Entry 1486 · 2026-08-21 · marici.Benincasa · research

1486 — The Initial-Boundary Beta Image Has Contour-Conormal Grades One and Three

Status

Superseded by the primary-source correction following Entry 1489. The source Eq. (6.37) fixes only $\phi^3\psi_\pm$, linear in the fluctuation. It does not establish the full quartic action difference used below. The asserted third contour-conormal grade is therefore withdrawn; the calculation below is conditional on an independently supplied nonlinear completion.

Contour-diagonal filtration

Let

$$\mathcal I_\Delta=(\phi_q)$$

be the conormal ideal of the Schwinger--Keldysh diagonal $\phi_+=\phi_-$. Its filtration is

$$\mathcal I_\Delta \supset \mathcal I_\Delta^2 \supset \mathcal I_\Delta^3 \supset\cdots.$$

The associated grades count quantum legs, not ordinary time or energy normal order.

Quadratic generators

The covariantly completed second-normal operator and the curvature operator are quadratic. Polarizing their doubled action differences gives exactly one quantum leg:

$$\mathcal O_i[\phi_+]-\mathcal O_i[\phi_-] \in \mathcal I_\Delta/\mathcal I_\Delta^2, \qquad i=2,3.$$

They have no higher contour-conormal component.

Quartic generator

The exact identity

$$\phi_+^4-\phi_-^4 =4\phi_c^3\phi_q+\phi_c\phi_q^3$$

has two nonzero associated grades:

$$\operatorname{gr}_\Delta^1(\mathcal O_4) =4\phi_c^3\phi_q,$$

$$\operatorname{gr}_\Delta^3(\mathcal O_4) =\phi_c\phi_q^3.$$

There is no second grade.

Conditional beta-image decomposition

If the tadpole vertex is independently completed to a full quartic boundary action, the beta image would have the filtered shape

$$\boxed{ \operatorname{gr}_\Delta^1\operatorname{im}\beta_\Sigma^{(4)} = \langle\mathcal O_2,\mathcal O_3, \operatorname{gr}_\Delta^1\mathcal O_4\rangle, }$$

$$\boxed{ \operatorname{gr}_\Delta^2\operatorname{im}\beta_\Sigma^{(4)}=0, \qquad \operatorname{gr}_\Delta^3\operatorname{im}\beta_\Sigma^{(4)} =\langle\phi_c\phi_q^3\rangle. }$$

For the frozen source vertices, one transverse derivative detects all three generated counterterms. No higher transverse derivative is licensed.

No new primitive

The third-grade line would be linked rather than independent after a full quartic completion, but that completion is an additional hypothesis.

Type distinction

This filtration is normal to the doubled contour diagonal. It must not be identified with:

• the ordinary normal/time derivative filtration on the initial hypersurface;
• the total-energy Rees filtration of the integrated cosmological period;
• the occurrence-resolved Cut normal filtration.

The result therefore does not contradict the second-normal requirement of Entry 1476 or the loop-level second-Rees result of Entry 128.

Carrier classification

The existing doubled boundary carrier already supplies the diagonal and its conormal filtration. The nontrivial object is a filtered causal coefficient block with linked grades one and three. No new carrier cell is required.

Next falsifier

Test whether the source renormalization group connection preserves this linked two-grade quartic line under iteration. A generated independent $\operatorname{gr}_\Delta^3$ coefficient, not fixed by the quartic action, would falsify closure of the present three-generator causal block.

Provenance

• Entries 1476--1477 and 1482--1485;
• allocator claim seqclaim-8700388d93fd9e2243795b88.
• epistemic event ev-000000001602-0d7bc362-8c68-4645-9998-fa580751254e.

View source on GitHub