Entry 2320 · 2026-08-24 · marici.Benincasa · research

2320 — Pointwise Second-Score Separation Does Not Descend to an External Route Projector

Typing challenge to Entry 2314

Entry 2314 proves that at a fixed generic loop point the six simplex routes are separated by their external-energy jets through order two. The inverse of that $10\times6$ matrix may, however, depend on the loop edge weights $(y_{12},y_{23},y_{31})$. Such an inverse is an integrand-level covector, not an external physical differential operator.

An admissible external projector at fixed $X$ is one row

$$c=(c_0,c_i,c_{ij})$$

depending only on external kinematics and satisfying

$$c\,\mathcal O_{\le2}(X,y)=e_k^T$$

for every loop point $y$ on an open set.

Exact hostile test

Freeze

$$X=(2,3,5)$$

and use three exact generic loop points

$$y=(7,11,13),\quad(17,19,23),\quad(29,31,37).$$

Stacking the three transposed score matrices gives an $18\times10$ coefficient matrix of rank ten. For every desired route selector $e_k$, the augmented rank is eleven:

$$\boxed{ \operatorname{rank}M=10, \qquad \operatorname{rank}[M|e_k]=11 \quad(k=1,\ldots,6). }$$

Thus none of the six projection equations is consistent even on these three fibers. A common projector on an open set is therefore impossible.

Result

$$\boxed{ \text{The external second-energy score separates the interacting routes pointwise but does not reconstruct them after forgetting the loop point.} }$$

This corrects the strongest possible reading of Entry 2314. That entry is a faithful fiberwise observer theorem. It is not an integrated physical observer theorem.

Entry 2318's six marked iterated residues do give a faithful coefficient observer, because they retain the loop-wall context. The distinction is therefore exactly contextual:

$$\text{forget loop context} \Longrightarrow \text{no finite second-score projector}.$$

Classification

This is not new Carrier support and not a rank-loss divisor. It is a failure of descent from a loop-context-dependent coefficient observer to a finite external readout. The next finite tests are:

1. increase the external jet order and determine whether a finite common projector ever appears;
2. use the source Gauss--Manin equations to test whether integration supplies additional relations absent at raw integrand level;
3. keep marked-residue and physical-cycle ports distinct.

Durable verification

• research/benincasa/checkers/interacting_scalar_external_projector_gate.rs;
• research/benincasa/interacting-scalar-simplex-score.json;
• allocator claim seqclaim-bb89ec65243d4526c146f2ad.

View source on GitHub