docs/DATA_SCHEMA.md

Dataset schema and interpretation

All configurations use one train split for compatibility with dataset loading. There is no statistical train/test separation. Rows across configurations overlap.

Time, energy, helicity, viscosities and amplitudes follow the manuscript's nondimensional periodic normalization; no SI calibration or physical measurement is supplied. Original CSVs retain the source decimal text. Derived JSONL table views use machine-precision numeric conversion, normally loaded as float64, including columns such as wavenumber and order that are mathematically integral.

Structured wrapper configurations

• verification_checks: record_id, check_name, check_group, passed (boolean), detail (string; empty if absent), source_path, source_index (zero-based integer in original checks array), evidence_scope, source_version. Residual strings retain their textual precision. Group is only the first underscore-delimited name token.
• claims: claim_id, title, statement, scope_and_limitations, proof_status, novelty_status, source_path, source_label (LaTeX chapter label), audit_path, source_version. Curated summaries, not independently certified truth labels.
• research_corpus: record_id, document_id, chunk_index (zero-based), line_start, line_end (one-based inclusive), content_format, text, source_path, source_sha256, source_version, evidence_status. Text preserves raw Markdown/LaTeX, including whitespace and trailing line endings. Reassemble each document by sorting chunk_index and concatenating text. A chunk may omit proof context; original complete source is authoritative.
• artifacts: artifact_id, path, artifact_kind, media_type, size_bytes, sha256, source_version, license_status. Exactly one row per preserved source file; derived wrapper files are instead covered by the root hash manifest.
• nonlinear_trajectories: record_id, run_id (1–3), sample_index (0–100), time, energy, helicity, epsilon, delta, viscosity, fourier_cutoff, product_grid_size, integration_timestep, source_path, source_version, evidence_status. Flattened existing records, not new simulations. Samples spaced by 0.005; solver timestep 0.001. Viscosity 1. Original full solver diagnostics are in research/data/nonlinear_demo.json.

Numeric views of original CSV tables

event_divisor (901 rows)

Loadable view: data/tables/event_divisor.jsonl. Original source: research/data/01_event_divisor.csv.

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.

Generation: analytical_samples.

Parameters: {"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)"}

Field	Meaning
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.

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.

scissor_carriers (6 rows)

Loadable view: data/tables/scissor_carriers.jsonl. Original source: research/data/02_scissor_carriers.csv.

The six alternating signed helicity coefficients used in the event scaffold in 01_event_divisor.

Generation: analytical_samples.

Parameters: {"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}

Field	Meaning
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.

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.

triad_kernel (901 rows)

Loadable view: data/tables/triad_kernel.jsonl. Original source: research/data/03_triad_kernel.csv.

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).

Generation: analytical_samples.

Parameters: {"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"}

Field	Meaning
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.

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.

phase_gate (501 rows)

Loadable view: data/tables/phase_gate.jsonl. Original source: research/data/04_phase_gate.csv.

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.

Generation: analytical_samples.

Parameters: {"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"}

Field	Meaning
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.

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.

quadratic_retuning (3 rows)

Loadable view: data/tables/quadratic_retuning.jsonl. Original source: research/data/05_quadratic_retuning.csv.

Three saved nonlinear root solves showing the Euclidean norm of the correction to the axial carrier amplitudes as the transverse perturbation parameter changes.

Generation: galerkin_simulation.

Parameters: {"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)"}

Field	Meaning
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.

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.

interaction_orders (3 rows)

Loadable view: data/tables/interaction_orders.jsonl. Original source: research/data/06_interaction_orders.csv.

Raw initial norms of pressure-projected nonlinear interaction and vortex stretching, alongside carrier-amplitude retuning for the same three Galerkin root solves.

Generation: galerkin_simulation.

Parameters: {"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"}

Field	Meaning
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.

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.

nonlinear_calendar (101 rows)

Loadable view: data/tables/nonlinear_calendar.jsonl. Original source: research/data/07_nonlinear_calendar.csv.

Helicity samples for the final saved epsilon=0.4 Galerkin run, retuned at two event times and one later anchor.

Generation: galerkin_simulation.

Parameters: {"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}

Field	Meaning
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.

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.

energy_clock (101 rows)

Loadable view: data/tables/energy_clock.jsonl. Original source: research/data/09_energy_clock.csv.

The same epsilon=0.4 Galerkin trajectory as 07_nonlinear_calendar, expressed as energy-helicity pairs without a time column.

Generation: galerkin_simulation.

Parameters: {"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"}

Field	Meaning
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.

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.

common_clock (1,101 rows)

Loadable view: data/tables/common_clock.jsonl. Original source: research/data/10_common_clock.csv.

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.

Generation: analytical_samples.

Parameters: {"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]}

Field	Meaning
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.

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.

energy_optimizer (8 rows)

Loadable view: data/tables/energy_optimizer.jsonl. Original source: research/data/12_energy_optimizer.csv.

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.

Generation: floating_point_optimization.

Parameters: {"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"}

Field	Meaning
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.

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.

dual_certificate (8 rows)

Loadable view: data/tables/dual_certificate.jsonl. Original source: research/data/13_dual_certificate.csv.

Absolute dual evaluation |(A transpose z)_k| compared with the energy weight 1/(2k), using the saved numerical optimizer dual vector.

Generation: floating_point_optimization.

Parameters: {"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}

Field	Meaning
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).

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.

conditioning (10 rows)

Loadable view: data/tables/conditioning.jsonl. Original source: research/data/14_conditioning.csv.

Numerical condition numbers of square observation matrices for progressively higher-order initial zeros with a fixed later anchor.

Generation: analytical_samples.

Parameters: {"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"}

Field	Meaning
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.

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.

root_sensitivity (250 rows)

Loadable view: data/tables/root_sensitivity.jsonl. Original source: research/data/15_root_sensitivity.csv.

Reference power laws eta^(1/m) for root orders m=1,2,4,8, illustrating the characteristic sensitivity exponent of a multiple zero.

Generation: illustrative_scaling.

Parameters: {"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"}

Field	Meaning
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.

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.

exact_example (701 rows)

Loadable view: data/tables/exact_example.jsonl. Original source: research/data/16_exact_example.csv.

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.

Generation: analytical_samples.

Parameters: {"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"}

Field	Meaning
time	Nondimensional time t with 2 nu=1.
helicity	Float64 evaluation of the rational four-carrier helicity sum normalized by c_1=1.

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.