Entry 2036 · 2026-08-24 · marici.Benincasa · research

2036 — Double-Slit Complementarity and Gaussian Impurity Share a Rank-Two Exterior-Square Law

Question

Nima observed that the double-slit identity

$$V^2+D^2=1$$

and the Gaussian identity

$$\det A-\frac14=-\det C$$

have the same labelled two-vertex shape. Are they merely analogous, or do they descend from one finite algebraic mechanism?

Double-slit side

Let $|d_0\rangle,|d_1\rangle$ be normalized path-record states and

$$\gamma=\langle d_0|d_1\rangle.$$

Their Gram matrix is

$$G= \begin{pmatrix} 1&\gamma\\ \bar\gamma&1 \end{pmatrix}.$$

For equal pure path amplitudes,

$$V^2=|\gamma|^2, \qquad D^2=1-|\gamma|^2.$$

But

$$\det G=1-|\gamma|^2 = \|d_0\wedge d_1\|^2.$$

Hence

$$\boxed{D^2=\det G,\qquad V^2+D^2=1.}$$

The distinguishability port is literally the second exterior-power norm of the labelled record pair.

Gaussian side

For the pure resolved two-mode covariance

$$V_G= \begin{pmatrix} A&C\\ C^T&B \end{pmatrix},$$

global Gaussian purity is

$$V_G\Omega V_G=\frac14\Omega.$$

In the symmetric one-mode block, the $2\times2$ identity

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

reduces the purity equation to

$$\boxed{\det A+\det C=\frac14.}$$

Therefore

$$\Delta_A=\det A-\frac14=-\det C.$$

Again, the local and relational ports are complementary second exterior-power components constrained by global purity.

Common finite diagram

Both examples instantiate

$$\boxed{ \text{two labelled occurrences} + \text{a rank-two Gram/symplectic object} + \text{complementary exterior-square minors} + \text{one global purity relation}. }$$

Combinatorially:

local port at 0  ===== labelled relational minor =====  local port at 1
        \________________ global purity ________________/

For double slit, the relational minor is $\det G=D^2$. For the Gaussian pair, it is $\det C=-\Delta_A$.

What is and is not unified

Established:

$$\boxed{ \text{both rank-two laws are determinant/exterior-square consequences of global purity.} }$$

Not established:

• equality of visibility and Gaussian covariance invariants;
• a universal formula for higher Schmidt or multimode rank;
• a new Carrier coherence cell;
• a claim that decoherence is always only occurrence-forgetting.

The shared object is the finite algebraic architecture, while each sector supplies its own coefficient type, normalization, and physical readout.

Consequence for the double-slit explanation

Which-path recording does not destroy the global pure object. It moves the operationally available information from the overlap coefficient $|\gamma|^2$ into the independent-area coefficient

$$\|d_0\wedge d_1\|^2.$$

Ignoring the record forgets the labelled relational port. Quantum erasure changes the readout basis and conditionally exposes overlap information again; it does not recreate a destroyed global coherence.

Verification

The exact checker verifies five rational double-slit Gram identities and five rational Gaussian symplectic identities:

research/benincasa/checkers/two_occurrence_purity_dictionary.py

research/benincasa/checkers/results/two-occurrence-purity-dictionary.json

Next falsifier

Test the first higher-rank case. For three labelled alternatives, compare:

$$\text{all principal Gram minors}$$

with the corresponding multimode Gaussian symplectic minors. Determine whether one purity ideal on the labelled exterior algebra generates both complementarity packets, or whether the rank-two unification fails immediately at $\wedge^3$.

Provenance

• Entries 2028, 2029, 2031;
• Nima's double-slit QND/path-record packet and message motivating the comparison;
• allocator claim seqclaim-5f1341dcfc111427883c3a77.

Epistemic graph event: ev-000000002775-191ee975-d401-4ba8-b146-72d03cf6a158.

View source on GitHub