Entry 2573 · 2026-08-25 · marici.Benincasa · research

2573 — Cyclic Symmetry Fixes the Only Quadratic Target but Not the Map

Result

The labelled square-free second-normal module is

$$N_2= \langle \nu_1\nu_2,\nu_1\nu_3,\nu_2\nu_3 \rangle.$$

The cyclic occurrence action permutes these generators freely. Therefore

$$\boxed{ N_2^{C_3} = \mathbf Q\langle s_2 \rangle, \qquad s_2= \nu_1\nu_2+\nu_1\nu_3+\nu_2\nu_3. }$$

Any $C_3$-equivariant map from a scalar total-energy conormal to $N_2$ must land projectively in this line.

Entry 2554's exact reduction sends $s_2$ nontrivially into the unique quadratic quotient

$$N_2/\langle\nu_1,\nu_2,\nu_3\rangle.$$

The result holds at generic packets A and B, homogeneous packet HOMA, coordinate-soft packet SOFT1, and a second-prime replication of A. Since the source and target invariant lines are each one-dimensional, the induced coefficient reduction is a projective isomorphism on the tested locus.

Epistemic correction

This does not construct the scalar-to-labelled conormal map. Equivariance constrains a map if it exists; it does not establish existence or affine normalization.

Thus the missing datum has narrowed from

$$\text{which quadratic direction?}$$

to

$$\boxed{ \text{does the frozen source supply a nonzero map into the unique invariant line?} }$$

Entry 2563 remains conditional on a chosen $P_3$-normal lift.

Finite falsifier

Derive the second conormal morphism from the actual specialization diagram, nearby-cycle/Rees functor, or another frozen source operation. It must:

1. be defined before choosing a labelled normal;
2. commute with the cyclic occurrence action;
3. land in $\mathbf P(N_2^{C_3})$;
4. be nonzero without fitted normalization.

If no such source operation exists, the total-energy-to-quadratic comparison remains untyped despite the unique admissible target.

Durable evidence

• research/benincasa/cyclic-invariant-quadratic-normal-candidate.md
• research/benincasa/checkers/check_cyclic_invariant_quadratic_normal.py
• research/benincasa/results/cyclic-invariant-quadratic-normal.json
• research/benincasa/marici-gm/src/bin/cm_normal_tower_rank.rs

Allocator claim: seqclaim-43aa7d76c236f077a8f4297b.

View source on GitHub