Prescribed-gradient minimizers and coefficient stability
for homogeneous real-stable polynomials
Scott Schubr, October 22, 2026

Contents
  Prescribed_Gradient_Stable_Polynomials.pdf - complete research paper
  main.tex - complete editable LaTeX source, including bibliography
  verify_quartic.py - optional exact arithmetic for the quartic examples
  verify_local_geometry.py - optional exact local/global example checks
  verify_cubic_constants.py - universal cubic coefficient identities and constants
  build.py - portable Tectonic build with a fixed source-date epoch
  CITATION.bib - citation for this paper
  MANIFEST.sha256.json - SHA-256 identities of the other eight files

The paper proves the all-fixed-degree prescribed-gradient minimizer theorem
and coefficient-l1 stability near the balanced gradient, with optimal
distance order and a count-law interpretation. It also classifies strict
local minimality at every positive gradient and gives a cubic global
obstruction inside the favorable local region. For every feasible ternary
cubic the CLOSED gradient ball ||t-1||_1<=1/8 has deficit at least d/256
and coefficient-distance constant 1/3584, including singular dispersion.
Its elementary coefficient proof uses no external tensor-norm theorem.
General-degree global constants remain qualitative. The balanced uniqueness
theorem is a cited prior input. The proof has no computational premise.

Rebuild
  With a conventional LaTeX distribution:
    pdflatex -interaction=nonstopmode -halt-on-error main.tex
    pdflatex -interaction=nonstopmode -halt-on-error main.tex
  Alternatively:
    python -B build.py
  To specify an installed executable:
    python -B build.py --tectonic /path/to/tectonic
  Required packages: fontenc, amsmath, amsthm, newtxtext, newtxmath,
  geometry, hyperref. No external bibliography database is needed.
  The compiler output is main.pdf. PDF metadata and bytes may vary with
  compiler versions and build time; the manifest identifies the supplied PDF.

Optional arithmetic reproduction (Python 3, standard library only):
    python -B verify_quartic.py
    python -B -O verify_quartic.py
    python -B verify_local_geometry.py
    python -B -O verify_local_geometry.py
    python -B verify_cubic_constants.py
    python -B -O verify_cubic_constants.py
  The checker evaluates the defining quartic on the Boolean cube at six
  rational parameters. Interpolation certifies the two degree-four identities.
  Real stability and nondeterminantal representability are cited mathematical inputs. The local/global checker independently
  expands categorical products with exact rational arithmetic and checks the
  displayed cubic identities, strict positivity, local-region membership and
  the nonpower tie. 
  The cubic-constants checker verifies all coefficient identities on the
  four-dimensional translation-invariant cubic space by basis expansion,
  and compares the reserve thresholds with exact rational arithmetic.
  The universal stability, PSD and analytic inequalities are proved in the manuscript.
  No output files are created.

Integrity
  Recompute SHA-256 for the eight files named in MANIFEST.sha256.json.
  A matching digest establishes byte identity, not mathematical correctness.

This is a scholarly source bundle. The paper discloses AI assistance.
