Guide 05

What counts as reproduction?

A solver-resolved sequence of controlled finite truncations—not merely a flow with the theorem’s headline scaling and not a numerical crash.

Required gates

1. Initial and global geometry

The field begins identically at rest. Temporal and spatial localization preserve smoothness and incompressibility. The implicit (q,η,X) map and anisotropic scales are used directly.

2. Leading local field

The explicit axis data and a controlled numerical continuation of E, U, and V₀ must satisfy the profile identities, radial moment constraints, axial-outflow checks, and exact heat exterior to improving accuracy.

3. Finite all-order hierarchy

Background corrections through order N in powers of q^2h must each carry their own moment and residual-order audit.

4. Oscillatory stress realization

Both labelled pulse families must solve their phase and amplitude equations on the auxiliary torus, live inside the intended tails and annulus, and converge in averaged covariance to the prescribed stress.

5. Mean and residual corrections

Compactly supported mean corrections and the residual-improvement cycle must reach the same truncation order. The decisive observable is bounded, converging force derivatives near t = 1.

6. Coupled convergence

Grid size, timestep density, pulse frequency, and truncation order must increase together. On every resolved interval t ≤ 1 − ε, solution differences and residual-force derivatives should decrease while peak speed grows and kinetic energy remains bounded.

Status

Capability	Status
Exact initial rest interval and smooth activation	Implemented
Implicit paper coordinates and similarity scales	Implemented
Appendix-B explicit axis data	Implemented
Discrete divergence-free finite continuation	Implemented, surrogate
Static core-refined comparison grids	32³, 64³, and 128³ native archives audited; spatial gate unmet
Outer-band candidate spatial and temporal screen	Three early checkpoints pass the spatial and independent from-rest velocity timestep gates; the 64³ parent stopped before five later locked checkpoints and has no final archive audit
Dense pressure-free balance	64³ baseline, half-step, quarter-step, and matched dense 128³ fields audited; the spatial residual sequence is mixed and remains materially timestep-sensitive
Finite pulse hierarchy and grid corrector	Implemented, surrogate
Exact stress-matched pulse construction	Not implemented
All-order background and mean corrections	Not implemented
Smooth residual-force extension through t = 1	Not demonstrated

Finite claim. No computation reaches t = 1 or instantiates an infinite hierarchy. The current numerical result is a provisional finite-mesh candidate interval with local velocity timestep stability and unresolved residual qualification. Convergence of increasingly faithful finite truncations remains the numerical target; the analytical paper supplies the blowup result.

Failure conditions

Peak growth disappears under grid or timestep refinement.
The spectral tail fills before the geometric scale threshold.
The core-refined and full-domain trajectories disagree within their shared core.
The force or its derivatives diverge because a required analytical cancellation is absent.
A claimed result depends on target initialization instead of starting from rest.