through N+1",
      "row_scaling": "Each observation-matrix row divided by its maximum absolute entry",
      "matrix_conversion": "Arbitrary-precision entries converted to numpy float64 before np.linalg.cond",
      "condition_norm": "numpy default spectral (2-norm) condition number"
    },
    "limitations": "Floating-point conditioning diagnostics depend on the selected carrier library, time, and explicit row scaling. They do not quantify every event calendar or provide inverse-problem noise guarantees. Mathematical invertibility does not imply numerical robustness.",
    "fields": {
      "order": "Positive integer N: helicity derivatives of orders 0 through N-1 vanish at the initial event.",
      "row_scaled_condition_number": "Computed spectral condition number of the row-scaled (N+1) by (N+1) observation matrix."
    }
  },
  "15_root_sensitivity": {
    "title": "Illustrative multiple-root noise scaling",
    "generation_kind": "illustrative_scaling",
    "description": "Reference power laws eta^(1/m) for root orders m=1,2,4,8, illustrating the characteristic sensitivity exponent of a multiple zero.",
    "parameter_context": {
      "root_orders": [
        1,
        2,
        4,
        8
      ],
      "noise_range": [
        1e-12,
        0.01
      ],
      "sample_count": 250,
      "noise_grid": "logarithmically spaced",
      "formula": "eta^(1/m)",
      "normalization": "Unit leading coefficient reference scaling; no flow-specific constants"
    },
    "limitations": "Reference scaling curves, not measured root shifts, noise realizations, or a simulation of perturbed Navier-Stokes observations. Actual constants depend on the local root coefficient and error model.",
    "fields": {
      "noise": "Positive illustrative observation-error magnitude eta.",
      "order1": "Reference displacement eta for root multiplicity 1.",
      "order2": "Reference displacement eta^(1/2) for root multiplicity 2.",
      "order4": "Reference displacement eta^(1/4) for root multiplicity 4.",
      "order8": "Reference displacement eta^(1/8) for root multiplicity 8."
    }
  },
  "16_exact_example": {
    "title": "Rational four-carrier helicity example",
    "generation_kind": "analytical_samples",
    "description": "Samples of an exponential sum with rational coefficients, an exact double zero at t=0, and an exact simple zero at t=log(2), normalized so the first coefficient equals one.",
    "parameter_context": {
      "two_times_viscosity": 1,
      "viscosity": 0.5,
      "carrier_wavenumbers": [
        1,
        2,
        3,
        4
      ],
      "decay_rates": [
        1,
        4,
        9,
        16
      ],
      "coefficients": [
        "1",
        "-227464/27141",
        "331779/27141",
        "-131456/27141"
      ],
      "event_times": [
        "0",
        "log(2)"
      ],
      "event_multiplicities": [
        2,
        1
      ],
      "sample_time_range": [
        0,
        2.2
      ],
      "sample_count": 701,
      "formula": "sum_j c_j exp(-r_j t)",
      "normalization": "First coefficient c_1=1; not the anchor-normalized coefficients in data/exact_example.json"
    },
    "limitations": "The defining coefficients and root constraints are exact rational mathematics, while this CSV stores float64 evaluations. Samples do not prove zero multiplicities or the absence of further zeros. It is a shear/heat scaffold, not a nonlinear trajectory or experiment.",
    "fields": {
      "time": "Nondimensional time t with 2 nu=1.",
      "helicity": "Float64 evaluation of the rational four-carrier helicity sum normalized by c_1=1."
    }
  }
}

	{
	"01_event_divisor": {
	"title": "Compiled helicity event scaffold",
	"generation_kind": "analytical_samples",
	"description": "Samples of an exponential helical-shear scaffold with a double initial zero, a simple zero at 0.18, and a double zero at 0.33, together with its energy.",
	"parameter_context": {
	"viscosity": 1,
	"carrier_wavenumbers": [
	1,
	2,
	3,
	4,
	5,
	6
	],
	"event_times": [
	0,
	0.18,
	0.33
	],
	"event_multiplicities": [
	2,
	1,
	2
	],
	"anchor_time": 0.5,
	"anchor_helicity": 1,
	"overall_velocity_scale_delta": 1,
	"compiler_decimal_precision": 100,
	"sample_time_range": [
	0,
	0.8
	],
	"sample_count": 901,
	"formula_helicity": "sum_k c_k exp(-2 nu k^2 t)",
	"formula_energy": "sum_k \|c_k\| exp(-2 nu k^2 t) / (2 k)"
	},
	"limitations": "Unscaled heat/shear scaffold: nonlinear advection vanishes, and initial energy is approximately 2612.2564. It is not a small-data nonlinear realization, shock simulation, or experiment. Arbitrary-precision values are exported as floating-point samples; small residuals can remain at nominal zeros.",
	"fields": {
	"time": "Nondimensional operational time t.",
	"helicity": "Signed normalized total helicity H(t), with H(0.5)=1 for the compiled scaffold.",
	"energy": "Nonnegative kinetic energy E(t)=one half of the normalized spatial average of \|u\|^2, at overall velocity scale delta=1."
	}
	},
	"02_scissor_carriers": {
	"title": "Signed coefficients of the six-carrier scaffold",
	"generation_kind": "analytical_samples",
	"description": "The six alternating signed helicity coefficients used in the event scaffold in 01_event_divisor.",
	"parameter_context": {
	"linked_table": "01_event_divisor",
	"viscosity": 1,
	"carrier_wavenumbers": [
	1,
	2,
	3,
	4,
	5,
	6
	],
	"event_times": [
	0,
	0.18,
	0.33
	],
	"event_multiplicities": [
	2,
	1,
	2
	],
	"anchor_time": 0.5,
	"anchor_helicity": 1,
	"compiler_decimal_precision": 100,
	"coefficient_count": 6
	},
	"limitations": "Coefficients belong to the unscaled one-coordinate helical-shear scaffold, whose nonlinear advection is zero. Signed helicity coefficients are not velocity amplitudes; amplitudes are sqrt(\|c_k\|/k) before an overall scale. Floating-point CSV values approximate the arbitrary-precision solve.",
	"fields": {
	"wavenumber": "Positive integer Fourier carrier wavenumber k.",
	"coefficient": "Signed helicity coefficient c_k multiplying exp(-2 nu k^2 t); its sign selects carrier chirality."
	}
	},
	"03_triad_kernel": {
	"title": "Exact third mixed-amplitude triad response kernel",
	"generation_kind": "analytical_samples",
	"description": "Samples of K_nu(t)=exp(-6 nu t)[exp(4 nu t)-1-4 nu t]/(8 nu) for three viscosities. For the specified standing-wave triad, the third mixed amplitude derivative of helicity at zero amplitudes is K_nu(t) cos(Phi).",
	"parameter_context": {
	"viscosities": [
	0.2,
	1,
	3
	],
	"sample_time_range": [
	0,
	0.8
	],
	"sample_count": 901,
	"phase_factor": "cos(Phi) omitted; columns store K_nu itself",
	"source_formula": "src/make_figures.py and chapters/03_triad_clock.tex"
	},
	"limitations": "An exact response derivative evaluated numerically, not the entire helicity trajectory at finite amplitudes. Kernel positivity alone does not establish positivity of a full finite-amplitude trajectory. No physical experiment is represented.",
	"fields": {
	"time": "Nondimensional time t.",
	"nu_0p2": "K_nu(t) at viscosity nu=0.2.",
	"nu_1": "K_nu(t) at viscosity nu=1.",
	"nu_3": "K_nu(t) at viscosity nu=3."
	}
	},
	"04_phase_gate": {
	"title": "Triad phase dependence of initial helicity acceleration",
	"generation_kind": "analytical_samples",
	"description": "Samples of cos(Phi), the normalized initial acceleration H second derivative at zero divided by 2 nu a b c for the specified standing-wave triad.",
	"parameter_context": {
	"phase_range_radians": [
	"-pi",
	"pi"
	],
	"sample_count": 501,
	"formula": "H second derivative at zero = 2 nu a b c cos(Phi)",
	"phase_definition": "Phi = phi_1 + phi_2 + phi_3, the closed-loop phase"
	},
	"limitations": "All viscosity and amplitude factors have been normalized away. The plotted cosine is not a finite-amplitude helicity time history; the stronger all-time quadrature extinction uses the symmetry argument in the manuscript.",
	"fields": {
	"phase": "Closed-loop interaction phase Phi in radians.",
	"normalized_acceleration": "cos(Phi), equal to H second derivative at zero / (2 nu a b c) when the denominator is nonzero."
	}
	},
	"05_quadratic_retuning": {
	"title": "Galerkin carrier-amplitude correction versus perturbation",
	"generation_kind": "galerkin_simulation",
	"description": "Three saved nonlinear root solves showing the Euclidean norm of the correction to the axial carrier amplitudes as the transverse perturbation parameter changes.",
	"parameter_context": {
	"viscosity": 1.0,
	"fourier_cutoff_K": 3,
	"quadratic_product_grid": [
	12,
	12,
	12
	],
	"integrator": "classical fourth-order Runge-Kutta",
	"time_step": 0.001,
	"overall_velocity_scale_delta": 0.1,
	"carrier_wavenumbers": [
	1,
	2,
	3
	],
	"event_times": [
	0.0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity_target": 0.01,
	"domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
	"provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
	"epsilon_values": [
	0.1,
	0.2,
	0.4
	],
	"correction_definition": "Euclidean norm of a-a0 before multiplying velocities by delta; a_j=a0_j exp(theta_j)"
	},
	"limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. Three values support an illustrative quadratic scaling; they are not an independent proof of an asymptotic order.",
	"fields": {
	"epsilon": "Transverse parameter epsilon inside the velocity ansatz delta times [carrier field + epsilon transverse field]; actual transverse velocity coefficient is delta epsilon.",
	"amplitude_correction_norm": "Euclidean norm of corrected minus base carrier-amplitude vector before common velocity scaling by delta."
	}
	},
	"06_interaction_orders": {
	"title": "Initial nonlinear diagnostics and carrier retuning",
	"generation_kind": "galerkin_simulation",
	"description": "Raw initial norms of pressure-projected nonlinear interaction and vortex stretching, alongside carrier-amplitude retuning for the same three Galerkin root solves.",
	"parameter_context": {
	"viscosity": 1.0,
	"fourier_cutoff_K": 3,
	"quadratic_product_grid": [
	12,
	12,
	12
	],
	"integrator": "classical fourth-order Runge-Kutta",
	"time_step": 0.001,
	"overall_velocity_scale_delta": 0.1,
	"carrier_wavenumbers": [
	1,
	2,
	3
	],
	"event_times": [
	0.0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity_target": 0.01,
	"domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
	"provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
	"epsilon_values": [
	0.1,
	0.2,
	0.4
	],
	"measurement_time": 0,
	"csv_values": "Raw diagnostics; figure alone divides some curves by their first values"
	},
	"limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. Initial norm trends illustrate first-order interaction and second-order retuning over three epsilon values; the analytical argument supplies the asymptotic claims.",
	"fields": {
	"epsilon": "Transverse parameter epsilon; common velocity scale delta=0.1 is applied in the solver.",
	"advection": "Normalized L2 norm of the Leray-projected nonlinear term P(u cross curl u), equal in norm to projected advective acceleration.",
	"stretching": "Normalized L2 norm of (curl u dot gradient)u at the initial state.",
	"retuning": "Euclidean norm of corrected minus base carrier amplitudes before common delta scaling."
	}
	},
	"07_nonlinear_calendar": {
	"title": "Retuned Galerkin helicity trajectory",
	"generation_kind": "galerkin_simulation",
	"description": "Helicity samples for the final saved epsilon=0.4 Galerkin run, retuned at two event times and one later anchor.",
	"parameter_context": {
	"viscosity": 1.0,
	"fourier_cutoff_K": 3,
	"quadratic_product_grid": [
	12,
	12,
	12
	],
	"integrator": "classical fourth-order Runge-Kutta",
	"time_step": 0.001,
	"overall_velocity_scale_delta": 0.1,
	"carrier_wavenumbers": [
	1,
	2,
	3
	],
	"event_times": [
	0.0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity_target": 0.01,
	"domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
	"provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
	"epsilon": 0.4,
	"sample_time_range": [
	0,
	0.5
	],
	"sample_count": 101
	},
	"limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. The intermediate helicity curve is not prescribed. The analytic shear reference visible in the associated figure is not included as a CSV column.",
	"fields": {
	"time": "Nondimensional evolution time t; 101 increasing samples from 0 to 0.5.",
	"helicity": "Numerical signed total helicity, the normalized spatial average of u dot curl u."
	}
	},
	"09_energy_clock": {
	"title": "Energy-helicity curve from the retuned Galerkin run",
	"generation_kind": "galerkin_simulation",
	"description": "The same epsilon=0.4 Galerkin trajectory as 07_nonlinear_calendar, expressed as energy-helicity pairs without a time column.",
	"parameter_context": {
	"viscosity": 1.0,
	"fourier_cutoff_K": 3,
	"quadratic_product_grid": [
	12,
	12,
	12
	],
	"integrator": "classical fourth-order Runge-Kutta",
	"time_step": 0.001,
	"overall_velocity_scale_delta": 0.1,
	"carrier_wavenumbers": [
	1,
	2,
	3
	],
	"event_times": [
	0.0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity_target": 0.01,
	"domain": "mean-zero incompressible velocity on (R / 2 pi Z)^3; normalized spatial averages",
	"provenance": "src/nonlinear_demo.py -> data/nonlinear_demo.json -> src/make_figures.py",
	"epsilon": 0.4,
	"underlying_time_range": [
	0,
	0.5
	],
	"sample_count": 101,
	"row_order": "Increasing simulation time; decreasing energy"
	},
	"limitations": "Uncertified floating-point Fourier-Galerkin illustration, not an interval-certified infinite-dimensional PDE solution. The analytical small-data theorem has an existential perturbation interval; this numerical parameter vector is not separately certified to lie in that interval. Sampled curves do not prove the global absence of additional zeros. This table removes timestamps but is still derived from one finite-dimensional numerical trajectory. Clock invariance requires energy and helicity to share a common reparameterization.",
	"fields": {
	"energy": "Numerical kinetic energy E=one half of the normalized spatial average of \|u\|^2.",
	"helicity": "Numerical signed total helicity H=the normalized spatial average of u dot curl u, paired with the energy at the same time."
	}
	},
	"10_common_clock": {
	"title": "Externally prescribed common operational-clock pauses",
	"generation_kind": "analytical_samples",
	"description": "Samples of the unscaled three-carrier shear scaffold composed with an externally imposed operational clock T, with the same T applied to both energy and helicity.",
	"parameter_context": {
	"viscosity": 1,
	"carrier_wavenumbers": [
	1,
	2,
	3
	],
	"event_times": [
	0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity": 1,
	"overall_velocity_scale_delta": 1,
	"time_units": "nondimensional",
	"laboratory_time_range": [
	0,
	1.1
	],
	"sample_count": 1101,
	"clock_formula": "T(t)=t-clip(t-0.15,0,0.18)-clip(t-0.65,0,0.20)",
	"pause_intervals": [
	[
	0.15,
	0.33
	],
	[
	0.65,
	0.85
	]
	],
	"operational_time_range": [
	0,
	0.72
	]
	},
	"limitations": "An imposed operational reparameterization of an unscaled heat/shear scaffold; it is not ordinary fixed-viscosity dynamics in laboratory time and does not freeze physical time. Both observations must share the same clock. Normalization delta=1 is not a certified small-data nonlinear instance.",
	"fields": {
	"laboratory_time": "Nondimensional external observation-time parameter t.",
	"operational_time": "Nondimensional process time T(t), held constant on the two imposed pause intervals.",
	"energy": "Unscaled scaffold kinetic energy E(T(t)); initial value approximately 4.8589048.",
	"helicity": "Unscaled scaffold helicity H(T(t)); the operational anchor H(0.25)=1."
	}
	},
	"12_energy_optimizer": {
	"title": "Finite-library minimum-energy helicity coefficients",
	"generation_kind": "floating_point_optimization",
	"description": "Signed helicity coefficients returned by a weighted L1 linear program over a finite eight-carrier library for two simple zero constraints and a unit anchor.",
	"parameter_context": {
	"viscosity": 1,
	"carrier_wavenumbers": [
	1,
	2,
	3,
	4,
	5,
	6,
	7,
	8
	],
	"event_times": [
	0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity": 1,
	"overall_velocity_scale_delta": 1,
	"time_units": "nondimensional",
	"objective": "sum_k \|c_k\| / (2 k)",
	"solver": "scipy.optimize.linprog, method highs",
	"nonzero_carriers": [
	1,
	2,
	8
	],
	"saved_objective_value": 3.2862485434002364,
	"source_json": "data/optimal_score.json"
	},
	"limitations": "Floating-point optimum within this finite library only; not a global optimum over all spectra or all nonlinear Navier-Stokes states. Unit-anchor coefficients are unscaled. The certificate has numerical residuals and is not interval-certified.",
	"fields": {
	"wavenumber": "Positive integer carrier wavenumber k in the finite library.",
	"coefficient": "Optimized signed helicity coefficient c_k; exact-looking zero entries are solver output, not a symbolic certificate."
	}
	},
	"13_dual_certificate": {
	"title": "Numerical dual evaluations for the finite-library energy optimum",
	"generation_kind": "floating_point_optimization",
	"description": "Absolute dual evaluation \|(A transpose z)_k\| compared with the energy weight 1/(2k), using the saved numerical optimizer dual vector.",
	"parameter_context": {
	"viscosity": 1,
	"carrier_wavenumbers": [
	1,
	2,
	3,
	4,
	5,
	6,
	7,
	8
	],
	"event_times": [
	0,
	0.12
	],
	"event_multiplicities": [
	1,
	1
	],
	"anchor_time": 0.25,
	"anchor_helicity": 1,
	"overall_velocity_scale_delta": 1,
	"time_units": "nondimensional",
	"observation_rows": [
	"H(0)=0",
	"H(0.12)=0",
	"H(0.25)=1"
	],
	"energy_weights": "1/(2 k)",
	"source_json": "data/optimal_score.json",
	"saved_max_dual_violation": 4.163336342344337e-14
	},
	"limitations": "Numerical primal-dual evidence for a finite-library linear program, not an exact symbolic or interval certificate. Dual evaluations can exceed the weight by roundoff-scale residuals.",
	"fields": {
	"wavenumber": "Positive integer carrier wavenumber k.",
	"dual_absolute": "Absolute value of the k-th entry of A transpose z for the saved dual vector z.",
	"weight": "Allowed dual envelope and weighted-L1 objective weight 1/(2 k)."
	}
	},
	"14_conditioning": {
	"title": "Conditioning of row-scaled confluent observation systems",
	"generation_kind": "analytical_samples",
	"description": "Numerical condition numbers of square observation matrices for progressively higher-order initial zeros with a fixed later anchor.",
	"parameter_context": {
	"viscosity": 1,
	"vanishing_orders": [
	1,
	2,
	3,
	4,
	5,
	6,
	7,
	8,
	9,
	10
	],
	"initial_event_time": 0,
	"anchor_time": 0.2,
	"carrier_wavenumbers": "For order N, k=1 through N+1",
	"row_scaling": "Each observation-matrix row divided by its maximum absolute entry",
	"matrix_conversion": "Arbitrary-precision entries converted to numpy float64 before np.linalg.cond",
	"condition_norm": "numpy default spectral (2-norm) condition number"
	},
	"limitations": "Floating-point conditioning diagnostics depend on the selected carrier library, time, and explicit row scaling. They do not quantify every event calendar or provide inverse-problem noise guarantees. Mathematical invertibility does not imply numerical robustness.",
	"fields": {
	"order": "Positive integer N: helicity derivatives of orders 0 through N-1 vanish at the initial event.",
	"row_scaled_condition_number": "Computed spectral condition number of the row-scaled (N+1) by (N+1) observation matrix."
	}
	},
	"15_root_sensitivity": {
	"title": "Illustrative multiple-root noise scaling",
	"generation_kind": "illustrative_scaling",
	"description": "Reference power laws eta^(1/m) for root orders m=1,2,4,8, illustrating the characteristic sensitivity exponent of a multiple zero.",
	"parameter_context": {
	"root_orders": [
	1,
	2,
	4,
	8
	],
	"noise_range": [
	1e-12,
	0.01
	],
	"sample_count": 250,
	"noise_grid": "logarithmically spaced",
	"formula": "eta^(1/m)",
	"normalization": "Unit leading coefficient reference scaling; no flow-specific constants"
	},
	"limitations": "Reference scaling curves, not measured root shifts, noise realizations, or a simulation of perturbed Navier-Stokes observations. Actual constants depend on the local root coefficient and error model.",
	"fields": {
	"noise": "Positive illustrative observation-error magnitude eta.",
	"order1": "Reference displacement eta for root multiplicity 1.",
	"order2": "Reference displacement eta^(1/2) for root multiplicity 2.",
	"order4": "Reference displacement eta^(1/4) for root multiplicity 4.",
	"order8": "Reference displacement eta^(1/8) for root multiplicity 8."
	}
	},
	"16_exact_example": {
	"title": "Rational four-carrier helicity example",
	"generation_kind": "analytical_samples",
	"description": "Samples of an exponential sum with rational coefficients, an exact double zero at t=0, and an exact simple zero at t=log(2), normalized so the first coefficient equals one.",
	"parameter_context": {
	"two_times_viscosity": 1,
	"viscosity": 0.5,
	"carrier_wavenumbers": [
	1,
	2,
	3,
	4
	],
	"decay_rates": [
	1,
	4,
	9,
	16
	],
	"coefficients": [
	"1",
	"-227464/27141",
	"331779/27141",
	"-131456/27141"
	],
	"event_times": [
	"0",
	"log(2)"
	],
	"event_multiplicities": [
	2,
	1
	],
	"sample_time_range": [
	0,
	2.2
	],
	"sample_count": 701,
	"formula": "sum_j c_j exp(-r_j t)",
	"normalization": "First coefficient c_1=1; not the anchor-normalized coefficients in data/exact_example.json"
	},
	"limitations": "The defining coefficients and root constraints are exact rational mathematics, while this CSV stores float64 evaluations. Samples do not prove zero multiplicities or the absence of further zeros. It is a shear/heat scaffold, not a nonlinear trajectory or experiment.",
	"fields": {
	"time": "Nondimensional time t with 2 nu=1.",
	"helicity": "Float64 evaluation of the rational four-carrier helicity sum normalized by c_1=1."
	}
	}
	}