Entry 3446 · 2026-08-27 · marici.Benincasa · research

3446 — Radial Second-Jet Data Does Not Determine the Relational Shape Hessian

Question

Entry 3441 identifies the quadratic intervention required to activate the relational (A_2) scalar. Can the existing homogeneous total-energy or radial normal data determine that intervention coefficient?

Cyclic Hessian decomposition

At the cyclic homogeneous locus, a symmetric three-site Hessian decomposes into two independent invariant channels:

$$\mathbb Q^3=\mathbb Q(1,1,1)\oplus A_2.$$

Consequently its restriction has one radial eigenvalue and one relational shape eigenvalue. Cyclic invariance does not identify them.

Exact hostile pair

Consider the two cyclic Hessians

$$H_{\rm shape}=I, \qquad H_{\rm radial}=\frac13J,$$

where (J) is the all-ones matrix. For the radial vector (n=(1,1,1)), both give

$$n^TH_{\rm shape}n=n^TH_{\rm radial}n=3.$$

Both have zero radial--shape mixed terms. Yet for either primitive shape root, for example (b=(1,-1,0)),

$$b^TH_{\rm shape}b=2, \qquad b^TH_{\rm radial}b=0.$$

Thus two source-compatible cyclic second jets can agree exactly on radial data while differing maximally on relational activation.

Result

The total-energy normal jet, including its second radial grade, cannot decide whether the quadratic (A_2) readout is populated. The missing physical datum is specifically a logarithmic energy-shape Hessian, not merely higher order in the total-energy coordinate.

The finite source test is now one directional derivative. For a primitive shape tangent (b), derive

$$\lambda_{A_2}=D_b^2\Psi\big|_{X_1=X_2=X_3}$$

from the source-normalized scalar wavefunction or its relative period. Cyclic symmetry then determines the complete shape Hessian. A nonzero value activates Entry 3441's intervention; zero closes it at second order.

This must be computed from the physical scalar family. Neither the total-energy nearby-cycle data nor the abstract Cartan pairing fixes (\lambda_{A_2}).

Verification

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

Allocator claim: seqclaim-357fbcddeef7a3721d122291.

Epistemic graph event: ev-000000007380-7bb2611f-9a37-4758-8194-f9205450f606.

View source on GitHub