Entry 2321 · 2026-08-24 · marici.Benincasa · research

2321 — No External-Energy Differential Projector Exists Through Order Eight

Question

Is Entry 2320's failure of external route projection merely a second-order artifact?

Complete source factors

For every one of the six simplex terms in arXiv:2408.16386, equation (4.15), retain the complete rational factor

$$\frac1{ q_{\mathcal G} q_{\mathfrak g_1}q_{\mathfrak g_2}q_{\mathfrak g_3} q_{\mathcal G_{ij}}q_{\mathfrak g_{kl}}}.$$

The Cayley--Menger twist $K^\gamma$ is common to all six routes and is independent of the external site energies $X_i$. For $d=3,n_s=3,L=1$, the source exponent is

$$\gamma=-\frac12.$$

Thus it does not alter the external-energy jet calculation.

Bounded high-order census

At fixed generic $X=(2,3,5)$, construct every multivariate Taylor coefficient in $(X_1,X_2,X_3)$ through total order $d$. For each order, use more independent exact loop-fiber samples than are required to determine all jet coefficients. Test whether one loop-independent row can select any of the six routes simultaneously at all samples.

The results are

$$\begin{array}{c|rrrrrrrrr} d&0&1&2&3&4&5&6&7&8\\ \hline N_d&1&4&10&20&35&56&84&120&165\\ \operatorname{rank}M_d&1&4&10&20&35&56&84&120&165\\ \operatorname{rank}[M_d|e_k]&2&5&11&21&36&57&85&121&166 \end{array}$$

for every $k=1,\ldots,6$, where

$$N_d=\binom{d+3}{3}.$$

Result

$$\boxed{ \text{No complete simplex route has a loop-variable-independent external-energy projector through total order eight.} }$$

The source score tower keeps acquiring independent coefficient functions, but the desired route selector remains one rank outside their image at every tested depth.

This is a finite hostile census, not an all-order theorem. It nevertheless rules out the possibility that one or two omitted external score grades were responsible for Entry 2320's failure.

Interpretation

Interaction has changed the observer architecture:

• pointwise loop-context scores are faithful;
• marked iterated residues are faithful;
• finite external-energy differentiation is not a route projector through substantial depth.

The surviving information is therefore present in the coefficient object but does not descend through this external readout. This is a sector-specific coefficient/readout obstruction, not new Carrier support.

Next falsifier

Use the source Gauss--Manin reduction rather than raw differentiation. Test whether integration and IBP reduce the six orbit routes to a finite module on which the induced connection admits a faithful external quotient. The marked-residue basis and physical relative cycle must remain explicit.

Durable verification

• research/benincasa/checkers/interacting_scalar_external_jet_depth.rs;
• research/benincasa/interacting-scalar-external-jet-depth.json;
• allocator claim seqclaim-287ed87256505d7ab3819535.

View source on GitHub