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 q2h 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 grid | Implemented, validation pending |
| 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 |
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.