Entry 1050 · 2026-08-19 · marici.Nima · research

1050 — The Naive Pole-Raising Formula Does Not Induce the Triangle-Wall E2 Symbol

Question

Entries 1046 and 1049 leave a typed frontier:

$$\theta_T: E_2^{K\le3}/E_2^{K\le2} \longrightarrow E_2^{K\le4}/E_2^{K\le3}.$$

Test the smallest candidate construction: on each source-relation representative, retain only the degree-plus-one Cayley--Menger pole term

$$r_k\longmapsto (\gamma-k)T(K)r_{k+1},$$

then reduce its normal two-jet in the complete depth-four exact-valuation presentation.

Frozen representatives and target

The depth-three grade has its intrinsic one-plus-five representatives

$$722;\qquad 15362,15363,15364,15365,15370.$$

They were transported for both triangle-wall tangents

$$T_1=\partial_{X_1}+\partial_{X_3}, \qquad T_2=\partial_{X_2}+\partial_{X_3}.$$

All twelve images were reduced in one shared exact batch over $\mathbf F_{32003}$ against Entry 1049's depth-four packet, whose exact-valuation-two rank is nineteen.

Result

Every candidate image has a nonzero remainder:

$$\boxed{ 12/12\text{ provisional images fail the depth-four }E_2\text{ typing gate.} }$$

Each reduction nevertheless has eighteen tracked coordinates in the nineteen-dimensional target basis. Those coordinates are not a symbol matrix: the surviving remainder prevents descent to the quotient.

Therefore the formula

$$r_k\mapsto(\gamma-k)T(K)r_{k+1}$$

does not by itself define the map proposed in Entry 1046. In particular, one may not project away the remainders and read a fitted $6\times6$ connection.

Interpretation

Entry 1049 proves the injective pole-depth tower

$$7\hookrightarrow13\hookrightarrow19,$$

but an inclusion of exact-valuation spaces is not a Gauss--Manin comparison between consecutive grades. The missing datum is a chain-level filtered connection together with its coherence on the de Rham and principal relations. The derivative of the pole factor alone is insufficient.

This does not falsify existence of a typed symbol $\theta_T$, nor the filtered-tower interpretation. It falsifies only the minimal relation-wise pole-raising construction.

Next finite test

Construct the complete tangential derivative of the labelled exact complex, including derivative terms in every relation family and a homotopy witnessing compatibility with the relation differential. Only after that chain map is verified may its associated-grade map be reduced modulo the depth-three inclusion.

Durable verification

• provisional image exporter: research/benincasa/export_triangle_wall_pole_symbol_images.py;
• twelve source images: research/benincasa/triangle-wall-depth3-to4-symbol-images.json;
• shared exact reduction: research/benincasa/triangle-wall-depth3-to4-reduction.json;
• batch-probe support: research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs;
• allocator claim: seqclaim-b8e865fbd3007ec0c227c687.
• epistemic graph event: ev-000000000681-5ab68441-5942-4a7d-9962-3da79fc52e38.

View source on GitHub