Dimension thresholds for log-concavity of normalized mixed moments
Scott Schubr
October 24, 2026

Main results
For d>=2 and arbitrary nonnegative coordinatewise increasing integrable
weights of positive finite mass on a positive finite box, mixed moments
normalized by product Gamma(alpha_i+1) are JOINTLY log-concave when
alpha_i>=d/4. For finite products of nonnegative ordered log-concave
coordinate-value lists, positive top nodes and an index-increasing
nonzero weight array, alpha_i>=d/4+4 suffices. Repeated coordinate values
are allowed and retain their multiplicity. Varied exponents are positive,
so zero grid nodes contribute zero before logarithms are differentiated.

These theorems require neither log-concavity nor independence nor
positive association of the weight. Actual increasing box densities and
finite arithmetic-grid weights give counterexamples at d/4-O(1). Thus
the separate universal thresholds in both classes have optimal leading
coefficient 1/4.  Scalar normalized-Mellin ancestry is classical
and explicitly credited. The Mellin assertions concern finite boxes and finite grids.

Spatial power smoothing
For fixed nonzero nonnegative coordinatewise decreasing inputs, one-sided
product-power truncations are jointly log-concave in their spatial
endpoint A when every power is at least d/4+4 for locally integrable
continuous inputs, or d/4+7 for finite-valued decreasing arrays on the
nonnegative integer lattice. Here d>=1. Infinite global mass or array
support is allowed because each active integration region is bounded
and each active sum is finite. These are distinct from the finite-grid
Mellin moment hypotheses above. The spatial common-power thresholds also have optimal leading coefficient 1/4. 

The explicit spatial witnesses use power 3 and dimension 15. On the
integer grid at A_i=100 the positive spatial Hessian eigenvalue times
100 squared is 5001/40804. An honest decreasing lower-box density at
A_i=1 has a positive eigenvalue at least 74686/332667. They concern general decreasing weights. The simplex
application instead proves the stated high-power all-dimensional bound.

Binary structure
An explicitly attributed consequence of classical matroid polynomial
log-concavity gives the dimension-free endpoint p_i>=2 for binary matroid
supports with activities in [0,1], including independent real powers.
The same compensated-transform calculation is stated with its exact
positive C2/log-gradient hypotheses. Unweighted hereditary binary
supports behave differently: the standard K_(m,m) family fails at every
fixed power for sufficiently large m, and m=ceil(50p) gives a failure
for every real p>=2. Its universal power threshold has order d. The exact quadratic
52-dimensional witness has positive eigenvalue
8*9^25/(2*10^26-9^26).

These are statements about raw REAL spatial interpolation.
The matroid input and the complete-bipartite graph family are classical.

Public files
main.tex           Complete editable LaTeX manuscript with embedded bibliography.
Paper.pdf          Compiled manuscript, when supplied in the prepared package.
verify_moments.py  Self-contained exact rational certificate checker.
README.txt         This document.
CITATION.bib       Citation metadata for this manuscript.
SHA256SUMS.txt     Exact public-file identities, when supplied in the package.

Verification
Use Python 3 from this directory:
  python -B verify_moments.py
  python -O -B verify_moments.py
Both runs must succeed and print identical JSON with status PASS. The
checker uses only the standard library, reads no external files and
writes no files. It reconstructs the rational finite-grid witness with
positive normalized-Hessian eigenvalue greater than 773/60000, the honest
continuous witness with gap greater than 2287/39204, mass balances,
dimension identities, spatial witnesses with their coordinate scale,
the exact binary witness and its scalar polynomial algebra, and threshold
arithmetic. Its finite identities supplement the analytic proofs of the general theorems, conditional laws, approximation limits and asymptotic embeddings.

Compile main.tex with an ordinary LaTeX installation supporting its
standard packages. Run enough passes to resolve references and citations.
The bibliography is embedded; no bibliography download is required.

Provenance and scope
The mixed-moment question arose in the author's unpublished prefix and
toric companion investigations. This paper develops joint exponent
concavity and spatial power smoothing with matching leading dimension
thresholds; the companion work supplies the cited kernel and geometric classification results.
Classical scalar IFR/Borell, normalized-Mellin, convolution and fractional
integration and matroid log-concavity results are credited. Scalar
integration order is separated
from joint spatial endpoint concavity.
 

AI-assisted development, source comparison and internal verification are
disclosed in the manuscript. 


Building the manuscript
-----------------------
From the directory containing main.tex, run `tectonic main.tex`. The packaged primary PDF contains the same manuscript as these editable sources. The accompanying mathematical code and certificate data are retained with their original scope.
