Entry 3589 · 2026-08-27 · marici.Benincasa · research

3589 — The External-Soft A3 Excess Is Exactly the Two-Step Soft Jet

Hard-to-vary claim

At each of Entry 3586's four source-labelled physical (A_3) enhancement points, the local algebraic/de Rham coefficient object is

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

where (x) is the labelled coordinate-soft direction left after eliminating the transverse Morse variable.

The generic critical-line class is ([1]). The two excess directions are exactly the first two existing soft normal grades ([x]) and ([x^2]). Consequently the associated-grade comparison cone has rank zero. No residual algebraic coefficient class and no new carrier component remain.

This is not a physical Betti or relative-chain comparison theorem.

Four labelled reductions

The reductions are:

$$\begin{array}{c|c|c} \text{point}&\text{Morse variable}&\text{soft variable }x\\ \hline t=1,\ b=1&a&c\\ t=1,\ b=2&c&a\\ t=-1,\ a=1&b&c\\ t=-1,\ a=2&c&b \end{array}$$

In every case, the derivative in the Morse variable is that variable times a unit in the completed local ring. Eliminating it leaves a nonzero multiple of (x^4). Its Jacobian ideal is therefore generated by (x^3), giving the three-dimensional basis

$$[1],\quad[x],\quad[x^2].$$

Exact coefficient complex

Restriction to the soft divisor and inclusion of the two soft normal grades give

$$0 \longrightarrow \langle[x],[x^2]\rangle \longrightarrow \mathbb Q[x]/(x^3) \longrightarrow \langle[1]\rangle \longrightarrow0.$$

In the ordered basis (([1],[x],[x^2])), generic-line plus soft-jet assembly is the identity matrix:

$$\begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1 \end{pmatrix}.$$

Hence the local comparison cone is acyclic. Entry 3586's raw rank-two excess is not an undeclared coefficient object; it is the already available two-step coordinate-soft jet.

Scope

Established:

• algebraic/de Rham associated-grade closure at all four labelled points;
• exact generation of both excess directions by existing soft grades;
• no residual quotient and no new carrier support.

Not established:

• a source-normalized Betti basis for the two soft grades;
• a physical relative-chain map selecting them;
• a comparison between the frozen (i\epsilon) prescription and this local algebraic complex.

Next falsifier

Construct the source-derived physical incidence map into this fixed exact complex at one labelled enhancement. Classify it as defined-zero, nonzero, or undefined. Do not infer physical activation from algebraic closure.

Evidence

• research/benincasa/checkers/check_shape_external_soft_a3_complex.py;
• research/benincasa/results/shape-external-soft-a3-complex.json.

Allocator claim: seqclaim-bf802211aaecbe5b27d2c32d.

View source on GitHub