OMEGA object exploration 02

Definitions follow Paper VII v1.6 and paper 8, Section 2: $\varphi=(1+\sqrt5)/2$, $s=\sin(\varepsilon/2)$, $A=s+\varphi^2$, $B=s+1$, nodes $(-A,-B,-s,s,B,A)$. Historical priority of these applications is not assessed.

1. Four outer marks have ordered cross-ratio $R=1+\varphi^2/(4(s+\varphi^2)(s+1))$, invariant under a nondegenerate line projectivity. Within the declared family, $s=\frac{\varphi}{2}\bigl(\sqrt{R/(R-1)}-\sqrt5\bigr)$. This does not recover scale, pose, or the full outline, and is not an experimental recognition accuracy claim.
2. Affine records $p_i=m+z_i q$ with $m=\alpha+ac\,u$ and $q=av$ have minimum distance five: two corruptions or four erasures are uniquely recoverable; three corruptions need not be. Projective image coordinates have minimum distance four: one corruption or three erasures. Exact remaining records and trusted labels are required. Ordinary measurement noise is a separate problem. The decoder rejects truncated, empty, scalar, and dimension-inconsistent positions.
3. Uniform baseline integration on $[-A,A]$ using only these six nodes admits a strictly positive degree-three rule and no nonnegative degree-four rule at the canonical seed. The obstruction polynomial $P(t)=(t^2-B^2)(t^2-A^2)$ is nonnegative on the nodes while $I(P)<0$.
4. At $s=(\sqrt5-1)/4=\sin(\pi/10)$ one has $\varepsilon=36^\circ$. The four outer nodes, normalized to $[-1,1]$, support the positive four-node Gauss–Lobatto rule of degree five with weights $1/12,5/12,5/12,1/12$. Both inner nodes receive zero weight. That changes the seed angle; it does not replace the one-arcsecond kernel. Scalar readouts $G=a\varphi$ and $C=a\pi$ can stay the same while the integration property changes.

• OMEGA_Object_Exploration_02.md — 20,104 bytes, SHA-256 12f3106a7980ce56e328c3ae29392f306c1c321e4c7c92decbce38fa00560513Copy. Restored notation, including the inner-weight equality $I(P)/P(s)=2A^2(5B^2-A^2)/(15(s^2-B^2)(s^2-A^2))$.
• OMEGA_Object_Exploration_02.py — 15,581 bytes, SHA-256 4c1b3cfa46262e21d19ece458d4dcb8fccd617d15ca4579a2f013108396c13a6Copy. Spec-applied 21-group checker with truncated-coordinate rejection.
• this_archive_exploration_results.json · optimized results · this_archive_exploration.log · ENVIRONMENT.json · HOSTING.md

The corrected checker completed 21 check groups: the original 19 mathematical groups and two groups verifying malformed-input rejection. Normal and optimized Python runs both passed. The decoder rejects inconsistent or empty coordinate vectors, incorrect record counts, and requested recovery budgets outside the proved bounds. The accompanying note restores the damaged mathematical notation.

The instruction named author files of 15,034 bytes (6c35365d…) and 19,434 bytes (e50f3443…). Those exact bytes were not in the upload, so public downloads match the hosted spec-applied files above, not those two hashes.

Python 3.10.21 / SymPy 1.14.0 / mpmath 1.3.0. No author OMEGA modules imported. 21 check groups PASS from len(checks): the original 19 mathematical groups and two malformed-input groups. Results JSON script_sha256 equals the hosted checker.

python3 OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.json
python3 -O OMEGA_Object_Exploration_02.py --output this_archive_exploration_results.optimized.json

A prior-run exploration_results.json was not attached. These this-archive files are not a substitute for a missing author transcript.

Not an additional source paper, not peer review, not a proof-assistant certificate, not an enlargement of Paper VII or paper 8, not a blowup theorem, and not a practical camera-noise claim. Frozen G₆ is unchanged. Related pages: geometry, sampling extension, boundary reconstruction, Paper VII, paper 8.