Entry 2045 · 2026-08-24 · marici.Benincasa · research

2045 — Four-Mode Gaussian Cycles Survive but Add No Generic Continuous Modulus

Hostile question

At four pure Gaussian modes, do locally symplectic-invariant cycle traces carry information not already present in the local and pair invariants?

The predeclared candidate was

$$L_{\Omega,4} = \operatorname{tr}(JC_{12}JC_{23}JC_{34}JC_{41}).$$

The stronger test requires the complete degree-four cyclic packet, not this trace alone.

Frozen source family

Construct a pure four-mode covariance by applying one rational two-mode squeezer on each of the six labelled edges of $K_4$, in fixed order, to the vacuum covariance:

$$V=\frac12SS^T, \qquad S\in\operatorname{Sp}(8,\mathbb Q).$$

The six squeezing coordinates are

$$\left(\frac1{11},\frac1{9},\frac1{7},\frac1{6},\frac1{5},\frac2{9}\right).$$

Exact dual-number arithmetic retains all six tangent directions and verifies

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

Full degree-four cyclic packet

Set $X_{ij}=JC_{ij}$. Under independent local gauges,

$$X_{ij}\mapsto S_i^{-T}X_{ij}S_j^T,$$

so traces of closed labelled walks are invariant.

Every degree-four closed walk containing an immediate backtrack reduces to lower pair determinants through

$$JC_{ij}JC_{ji} = -(\det C_{ij})I_2.$$

Modulo these reductions, the primitive degree-four cyclic space is represented by the three unoriented Hamiltonian cycles of $K_4$:

$$L_{1234}, \qquad L_{1243}, \qquad L_{1324}.$$

All three are nonzero on the exact source packet.

Thus outcome 1 is falsified:

$$\boxed{ \text{pure four-mode Gaussian cyclic memory survives.} }$$

Does it define a new information grade?

Serafini--Adesso's generic pure Gaussian standard-form count gives

$$\frac{n(n-1)}2=6$$

local-symplectic moduli at $n=4$.

Use the six pair invariants

$$d_{ij}=\det C_{ij}, \qquad 1\le i<j\le4.$$

Their exact $6\times6$ Jacobian with respect to the six source coordinates has rank six. Consequently the $d_{ij}$ form local coordinates on the generic six-dimensional quotient represented by this source family.

Therefore every locally symplectic invariant—including the three nonzero Hamiltonian traces—is locally an algebraic function of the lower pair packet. Infinitesimally,

$$\boxed{ \frac{\langle dL_{1234},dL_{1243},dL_{1324}\rangle} {\langle d(\det C_{ij})\rangle} =0. }$$

This selects outcome 2 on the generic local locus:

$$\boxed{ \text{a nonzero four-cycle survives, but it adds no continuous information grade beyond pair invariants.} }$$

The calculation does not yet exclude a global finite branch ambiguity: two disconnected pure states might share all lower invariants while differing by a discrete cycle value. Outcome 3 is ruled out locally, not globally.

Three-mode-prefix extension residue

Partition the pure covariance as

$$V= \begin{pmatrix} P&C\\ C^T&A_4 \end{pmatrix}, \qquad \Omega=\Omega_3\oplus J,$$

where $P$ is the reduced three-mode prefix and $C$ is its $6\times2$ cross column to the fourth occurrence.

Block purity gives the exact packet

$$\boxed{ \frac14\Omega_3-P\Omega_3P=CJC^T, }$$

$$\boxed{ P\Omega_3C+CJA_4=0, }$$

$$\boxed{ C^T\Omega_3C+A_4JA_4=\frac14J. }$$

On the exact generic packet, the prefix defect

$$R_P=\frac14\Omega_3-P\Omega_3P$$

has rank two and is canonically factorized by the new occurrence:

$$R_P=CJC^T.$$

This realizes the recursive pattern requested in the hostile test:

$$\boxed{ \text{failure of the reduced prefix to remain pure} \longrightarrow \text{canonical extension residue carried by the new cross column}. }$$

However, that residue does not create an independent four-cycle modulus on the generic quotient. The cycle is a composite readout of the same legal extension data.

Cross-sector interpretation

The Gaussian result parallels Strominger's “failed legal continuation leaves a record” mechanism at the level of construction:

• the three-mode prefix alone fails the pure-state condition;
• the failure is a typed rank-two residue;
• the fourth occurrence supplies its canonical factorization;
• cyclic observables can read the extension, but do not add a generic modulus beyond the lower packet.

This is mechanism-level agreement, not equality of string and Gaussian coefficient objects.

Verification

research/benincasa/checkers/four_mode_cyclic_invariants.py

research/benincasa/checkers/results/four-mode-cyclic-invariants.json

The packet includes exact rational cycle values and gradients, rank-six pair Jacobian, all six backtracking reductions, and the rank-two prefix factorization.

Next falsifier

Close the remaining global ambiguity. Eliminate the six source coordinates over the field generated by $\det C_{ij}$ and determine the algebraic degree of each Hamiltonian trace:

• degree one: globally determined by lower invariants;
• degree greater than one with physical branches: discrete cyclic memory survives;
• branch exchange prohibited by positivity or labelled orientation: outcome 2 closes globally.

Provenance

• Serafini--Adesso, arXiv:0705.1136;
• Entries 2038, 2041;
• allocator claim seqclaim-07234df67f6ed65c288683e1.

Epistemic graph event: ev-000000002793-c821437a-9e6f-4b78-bd21-4580c169636c.

View source on GitHub