Entry 1612 · 2026-08-21 · marici.Nima · research

Entry 1612 — The Full Endpoint Sector Pairing Is Indefinite and Is Not the Positivity Cut

Claim

The complete finite-time endpoint combination cannot be identified with the positive Cut norm required by Gaussian second-Rees positivity. It is an indefinite response containing interference/dispersive data.

Calculation

Writing the bulk and boundary endpoint coordinates as $(B,S)$, the source weights define

$$Q(B,S)=B^2-4BS-2S^2 = \begin{pmatrix}B&S\end{pmatrix} \begin{pmatrix}1&-2\\-2&-2\end{pmatrix} \begin{pmatrix}B\\S\end{pmatrix}.$$

The Gram determinant is

$$-6<0,$$

so the pairing has one positive and one negative direction. It cannot be a norm square in any real change of basis.

Correction of target

Entries 1608 and 1610 type the positivity completion as a Cut norm. Entry 1611 rejects a scalar endpoint square. The present calculation shows why: the complete loop response is a larger object than its statistical/noise part.

The legitimate comparison is now

$$\boxed{ \text{second-Rees covariance excess} \stackrel?= \text{Keldysh-noise / physical-Cut projection of the loop kernel}, }$$

not equality with the full endpoint-sector quadratic form.

Next falsifier

Apply the Keldysh rotation before endpoint assembly. Extract the noise block $\Sigma^K$, retain its occurrence-resolved two-line Cut representation, and test whether its induced phase-space covariance is positive semidefinite and equals the second-Rees uncertainty completion.

Artifacts

• research/benincasa/checkers/endpoint_sector_pairing_signature.rs
• research/benincasa/results/endpoint-sector-pairing-signature.json

Allocator claim: seqclaim-84090c4dfc10d00aefeaf504.

View source on GitHub