# Verification report for the separated polynomial bound

Version 1.0, 17 October 2026. Author: Alper Ferudun.

## Accepted mathematical scope

For every positive even k and m >= 1, the exact maximum number of real
solutions of f_i(x_i) = g_i(t), with all degrees at most k and a finite
entire real solution set, is k^2/4 for m = 1 and m(k-1)k^m/4 for m >= 2.
The contribution is the upper bound for the separated class. The attaining
family for m >= 2 is credited to DannyExperiments, DOI
10.5281/zenodo.21875290. The unrestricted extremal problem remains open.

The written proof was accepted after the originating researcher's detailed
self-audit. No known mathematical gap is recorded in this scoped argument.
It uses the local root charge mu - o/2, the even-cap degree budget, the
normalization of even-degree left-hand sides by their global minima, and
the fiber loss at zero levels. Constant left-hand sides, identically zero
right-hand sides, nonpositive selectors, odd actual degrees below the even
cap, multiple roots, and the m = 1 case are handled explicitly.

This is not proof-assistant verification or independent human review.

## Exact supplementary checks

- Python uses rational polynomial evaluation and explicit failure checks:
  553,421 checks, including 174,313 finite systems and 96 sharpness instances.
- C++17 uses integer root-parity sign transitions, not the Python polynomial
  expansion algorithm. Across eight catalogs, both implementations agree
  on feasible-system counts, largest weights, and maximizing-system counts.
- Normal and optimized Python runs have byte-identical expected receipts.
- Negative controls demonstrate why a nonpositive selector needs separate
  treatment, why the odd-degree cap cannot be substituted, and why the
  numerical bound cannot extend to arbitrary sums of squares.
- Positive quadratic factors test the unused nonreal-root degree budget.

The finite catalogs do not enumerate all real polynomials and do not prove
the universal theorem. Their parameters are recorded in the receipts.
The portable replay regenerates all receipts from an extracted source ZIP,
checks them against the retained expected files, and compares the two
implementations. See `README_submission.md` for the command.

## Presentation and provenance

The five-page PDF compiled successfully with both the built-in compiler
and Tectonic. All five rendered pages were directly inspected, with no
clipping, broken mathematics, or overfull boxes. The package validation
also rebuilds the extracted LaTeX and checks PDF text and content streams.

The prior construction is attributed in the abstract, introduction,
sharpness section, bibliography, and metadata. Its full source was read.
The bounded literature comparison does not certify absolute priority.
No third-party full text is included in the public source ZIP.