Certified spectral instability of the excited De Gregorio profile f3

(discrete level, interval arithmetic, one-command reproducible)

Theorem (discrete level). The m=560 finite-difference discretization (exact rational nodes) of the linearization of the De Gregorio dynamic-rescaling evolution about the explicit frozen approximation of the second excited Huang–Tong–Wei profile f3, restricted to the Chen–Hou–Huang perturbation class in the energy space E, has a certified unstable eigenvalue (Re mu >= 0.50938462, enclosure radius 5.9e-13), certified resolvent floors on a surrounding contour (whole-contour sigma >= 9.594e-4), and a certified parametrix residual delta <= 0.2188 at all twelve baseline contour points. All constants are produced in 128-bit ball / outward-rounded interval arithmetic.

Paper: paper/main.pdf (arXiv link to be added on posting).

Regenerate the certificates

Local rows (plain Python: numpy/scipy/sympy/mpmath==1.3.0):

python3 repro/run_all.py        # 4/4 rows, ~4 minutes

Ball-arithmetic rows (python-flint / Arb; free Colab hardware suffices):

# stages 1-3 (eigenvalue Krawczyk + contour floors):
python3 scripts/arb_certificate_f3_colab.py
# stage 4 (parametrix residual; needs results/t4_parametrix_payload.npz):
DG_HERE=. DG_PAYLOAD=results/t4_parametrix_payload.npz \
  python3 scripts/arb_certificate_f3_colab_t4stage2.py     # 498 s measured

Certified outputs shipped in results/ (JSON); the payload npz is the frozen parametrix (dyadic float64 = exact binary rationals).

Layout

• repro/ one-command regeneration harness + pinned environment
• scripts/ certificate scripts (ball arithmetic + rigorous mpmath.iv rows) and the float t3 parametrix scripts (evidence-grade, clearly non-certified)
• results/ certified outputs (per-z tables, stage-4 results, frozen payload)
• lemmas/ the mathematical working documents (Parts I-VI): reduction theorem, local models, kernel table, certified patch constants, profile bridge, residual-enclosure lemma
• paper/ LaTeX source + PDF

Use of AI tools

This work — both the research and this codebase — was carried out by the author in an interactive computational session with AI assistants (Anthropic's Claude: Fable 5 and Opus 4.8). The AI assisted with code implementation, numerical experiment design, diagnostics, literature positioning, and drafting, under the author's direction.

A note on the working documents

lemmas/ are working documents from the research record. Process language ("handover", session dates) is retained deliberately as provenance.