# Verification and scope report

The written elementary proof constructs a degree77 polynomial whose78
coefficients are strictly positive. Its original unweighted local quantities
c_k=a_k²−a_(k−1)a_(k+1), with zero outside coefficients, have a good-index list
with exactly one parity change. The polynomial has at least three distinct
negative real roots, disproving the proposed bound of one.

The proof uses a degree38 alternating block, an exact weighted Dirichlet-energy
identity, a strictly negative rational half-elasticity bound and reciprocal
gluing. The resulting upward-crossing root at−1, positive constant and negative
odd-degree signed leading term force two further roots by continuity.

Independent integer homogeneous Horner evaluations give signs +,−,+,− for
g(t)=f(−t) at0,999/1000,1001/1000,2. These certify roots in three disjoint
intervals. Clearing denominators and dividing by the gcd produces positive
primitive integer coefficients without changing roots or c_k signs.

The independent checker imports no constructor code. It checks every integer
c_k, the complete good-index list, rational interval signs, harmonic formula,
both energy expressions and the exact primitive coefficient scaling. It passes
under Python3.13.5 and3.9.6 with identical scientific output. The compact
negative bound checker also passes in both runtimes. A separate proof worker
reconstructed every coefficient and the c/sign signature from the formula.
These are reproducible computational cross-checks, not human referee approval
or a formal proof-assistant certification.

The exact source hypotheses were compared with the primary journal and arXiv
text, including the nonnegative cutoff and endpoints. The primary agent viewed
complete journal PDF pages5 and6. The related Forsgård–Novikov–Shapiro tropical
obstruction is credited; it uses global tropical indices, not this local c_k
statistic. Its arXiv PDF page12 containing Lemma28 and Theorem11 was actually
viewed by the primary agent; page13 is a separate quartic section, not the
Theorem11 proof page.

The bounded primary-source search located no prior exact resolution of the
precise local-parity inequality, but does not certify exhaustive historical
novelty or priority. No minimal degree, exact total number of real roots,
weighted-neighbor resolution, human-refereed status or external publication
is claimed by this report. All failed searches and shortcuts are retained.