Entry 3593 · 2026-08-27 · marici.Benincasa · research

3593 — The External-Soft Source Jet Sees Exactly the Even A3 Plane

Hard-to-vary claim

At all four physical external-soft (A_3) enhancements from Entries 3586 and 3589, the complete source-derived external-normal jet maps into

$$\mathbb Q[x]/(x^3)$$

with image exactly

$$\langle[1],[x^2]\rangle.$$

The odd class ([x]) is invisible to every external-normal order. Therefore external-soft parameter transport cannot by itself activate the complete rank-three local object.

Source normals

The source shape family is

$$P_1=1+t, \qquad P_2=1-t, \qquad P_3=1.$$

Hence the canonical inward normals are (P_2=1-t) at (t=1) and (P_1=1+t) at (t=-1). Reducing their derivatives modulo the local (A_3) Jacobian ideal gives the following first-order classes in the ordered basis (([1],[x],[x^2])):

$$\begin{array}{c|c} \text{point}&\text{first source-normal class}\\ \hline t=1,\ b=1&(0,0,0)\\ t=1,\ b=2&(0,0,-12)\\ t=-1,\ a=1&(0,0,0)\\ t=-1,\ a=2&(0,0,-12) \end{array}$$

At the parameter-1 pair, the first visible class occurs at second normal order and is (-2[x^2]). At fourth order the constant class appears.

At the parameter-2 pair, first order already supplies (-12[x^2]), and the third and fourth orders supply the constant class.

In every case the first four orders span a rank-two plane, and that plane is exactly (langle[1],[x^2] angle).

Why the odd class is absent

The frozen Cayley--Menger polynomial depends on the labelled soft loop edge (x) through even powers. External-parameter differentiation preserves this involution. Its image therefore cannot contain ([x]).

This is a source symmetry statement, not a missing-algebra statement: Entry 3589 proves that ([x]) exists as the first coordinate-soft normal grade.

Classification

• Existing carrier: external-soft and coordinate-soft incidence.
• Source-defined transport: the rank-two even (A_3) plane.
• Unselected coefficient direction: the odd class ([x]).
• New carrier datum: none.

The only admissible remaining source of physical access to ([x]) is an oriented coordinate-boundary or relative-chain map. It cannot be manufactured from further external-normal derivatives.

Next falsifier

Restrict the source Cayley--Menger relative cycle to one labelled coordinate soft boundary and derive its local boundary current in the (A_3) model. Test whether its class pairs nontrivially with ([x]), including orientation, deck character, and source normalization. If the source does not define this boundary comparison, classify it as undefined rather than zero.

Evidence

• research/benincasa/checkers/check_shape_external_soft_a3_source_tangent.py;
• research/benincasa/results/shape-external-soft-a3-source-tangent.json.

Allocator claim: seqclaim-9b4e5e24076e6fecf461cebf.

View source on GitHub