# Verification and scope

This self-contained proof answers the exact numerical-coordinate feasibility
question in Shapiro2015 §8 Problem6 for source-defined polynomial-like
functions. It covers every n and permitted cross-row coincidences. A finite
trace certificate and real-algebraic quantifier elimination provide a
terminating procedure for encoded algebraic coordinates. Ordinary-polynomial
witnesses may have unspecified larger algebraic degree. No practical
complexity, arbitrary-real encoding or minimum witness degree is claimed.

The full analytic proof eliminates integration constants, proves kernel
full span using compact smooth jets, constructs an exact positive density
from interior moment vectors with a mandatory constant baseline, and uses
repeated Rolle to certify exact simple global derivative roots. Isolated
jump values are excluded; both one-sided traces are retained.

Finite controls corroborate implementation, not the theorem. Two Python
runtimes gave byte-identical exact outputs. A separate JavaScript rational
implementation passed5912assertions on49arrays of degrees2--8. Primary
cross-comparison matched570baseline identities and13035piece vectors. No
numerical root solver or sampled positivity establishes the analytic proof.
Checkers and actual receipts accompany the source archive.

Kakeya1915, di Dio2019/2025 and Pinkus--Wulbert2005 supply classical generic
and interpolation-specific moment-cone methods. Exact cubic results are
credited to Shapiro--Shapiro; arrangement results are distinguished from
positions. The inaccessible1979 Hermite--Birkhoff/monotone-spline source
remains an overlap risk. Whether this application is already-known is
uncertain; no absolute priority or exhaustive novelty clearance is claimed.

This is an AI-assisted, unrefereed preprint accepted after detailed
originating-researcher self-audit. There is no independent human review or
formal proof-assistant certification. Native LaTeX compilation succeeded,
installed cached Tectonic export succeeded, all five pages were visually
inspected without clipping, overlaps or broken mathematics, and final logs
contain no TeX warning. These artifact checks do not certify novelty.