Entry 3432 · 2026-08-27 · marici.Benincasa · research

3432 — Cyclic Scalar Descent First Sees the Relational Grade Quadratically

Deutschian prediction

Let the labelled relational sector be the rational (A_2) representation of the cyclic occurrence group (C_3). If a scalar readout is analytic at the zero relational class and introduces no asymmetric probe, then its linear term vanishes. Its first possible nonconstant term is quadratic and is proportional to the unique cyclic-invariant Cartan, equivalently Clifford, norm on (A_2).

This is hard to vary: changing the conclusion requires either breaking cyclic descent, adding an independently typed probe, or changing the representation.

Finite falsifiers

The prediction fails if any of the following occurs:

1. the invariant dual ((A_2^*)^{C_3}) is nonzero;
2. the invariant symmetric-square space ((\operatorname{Sym}^2 A_2^*)^{C_3}) has dimension other than one;
3. the source cyclic observer fails to recover that quadratic norm;
4. an anisotropic quadratic form survives cyclic transport.

Exact census

In a simple-root basis, cyclic rotation is

$$R= \begin{pmatrix} 0&-1\\ 1&-1 \end{pmatrix}, \qquad R^3=I.$$

Exact rational elimination gives

$$\dim (A_2^*)^{C_3}=0, \qquad \dim (\operatorname{Sym}^2 A_2^*)^{C_3}=1.$$

The surviving quadratic line is generated by

$$C= \begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}, \qquad R^TCR=C.$$

Every root in the cyclic orbit has (C)-norm (2). A hostile anisotropic form fails the invariance equation.

In the three-site sum-zero realization, Entry 3413's observer is

$$L=3\pi_{A_2}.$$

For every (b\in A_2), with (y=Lb),

$$\sum_i y_i=0, \qquad \sum_i y_i^2=9\sum_i b_i^2.$$

Thus invariant linear aggregation is blind, while quadratic aggregation recovers the relational magnitude with a fixed normalization.

Result and scope

All four falsifiers are rejected. The first cyclic scalar channel available to the relational occurrence sector is uniquely quadratic.

This does not prove that the physical cosmological observable has a nonzero coefficient in this channel. It proves the allowed form of its first possible source-natural scalar response. The next physical falsifier is to derive the source-normalized second response and determine whether its coefficient is nonzero; it must not be inferred from this invariant census.

The result also does not identify this norm with the three-site quartic (\mathcal Q) or with the second total-energy Rees grade. Such an identification requires a typed source map.

Verification

Checker: research/benincasa/checkers/audit_a2_quadratic_scalar_prediction.py.

Allocator claim: seqclaim-714914b016e163c93fbd59c4.

Epistemic graph event: ev-000000007343-8bb6dff2-e8d9-4aef-9889-b7423109ee2f.

View source on GitHub