Sharp Quadratic Smoothing for Truncated Simplex Sums
Scott Schubr — October 24, 2026

The paper proves joint log-concavity of the raw truncated simplex lattice sum
in three positive REAL spatial endpoints, for every nonnegative integer radius
and independent real powers at least two. The universal power endpoint two is
sharp. Complementary theorems treat up to TWO arbitrarily repeatable occupancy
coordinates with any number of MATROID binary companions and binary activities
in [0,1], uniformly in the radius and dimension. All independent real powers
are at least two. The unrestricted higher-dimensional simplex square kernel
is open; no three-free-coordinate/matroid-companion theorem is asserted.

Integer-endpoint midpoint log-concavity has prior coverage through unimodular
fiber theorems. The contribution concerns the prescribed real positive-part
function; integer anchor inequalities do not alone establish its real joint
concavity. The two-dimensional baseline and monotone fractional lift are
credited to the author's earlier prefix manuscript. Classical ALOV complete
log-concavity, Branden-Huh homogeneous product/equivalence and Prekopa
marginalization are tools, not new principles here. The abstract matroid
transfer uses classical contraction and rank-two parallel-class structure.

Files
  Paper.pdf / main.tex: manuscript and editable scholarly source.
  verify_cells.py: all fourteen complete three-dimensional cell cones,
    ninety-eight principal-minor identities and forty-two direct-sum guards.
  verify_deep.py: the uniform three-dimensional deep-region component.
  verify_lorentzian.py: all fifteen uniform one-row quadratic types and N=0.
  verify_two_rows.py: eight symbolic two-row chambers, seventy-two quadratic
    derivative types, all seven bordered signs and the N=0 cases, including
    explicit positive polynomial multipliers when needed.
  verify_baseline.py: the previously established two-dimensional identities.
  CITATION.bib: citation metadata for this manuscript.
  SHA256SUMS.txt: SHA256 identities of the other public files.

Reproduce with Python 3. The baseline checker uses only the standard library.
The deep, Lorentzian and two-row checkers require SymPy; the cell checker requires
python-flint. The exact replays used Python 3.12, SymPy 1.14.0, mpmath 1.3.0
and python-flint 0.9.0. These are ordinary publicly available dependencies:

  python -m pip install sympy==1.14.0 python-flint==0.9.0
  python -B verify_baseline.py
  python -B verify_deep.py
  python -B verify_lorentzian.py
  python -B verify_two_rows.py
  python -B verify_cells.py all

Repeat each command with -O to retain and check the explicit exception gates.
Every checker uses exact integer/rational polynomial arithmetic and reads no
data files, downloads nothing and writes no files. Each prints one JSON object.
The cell checker also accepts any case name displayed in its EXPECTED table,
so its finite identities can be inspected separately. Its complete expansions
contain millions of terms; memory and runtime are substantially greater than
for the short baseline checker.

The complete coefficient signs certify unbounded parameter families, not a
finite numerical grid. Direct original-sum guards catch transcription errors;
they are not the source of the uniform theorem. Analytic case coverage,
dimension reduction, gluing, complete-log-concavity hypotheses and fractional
marginalization must also be checked in the manuscript. Internal exact and
mathematical checks are not external human refereeing or proof-assistant
verification. No claim to worldwide priority or to solving the unrestricted
higher-dimensional kernel question is made.
