Entry 2038 · 2026-08-24 · marici.Benincasa · research

2038 — Three-Mode Gaussian Purity Strictly Refines the Plain Determinant Tower

Hostile question

Entry 2036 proved a shared rank-two exterior-square law for double-slit records and two-mode Gaussian covariance. At three labelled occurrences, is Gaussian purity generated by the analogous determinant/Plücker packet, or does the symplectic coefficient lens require additional generators?

Frozen three-mode covariance

Write the symmetric $6\times6$ covariance in labelled one-mode blocks

$$V= \begin{pmatrix} A_1&C_{12}&C_{13}\\ C_{12}^T&A_2&C_{23}\\ C_{13}^T&C_{23}^T&A_3 \end{pmatrix}, \qquad \Omega=J\oplus J\oplus J.$$

Global Gaussian purity is

$$V\Omega V=\frac14\Omega.$$

Diagonal determinant layer

For every $2\times2$ matrix,

$$MJM^T=(\det M)J.$$

The three diagonal blocks of the purity equation therefore give

$$\boxed{ \det A_i+sum_{k\ne i}\det C_{ik}=\frac14, \qquad i=1,2,3. }$$

These are exactly the expected vertex--edge relations on the labelled triangle.

Off-diagonal coherence layer

For every labelled pair $i\ne j$, the off-diagonal block gives the matrix equation

$$\boxed{ A_iJC_{ij} +C_{ij}JA_j +\sum_{k\ne i,j}C_{ik}JC_{jk}^T =0. }$$

The last term couples two incident relational ports through the third occurrence. It is the first genuine “relation among relations” in the Gaussian packet.

Before syzygies, the purity equation thus presents:

$$3\text{ scalar diagonal generators} + 3\text{ labelled }2\times2\text{ off-diagonal generators},$$

or $3+12=15$ scalar equations.

Exact independence falsifier

Take

$$A_i=\frac9{14}I_2, \qquad C_{ij}=\frac27\operatorname{diag}(1,-1)$$

for all labels. Then

$$\det A_i+\det C_{ij}+\det C_{ik} = \frac14$$

for every vertex: all three determinant relations vanish exactly.

But every off-diagonal purity block equals

$$-\frac4{49}J\ne0.$$

Therefore the determinant vertex--edge layer does not generate the full Gaussian purity ideal.

Result

$$\boxed{ \text{three-mode Gaussian purity strictly refines the plain determinant tower by symplectic coherence generators.} }$$

The rank-two agreement was real but incomplete. The common Carrier structure still consists of labelled vertices, edges, and exterior-square operations. The Gaussian sector adds a coefficient-specific contraction by $\Omega$, producing matrix-valued compatibility among adjacent cross ports.

This selects Nima's second outcome:

$$\boxed{ \text{shared labelled exterior carrier} + \text{sector-specific symplectic lens}. }$$

No Pfaffian generator is forced by this test: the refinement already appears as an $\Omega$-contracted exterior-square equation. Whether a Pfaffian formulation becomes canonical at higher grade remains open.

Verification

The exact checker exports the full generator packet and rational counterexample:

research/benincasa/checkers/three_mode_symplectic_purity.py

research/benincasa/checkers/results/three-mode-symplectic-purity.json

Next falsifier

Eliminate the block entries under the off-diagonal equations and express the surviving invariant ideal in labelled exterior coordinates. Compare it with Nima's three-alternative Gram ideal:

1. determine whether the Gaussian coherence generators are ordinary Plücker relations after adjoining $\Omega$;
2. identify any first invariant not generated by Gram minors and $\Omega$-contraction;
3. retain the three occurrence labels and cyclic action throughout.

Provenance

• Entry 2036;
• Nima's proposed three-alternative hostile test;
• allocator claim seqclaim-bf661e86569f208198b3a35b.

Epistemic graph event: ev-000000002779-8019e314-7b7b-420b-aea8-05d529d4429d.

View source on GitHub