Entry 3441 · 2026-08-27 · marici.Benincasa · research

3441 — The Minimal Quadratic Readout Factors Through a Rank-Three Intervention Module

Question

Entry 3437 proves that linear preparation and readout cannot activate the unique quadratic scalar on the relational (A_2) sector. Does inserting one typed intervention stage suffice, and what is its minimal universal object?

Three-stage factorization

Work in the category of graded polynomial laws. The preparation supplies (b\in A_2). The degree-two intervention is the Veronese map

$$\nu_2:A_2\longrightarrow\operatorname{Sym}^2(A_2), \qquad b\longmapsto b^{\odot2}.$$

The readout is contraction with the Cartan metric,

$$q_C:\operatorname{Sym}^2(A_2)\longrightarrow\mathbb Q.$$

Thus the scalar factors as

$$A_2\xrightarrow{\nu_2} \operatorname{Sym}^2(A_2) \xrightarrow{q_C}\mathbb Q.$$

The first arrow is polynomial of degree two; the second is linear. This keeps the nonlinear authority in the intervention rather than hiding it in readout.

Exact hostile test

In the simple-root basis, the cyclic action on (A_2) induces an exact three-dimensional action on

$$\operatorname{Sym}^2(A_2) =\langle x^2,xy,y^2\rangle.$$

The audit establishes:

1. (\nu_2) is cyclic-equivariant;
2. its images span all three dimensions of the symmetric square;
3. the cyclic-invariant readout space on that middle object has dimension one;
4. its generator is the Cartan contraction (2x^2-2xy+2y^2);
5. deleting the intervention gives the Entry 3437 linearity contradiction.

Hence the rank-three middle object is universal for quadratic laws. Replacing it directly by its invariant scalar quotient would presuppose the final readout and lose non-invariant quadratic responses.

Architectural result

The notation (3(3+2+1)+2+1) has a coherent operational interpretation:

• preparation module;
• intervention or jet module;
• readout module;
• two typed interfaces between them;
• one final event closure.

These are three situated modules over one shared Carrier, not three separate Carriers. At quadratic order, the intervention module is concretely (\operatorname{Sym}^2(A_2)).

The result proves sufficiency and minimal universality at the algebraic level. It does not prove that the frozen Bunch--Davies source realizes the Veronese intervention physically. That remains a second-jet or two-copy preparation question.

Verification

Checker: research/benincasa/checkers/audit_three_module_quadratic_factorization.py.

Allocator claim: seqclaim-9799eff99206ee21b8b57e64.

Epistemic graph event: ev-000000007362-5a8b3e26-0602-4dfb-80aa-c0ded2dcfa8b.

View source on GitHub