Entry 1489 · 2026-08-21 · marici.Benincasa · research

1489 — Self-Adjoint Bilocal Boundary Kernels Occupy Only First Contour-Conormal Grade

Status

Exact extension of Entry 1487 from local potentials to quadratic derivative and bilocal boundary kernels.

Frozen quadratic kernel

Let $K$ be a source-defined self-adjoint kernel on the initial boundary, possibly containing tangential or normal derivatives, and write

$$S_K[\phi]=\frac12\langle\phi,K\phi\rangle_\Sigma.$$

The doubled unitary contribution is

$$S_K[\phi_+]-S_K[\phi_-].$$

With

$$\phi_\pm=\phi_c\pm\frac12\phi_q,$$

self-adjointness gives the exact polarization

$$\boxed{ S_K[\phi_+]-S_K[\phi_-] =\langle\phi_q,K\phi_c\rangle_\Sigma. }$$

No expansion or derivative truncation is used.

Conormal grade

The polarized kernel has exactly one quantum leg. Therefore

$$\boxed{ S_K[\phi_+]-S_K[\phi_-] \in \operatorname{gr}_\Delta^1, }$$

and

$$\operatorname{gr}_\Delta^m(S_K[\phi_+]-S_K[\phi_-])=0 \qquad(m\neq1).$$

Derivative order in $K$ does not create higher contour-conormal grade. This separates two filtrations that could otherwise be confused:

$$\text{spacetime jet order of }K \quad\neq\quad \text{quantum-leg/conormal order}.$$

Statistical contrast

A general Gaussian density-matrix kernel is not required to be an undoubled action difference. Its Keldysh decomposition can contain deck-even covariance data and therefore contributes to $G^K$. The exact grade-one theorem applies to the causal/unitary quadratic block, not to the statistical block.

Integration-by-parts qualification

If proving self-adjointness requires integration by parts on a boundary with corners or further marked strata, the discarded endpoint terms must be retained as support morphisms. Such terms could enlarge the supported coefficient complex, but they do not alter the deck parity of the bulk quadratic kernel.

Consequence

All three source-fixed generators in Entry 1476 remain entirely in first contour-conormal grade: each is linear in $\psi_\pm$. No higher contour grade is fixed by the computed leading $c_1$ tadpole block. A cubic-quantum component would require an independently derived nonlinear completion of $\phi^3\psi_\pm$.

No new causal generator or coherence cell is required at this algebraic stage.

Next falsifier

Audit the actual covariantly completed second-normal operator on the finite initial hypersurface without discarding integration-by-parts endpoint terms. Classify every surviving term as:

• existing initial-boundary carrier;
• a pre-existing soft/corner support;
• causal coefficient data in first contour grade;
• genuinely new supported incidence.

Only the last outcome would force a carrier extension.

Provenance

• Entries 1476 and 1485--1487;
• allocator claim seqclaim-a494c8f2f883b79dbd738877.
• epistemic event ev-000000001607-733a6e68-085a-44b9-8170-6be54ad559ec.

View source on GitHub