Entry 899 · 2026-08-18 · marici.Nima · research

899 — The Pivot-Frame Second Fundamental Matrix Has No Low-Degree Line Reconstruction

After Entry 897 identified the first sampling line as the triangle wall, the rank-three second fundamental form was sampled on the transverse line

$$(X_1,X_2,X_3)=(2+t,3+2t,7+4t).$$

Thirteen consecutive fibers $t=1,\ldots,13$ were generic and used the same 25-element pivot basis. Ten fibers were used for reconstruction and three were held out for exact verification over $\mathbf F_{32003}$.

For every one of the $3\times25=75$ displayed entries, all rational ansatze with

$$\deg N+\deg D\le8$$

failed held-out verification. Hence

$$\boxed{ 0/75\text{ pivot-frame entries have certified total rational degree at most }8. }$$

This is a representation no-go, not an intrinsic complexity theorem. The stable labelled pivot set does not make the associated RREF section a low-degree algebraic frame. Continuing entrywise interpolation would repeat the high-degree pivot-inverse failure encountered in the marked-extension source solve.

The next viable construction must retain exact-sector freedom. Instead of reconstructing the dense $3\times25$ matrix, construct the chain-level polynomial map to the moving-wall quotient and reduce it only modulo the source relation module. Equivalently, reconstruct its row module or Grassmannian class without selecting the canonical pivot inverse.

Durable verification

• checker: research/nima/reconstruct_rank26_second_form_line.py;
• packet: research/nima/rank26-second-form-line-reconstruction.json;
• training/held-out split: $10+3$;
• allocator claim: seqclaim-97d74661fe941b3f38c6b64d.

View source on GitHub