Additional note, not a source paper

Affine dynamics, sampling, and quadrature

Complete positive-quadrature classification for the symmetric six-node family, a cross-ratio criterion, and the sharp amplification of bounded sample errors. Canonical seed remains one arcsecond.

Classification

Nodes $(-A,-B,-s,s,B,A)$ with $A=s+\varphi^2$ , $B=s+1$ , normalized by $A$ . Uniform average $I(f)=\tfrac12\int_{-1}^{1}f$ . Unique degree-five pair masses are (3) in the note. Sign of the inner mass is the sign of $5\beta-1$ .

Largest nonnegative exactness degree:

$d_+(s)=3$ for $0<s<s_*$ , and $d_+(s)=5$ for $s_*\le s<1$ , where $s_*=(\sqrt5-1)/4=\sin 18^\circ$ . Signed exactness is five throughout; six is impossible even with signed weights. The one-arcsecond seed is in the first row. At the threshold the inner pair has zero weight and the four outer nodes are Gauss–Lobatto.

Within the family, nonnegative degree-five weights exist iff $R\le R_*=(5+3\sqrt5)/10$ . Sharp amplification of additive sample errors of size $\delta$ is $\delta\sum|w_i|$ . Below the threshold this factor is strictly larger than one.

Hosted files

PROOFS.md — 28,146 bytes, SHA-256 99d6b58e29940ea9e06d8585cccf0162a4f99a864c6748d416f393ec3cf2e636
verify.py — 18,356 bytes, SHA-256 381d091c92928a42a970f79c08f366a3681f74352b4780b4a16b93d862e60ce0
Verify.java — 23,485 bytes, SHA-256 4b00c0ee5edd80376d7179ed93721d53fb5fd3cfd5aeca31404e64acef246517
expected.txt — 5,683 bytes, SHA-256 7ef550cb3f03d5032769024b9cbeb47f5c143ef45f709a0f1b785128684fd662
README.md, REPRODUCTION_REPORT.md, run_all.py, SHA256SUMS.txt, HOSTING.md

Author report: eight executions matched expected.txt. Hosted author evidence includes Python, Java, and Rust stdout transcripts (all SHA-256 7ef550cb3f03d5032769024b9cbeb47f5c143ef45f709a0f1b785128684fd662) plus rust-build stderr and cargo/java version files. This archive has not rerun Julia or Rust.

This-archive run

Python 3.10.21, OpenJDK 17.0.20. Stdout of all four commands below is byte-identical to expected.txt. TOTAL 21 PASS

python3 verify.py
python3 -O verify.py
javac Verify.java && java Verify
java -da Verify

Missing

Not received, not reconstructed: verify.jl, the Rust crate (src/main.rs, lockfile, vendor), RUNTIME_SOURCES.tsv, and the Julia/Rust version transcripts. Author rust stdout is hosted; the crate needed to rebuild it is not. Full list: MISSING.txt.

What this is not

Not an additional source paper, not peer review, not a proof-assistant certificate, not a four-language this-archive completion, and not an enlargement of Paper VII or paper 8. Related: exploration 02, geometry, reproduce.