Finite-order positivity and growing negative inertia for averaged Riemann theta products
Scott Schubr

Read main.pdf for exact statements, complete proofs, attribution and limitations.
The principal new result is a common negative-subspace theorem for the complete
theta upper-tail form at A=23/2: each parity through actual degree N has inertia
at least 3 log(N)/(8 pi)-O(log log N). The full form requires correction rank at
least 3 log(N)/(4 pi)-O(log log N), even for degree-dependent corrections that
couple parities. The effective conservative count can be vacuous at enormous
finite degrees. Negative reserves use the stated reference metric, not raw
monomial-coordinate eigenvalue floors. No physical all-order positivity, Riemann
hypothesis, optimal inertia asymptotic or practical failure-onset claim is made.

This is a new paper in the same theta research family. Previously published
finite positivity through degree eight, terminal positivity, exact rational
witnesses and eventual degree bounds are explicitly attributed to:
Scott Schubr, Monotone center weights and the averaged odd-quintic theta gate,
published edition 6, October 24, 2026:
https://research.luminaryoddities.com/monotone-center-weights-averaged-odd-quintic-theta-gate/v/6
The old paper retains its publication identity. Reused results are not claimed
as new independent contributions here. Classical tools are credited in the paper.

Source organization
  main.tex                        document driver
  introduction.tex                question, contribution and attribution
  framework.tex                   common definitions and essential analytic tools
  negative_inertia.tex            growing-inertia theorem and repair obstruction
  cutoff_obstruction.tex          inherited complete Fourier obstruction
  constructive_obstruction.tex    inherited rational witnesses and degree bound
  conclusions.tex                 conclusions and remaining mathematical questions
  negative_band_certificate.tex   complete new closed-band enclosure proof
  finite_order_theory.tex         full inherited finite positive theory
  arithmetic_certificates.tex     inherited detailed arithmetic certificates
  references.tex                  bibliography

Public computational supplements include the inherited twenty Python files,
six inherited exact witness JSON files, and the new complete-band and inertia
checkers with theta_negative_band_nodes.json. The inventory and every file hash
except the manifest's own hash are in MANIFEST.sha256. README.txt and CITATION.cff
provide reader instructions and citation metadata. There are no private review,
assessment, correspondence, publication-control or privacy-receipt documents.

New closed-band and inertia checks
  python -m pip install python-flint==0.9.0
  python verify_theta_negative_band.py
  python verify_theta_inertia.py
Repeat with python -O. Explicit exception acceptance gates remain active.
The band consumer reconstructs all product and integration tails, complex-disk
majorant and interpolation error. It uses the eight supplied outward node
integrals. To regenerate them independently from the full Mellin-series evaluator:
  python verify_theta_negative_band.py --recompute-nodes > fresh_nodes.json
  python verify_theta_negative_band.py --nodes-file fresh_nodes.json
Keep regenerated output in a separate working directory. The inertia checker
verifies constants for the analytic common-space theorem, not a finite-dimensional
numerical simulation of its asymptotic conclusion. The manuscript supplies the
concentration, reflection, projection and rank arguments.

Inherited supplements and reproduction
The following reader instructions apply to the verbatim inherited code and data.
Their finite computations do not replace the analytic proofs or imply a fresh
independent mathematical implementation. No reuse license is specified.

Build with a current LaTeX installation:
  tectonic main.tex
or run pdflatex main.tex twice. Standard packages: geometry, fontenc, amsmath,
amssymb, amsthm, booktabs, microtype and hyperref. No external bibliography,
images or private files are required to compile the PDF. The six inherited witness JSON
files are required separately for the exact polynomial witness verification.
Tectonic can download its normal support bundle on first use.

The sixteen retained arithmetic scripts use only the standard library and reconstruct their
own arithmetic. Run:
  python verify_prefix.py
  python verify_tail.py
  python verify_even.py
  python verify_separated.py 1024
  python verify_counterexample.py
  python verify_sextic_initial.py 256
  python verify_sextic_tail.py
  python verify_sextic_mean.py 1024
  python verify_septic_tail.py
  python verify_septic_compact.py 1024
  python verify_septic_residual.py 1024
  python verify_octic_tail.py
  python verify_octic_compact.py 1024
  python verify_octic_residual.py 1024
  python verify_hermite_tail.py
  python verify_finite_order_tail.py 0
  python verify_finite_order_tail.py 8
  python verify_finite_order_tail.py 10
Repeat with python -O in place of python. Explicit exception acceptance gates
remain active. Each corresponding normal/optimized output should agree.
The complete-cell integrators are slower than the polynomial checks and print
progress on stderr. Specified grids reproduce the manuscript's finite checks.

Both septic and both octic compact checkers cover every real cutoff in
[0,7/100] by signed whole-cell loss intervals, not by sampling cutoff values.
Their shared outward rational arithmetic engine is not a collection of
independent arithmetic implementations. Residual integrators split the cell
containing7/100 and retain the full inverse penalty and output cross coupling.
Omitted theta atoms and all head-tail/tail-tail terms remain rigorously bounded.

The Hermite checker reconstructs121 generic formal coefficient pairs through
degree10 and exact rational constants. This bounded control is not an all-N
proof by testing; the analytic row/disc theorem in the paper supplies that proof.
The finite-order algorithm returns full halfline parity bounds. Its safe bound
is theoretical and can be very large; no efficient complexity is claimed for
arbitrarily high input degrees. Input degree0 is included; negative degrees
are rejected. Returned low-order thresholds are not new averaged theorems.

The inherited published compact fixed-center classification is a stated
mathematical dependency. Its historical complete computation is not rerun by
these shorter scripts. Complete-tail estimates and finite-to-global arguments
are proved in the manuscript. Normal/-O agreement repeats one algorithm and
is not external human refereeing or formal proof-assistant verification.

The fixed-center sextic obstruction and generic smooth-density quintic
counterexample do not refute the actual averaged theta conclusions. Arbitrary
nonmonotone weights are covered only in the explicitly stated supporting
variance or far-center statements. No reuse license is specified.

Complete-theta Fourier certificate reproduction
  python -m pip install python-flint==0.9.0
  python verify_theta_cutoff_fourier.py
  python verify_theta_cutoff_direct.py
Both require Python 3.10+ and write their certificate JSON to standard output.
The strict outward-enclosure gates, not requested quadrature tolerances,
establish negativity. Run with -O as an additional control: acceptance uses
explicit exceptions rather than removable assert statements. No finite atom
truncation or numerical plot is substituted for the paid complete tails.

Exact ordinary rational polynomial witnesses
-----------------------------------------
The files theta_witness_even.json and theta_witness_odd.json specify all
coefficients exactly in the rational Meixner-Pollaczek basis stated in the paper.
They have degrees4096 and4097. Coefficient selection is not a proof assumption.
Four theta_witness_grid_START_STOP.json files supply outward enclosures at all
4000 exact half-grid nodes x_k=(2k+1)/20, k=0,...,3999.

Install python-flint==0.9.0, then run:
  python verify_theta_polynomial.py
  python -O verify_theta_polynomial.py
The fast consumer checks exact polynomial association, every index, point-ball
arithmetic, exact reference/Jacobi norms, and both complete-integral reserves.
It uses the supplied reviewed Fourier point enclosures; it does not recalculate
all special-function values in this fast mode.

For complete independent regeneration, create a separate directory fresh and
redirect each command's stdout to the indicated file in that directory:
  python verify_theta_polynomial.py --recompute-grid 0 1000
    > fresh/theta_witness_grid_0_1000.json
  python verify_theta_polynomial.py --recompute-grid 1000 2000
    > fresh/theta_witness_grid_1000_2000.json
  python verify_theta_polynomial.py --recompute-grid 2000 3000
    > fresh/theta_witness_grid_2000_3000.json
  python verify_theta_polynomial.py --recompute-grid 3000 4000
    > fresh/theta_witness_grid_3000_4000.json
Each command and its redirection belong on one shell line. These four independent
ranges can run in parallel. Computation can take several minutes, depending on
the machine. Then run:
  python verify_theta_polynomial.py --grid-dir fresh
All regenerated enclosures include the full omitted-product and series tails.
The polynomial is evaluated at2048bits and Fourier quantities at1024bits.
The fast and full routes must both put each normalized integral strictly between
-3/10000000 and -2/10000000. Both sampling and infinite-sample reserves are paid;
no Fourier plot or finite moment prefix is substituted for the full integral.
The public mathematical proof explains all analytic constants. Neither result
asserts minimal degree, an effective universal onset, physical-weight failure,
RH, external human refereeing, or a new general quadrature principle.
