Navier--Stokes Cascade Research Sandbox

This is a finite-dimensional experimental sandbox for the Navier--Stokes cascade/leakage program developed in the chat.

It does not prove Navier--Stokes regularity or blow-up. It computes exact Fourier-symbol facts for finite mode fans:

• Leray projection and the exact projected bilinear symbol
• permutation-shell seed amplitudes
• forced first leakage fan
• radius fan growth
• path-collision reports
• velocity/vorticity line-bundle defect
• helical-coordinate diagnostics

Fresh-chat handoff

Read HANDOFF.md first. For a new ChatGPT project/chat, upload the zip and paste NEXT_CHAT_PROMPT.md.

Run

python experiments/first_fan.py

Main files

• navier_stokes_cascade/ns_symbol.py — projection, bilinear map, vorticity polarization, angle/line-defect metrics.
• navier_stokes_cascade/mode_sets.py — permutation shells and symmetric seed/fan amplitudes.
• navier_stokes_cascade/defects.py — interaction enumeration, collision reports, radius variance.
• navier_stokes_cascade/helical.py — helical basis and helicity diagnostics.
• experiments/first_fan.py — first experiment and sanity checks.

Current theorem target

Universal First-Handoff Defect Theorem:

Given arbitrary strong 3D first-shell transfer, the full Navier--Stokes leakage layer should force either radius variance, polarization/vorticity covariance defect, suppression loss, or geometric degeneration.

This repo currently verifies the symmetric seed case and surfaces the next computational questions.

Latest continuation notes

The experiments have now separated the first-handoff obstruction into regimes:

1. Real / non-helical cyclic geometries: robust comparable-collision vorticity miss stays near 0.05, with radius relative variance near 0.06.
2. Pure-helical line-bundle geometries: polarization/vorticity miss can collapse near zero, but the closure drifts toward a near-ray frequency set.
3. Full-vector helical closure: if the output is not projected back to a pure helical line, helicity purity decays and robust vorticity miss grows rapidly.

The new helical closure diagnostics support the following working trichotomy:

real/scalar cascade          -> robust line-bundle/vorticity defect
full-vector helical fan      -> growing polarization/vorticity defect
projected pure-helical fan   -> near-ray collapse and normalized coefficient depletion

Additional experiment files:

• experiments/robust_optimize_first_handoff.py — robust optimization using only comparable multi-path collisions.
• experiments/classify_minima.py — classifies low-defect optimizer minima against cyclic/off-cycle parent-polarization patterns.
• experiments/helical_line_bundle_scan.py — scans pure-helicity sign assignments on the first fan.
• experiments/helical_radius_closure.py — iterates projected/full-vector helical closures and reports radius/linearity/coefficient diagnostics.
• experiments/helical_closure_sweep.py — checks whether projected-helical near-ray collapse is stable across cutoffs and mode caps.
• experiments/near_ray_coefficient_scan.py — direct controlled scan showing normalized NS coefficients decay like N^-1 for bounded-defect near-ray mode pairs, while transverse control interactions remain O(1).

Key recent outputs:

• helical_closure_sweep.txt
• near_ray_coefficient_scan.txt
• helical_projected_nocap_rel1e4.txt

The strongest current theorem target is now the Helical Escape Trichotomy:

A strong leakage-absorbing first handoff either develops robust polarization/vorticity defect, or stays nearly pure-helical and collapses toward near-ray geometry, causing normalized triad coefficient depletion, or suppresses a significant fraction of forced output.

This is still far from a Clay proof. The immediate finite theorem target is to prove the controlled near-ray coefficient depletion formula and then connect it to the observed projected-helical closure collapse.