Guide 02 · implementation record

From the paper to a finite grid

This page is the audit trail for the profile now implemented in the solver: what is explicit in the paper, what is chosen numerically, and what remains outside the present reproduction.

The central constraint. The paper proves existence through parameter ranges, continuation, patching, moment matching, and an all-order correction scheme. It does not publish one unique table of complete numerical functions E(X,η), U(X,η), and V₀(X,η). A solver can reproduce the explicit axis data exactly, but any finite off-axis continuation must be named and recorded as a numerical choice.

1. Similarity coordinates

Set τ = 1 − t, D = 1/2 − h, and A = 1/2 + h. With normalized height ζ = z/R₀, the physical branch of the paper’s implicit map is solved cell-by-cell:

q − ζ²q^2h = τ
η = ζ / q^D
X = (r/R₀)² / (2q).

The implementation uses seven Newton steps from a positive asymptotic guess. Its coordinate-closure residual is measured at every saved frame. The cylindrical leading field is organized as u_θ = q^−AE, u_z = q^−AU, with the radial component constrained by incompressibility.

2. Explicit Appendix-B axis data

The updated solver no longer invents a generic odd axial curve. It evaluates the paper’s explicit axis construction on an 8,193-point η table. For d = 1 − η² and L = 1 − 2hη²:

U^*(η) = 4η + j₀
H^*(η) = Dη + dU^*(η)
ζ^*(η) = −LH^* / ((H^*)² + σ²)
φ^*(η) = exp(Λ ∫₀^η ζ^*(w) dw).

The integral is evaluated by cumulative trapezoids and shifted so φ^*(0) = 1. The default offset j₀ = 0.03 lies inside the paper’s stated interval 0 < j₀ ≤ 0.05.

Choice	Default	Status
h	0.008	Inside the theorem’s (0, 0.01) interval
j₀	0.03	Inside the published axis-data interval
Λ	2.0	Finite numerical choice; not certified as “sufficiently large”
σ	0.25	Finite numerical choice; not certified as “sufficiently small”
\|η\| plateau / taper	0.86 / 0.98	Compact numerical continuation

3. From axis data to a 3D field

The table supplies the near-axis axial datum and its positive auxiliary swirl profile. The current finite continuation carries those functions across the diagnostic core, applies a compact η taper only near |η| = 1, and combines them with radial inner/annular profiles. The radial profile has zero moment, so the meridional vector potential closes outside the annulus.

The paper gives an incompressibility identity of the form

V₀ = (X/L) [2ηU − 2Dη 𝒜_X(U) − d∂_η𝒜_X(U)],

with the radial operator 𝒜_X defined in its profile construction. Numerically, we enforce the same constraint more robustly by building a vector potential and applying the centered discrete curl. Thus the sampled V₀ analogue is determined by the same discrete incompressibility operator used by the solver, and ∇·u vanishes to roundoff.

This is a deliberate fidelity boundary: the explicit axis equations are reproduced; the complete analytic continuation and its profile-cone proof are not claimed.

4. Axial-outflow checks

A vertical “jet” is not expected to become dramatically slender at the times currently resolved. The similarity aspect ratio is

ℓ_z / ℓᵣ = τ^−h.

At t = 0.992 and h = 0.008, this is only about 1.04. The new core-frame GIF divides physical coordinates by ℓᵣ and ℓ_z, exposing the signed axial evacuation without asking a physical-frame image to manufacture visible slenderness.

Each frame now records the theoretical and energy-weighted measured aspect ratios, the fraction of core axial energy pointing away from the dividing plane, upper and lower signed axial fluxes, and their imbalance.

5. Static core-refined mesh

The solver’s FFT projection requires a uniform periodic grid, so the first refinement move is a fixed fine box around the known singular core—not dynamic AMR. The core-refined preset changes the half-width from 1.00 to 0.68 at the same N³ cell count:

Δx_full = 2/N
Δx_core = 1.36/N
spacing gain = Δx_full/Δx_core = 1/0.68 ≈ 1.47.

A 224³ core-box run therefore has the core spacing of roughly a 329³ full-box run without increasing the field-array count. The fixed spatial taper moves to 0.61–0.66. At activation, the midplane outer annulus estimate is

r_annulus = R₀√(2τX_b)
= 0.52√(2 · 0.45 · 1.45) ≈ 0.594.

That leaves about 0.066 between the midplane annulus and the taper edge—10.9 cells at 224³—plus 0.02, or 3.3 cells, between the cutoff and periodic boundary. Both clearances are written into the diagnostics rather than assumed.

Tradeoff. The core-refined box intentionally gives up part of the fixed exterior and high-|η| annulus. It is a local core-convergence experiment, not a replacement for the full-domain trajectory. Agreement must be judged inside their overlapping core and against a full-box control.

6. Remaining analytical gap

The finite three-level pulse hierarchy and bounded deconvolution corrector remain numerical surrogates. The exact stress matching, successive q^2nh corrections, mean corrections, moment constraints at every order, and smooth extension of the residual force through t = 1 are not yet instantiated. Those are the next gates, not details to hide behind a favorable picture.

Source and implementation

Equations above are transcribed from OpenAI’s paper, especially the leading-profile development and Appendix B. The implementation lives in profile.py; every finite parameter is serialized into run.json.