A calibrated spectral survey of the linearized Navier–Stokes operator at the Hou–Wang–Yang (HWY) self-similar profile (arXiv:2509.25116): symmetry eigenvalues, the anatomy of two pseudospectral false positives (the second new in the v1.2 draft), and calibrated residual-landscape surveys of both axisymmetric parity sectors on from-scratch decay-adapted divergence-free bases.
Paper (P. Salmond):
- v1.1, PDF of record — Zenodo, CC BY 4.0; source:
paper/preprint_survey_v4.tex. - v1.2 draft (adds the even-sector survey) — source:
paper/preprint_survey_v5.tex; pending independent verification (see below) before release. Changes:CHANGELOG.md.
| Quantity | Value |
|---|---|
| Landscape vertex (real axis) | λ = 0.113142 (known λ₁ = 0.1131420…, six-figure agreement) |
| Valley depth r(λ₁) | 9.3141 × 10⁻³, certified pointwise by an independent path |
| Instrument noise | 1.5 × 10⁻⁴ relative (cut-stable) |
| Second minimum, real axis [−0.10, 0.30] | none |
| Second minimum, complex window [0, 0.35] × [0.05, 0.60]i | none (strictly monotone in Im λ on every row) |
| Exact symmetry eigenvalues | −1/2 (translation), −1 (time-translation) — both stable |
| v₁ fit error of the tapered trial space (N = 5366) | 2.557 × 10⁻⁴ |
| Quantity | Value |
|---|---|
| Calibration eigenpair at −1 (with closed-form eigenpressure 2P + ξ·∇P) | pointwise residual 3.5 × 10⁻⁷ |
| Real-axis landscape [−0.10, +4.2] | strictly monotone, no local minimum; window exhausts the numerical-abscissa bound λ_max(K̃) = 4.1668 |
| Complex window [0, 0.35] × (0, 0.60]i incl. low strip | no off-axis local minimum; cut-stable to 7 digits |
| Depth standard (no known discrete eigenvalue → exactly planted eigenpair) | floor ε₀ ≈ 5 × 10⁻⁵; resolving-power law reproduced, measured 𝒜 ≈ 9 |
| Null sensitivity at λ = 0 | escapes only if eigenfunction span-fit δ > 2.5 × 10⁻² (measured 𝒜) / 2.2 × 10⁻³ (conservative 𝒜 = 10²); span fits shell-type fields at δ < 10⁻⁶ |
| The λ = −1/2 dip (passed ablation, controls, enrichment-depth) | artifact — a scaling-quasimode family at λ = (1−α)/2 manufactured by ρ^−α tail bases; closed by a tail-free double dissociation |
Conclusion (evidence-grade, floating point, axisymmetric sectors
only): no unstable eigenvalue beyond λ₁ at the instruments' stated
sensitivities; the instability order of the HWY profile within the
axisymmetric class is one. See the paper's Limitations section for
scope. The v1.2 even-sector gate ladder was re-executed from scratch
in a fresh environment on 2026-07-12 and reproduced every pinned value
(even/session6_verification/); the results nevertheless remain
pending independent human verification before release.
src/ all Python (flat module layout; scripts import by name)
ns_part12_gate.py gate-validated pointwise operator realization
ns_part3k.py, ns_part3f.py, ns_part3_spectrum.py
operator application (Lcols), geometry, quadrature
hdf5min.py pure-Python MATLAB v7.3 (HDF5) reader; no h5py/MATLAB needed
nsx_basis.py, nsx_op.py, nsx_x2_aa.py
Stokes-stream div-free basis; separable assembly
kernels (gates T4/T5)
nsx_x5.py production basis + grams + gates (modes: gate|full)
nsx_x5b.py Schur deflation + real-axis landscape
nsx_x5c.py complex-grid survey (checkpointed)
nsx_x5r.py pointwise certification of the landscape valley
mk_heatmap.py figures
nsx_diag*.py, nsx_x0*.py, nsx_r1*.py, nsx_x4*.py, nsx_fitres.py
forensic diagnostics: the open-domain quadrature
demonstrations, operator-domain divergence,
target spectroscopy, radial-family shootout,
resolving-power measurements (paper §6)
mk_even_figures.py even-sector figures (paper §7), from the
preserved arrays/logs under even/ only
even/ the even-sector campaign (paper §7), preserved VERBATIM:
every gate/assembly/probe/measurement script with its run
logs and small result arrays, one directory per session
(see even/README.md and docs/REPRODUCE.md part B)
data/ NOT included — fetched from the HWY public repository
(see data/README.md; no-redistribution note)
results/ small result arrays (landscapes) + figures; large Gram
caches are regenerable (see docs/REPRODUCE.md)
paper/ LaTeX source (v1.1 of record: preprint_survey_v4.tex;
v1.2 draft: preprint_survey_v5.tex)
docs/ REPRODUCE.md (the gate ladders, parts A and B) + campaign
lab notes, incl. the even-sector handover/lab record
pip install -r requirements.txt
# fetch HWY data (see data/README.md), then:
cd src
python ns_part12_gate.py # operator trust anchors
python nsx_x5.py gate # T6 assembly gate (machine precision)
python nsx_x5.py full # grams + X0r/X1 gates (~7 min, ~2 GB)
python nsx_x5b.py # Schur + real-axis landscape (~35 min)
python nsx_x5c.py # complex grid (~80 min, checkpointed)
python nsx_x5r.py # pointwise certification
python mk_heatmap.py # figuresFull expected outputs, runtimes, and tolerances: docs/REPRODUCE.md
(part A: odd sector; part B: the even-sector campaign from even/).
This repository and the accompanying preprint are the work of P. Salmond. The following are introduced here and should be attributed accordingly if reused or built upon (full statements and derivations in the preprint, Zenodo DOI above):
- the resolving-power law and calibration protocol for residual landscapes (attainable floor ≈ eigenfunction best-approximation error × operator amplification), calibrated against the known eigenvalue — the main methodological claim;
- the first spectral reconnaissance of this operator beyond λ₁, and the documented false-positive case study;
- the decay-adapted divergence-free trial space (Stokes-stream family, dual-width radial functions, spectrum-designed taper) and the gate-validated assembly/landscape pipeline;
- (v1.2 draft) the planted-eigenpair depth standard — an exact rank-one Gram-level plant that supplies the calibration protocol to a sector with no known discrete eigenvalue — and the identification of the λ = (1−α)/2 scaling-quasimode artifact family generated by algebraic-tail radial bases, with the object-resolved forensics and tail-free double-dissociation methodology that exposes it.
Of the three numerical-analysis pitfalls the paper documents, two are recorded but not claimed as novel — they specialize standard practice (weighted/open quadrature for axis–equator standoffs, cf. Bernardi–Dauge–Maday 1999; an operator-domain constraint on decay-adapted radial bases, cured by an origin mask) and are written up because they are silent at the Rayleigh–Ritz level yet fatal to the operator-norm survey. The third — the tail-generated artifact family — is claimed as an original identification (see above). The landscape instrument itself is the residual-pseudospectrum / ResDMD validation principle applied to this operator, not a new method.
Reuse of code is governed by the MIT LICENSE (which requires keeping
the copyright notice); reuse of results or methods should additionally
cite the preprint (CITATION.cff). Please also cite Hou–Wang–Yang,
arXiv:2509.25116, whose profile and verified eigenpair this work builds
on.
Theirs (Hou-Wang-Yang): the data files of their public repository
(profile, certified eigenpair, phi-families) — fetched at run time,
never redistributed here — and all the mathematics of
arXiv:2509.25116. Ours (MIT): every line of code in src/
(an independent Python implementation of their published operator,
plus the bases, instruments, and diagnostics), all derived results in
results/, and the paper. The operator-layer modules mirror their
data formats and conventions (facts/interfaces), but contain no code
from their repository. Per-file SPDX headers and NOTICE make this
explicit.
The HWY repository carries no license file; none of its data is
redistributed here. The pipeline reads it from the authors' public
repository as published scientific record, with attribution. All code
in this repository is MIT-licensed; results in results/ are derived
quantities.
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.
The distinction that matters for trust: the AI was part of the
production process, not part of the trust chain for the results.
Every numerical claim rests on computation that you can re-run and on
gates checked against external ground truth (the validated operator, the
HWY profile/eigenpair residuals, the symmetry-eigenvalue identities) —
never on a model's assertion. The code is conventional Python and is
meant to be read, checked, and reproduced; the gate ladder in
docs/REPRODUCE.md exists precisely so that nothing here has to be taken
on faith. The author is responsible for the scientific content, the
claims, and their verification. The mathematical correctness of the
derivations has not yet been independently peer-reviewed.
See CITATION.cff. Please also cite Hou–Wang–Yang, arXiv:2509.25116,
whose profile and verified eigenpair this survey builds on.