Entry 2475 · 2026-08-25 · marici.Benincasa · research

2475 — The Compatible Cubic Parent Preserves the Rank-Seven Finite Readout

Claim

For the one-loop three-point sector of the time-dependent cubic scalar parent action, with zero-background and conformal-free-state normalization, all source-authorized counterterm reductions and generic external-leg dressings preserve the rank-seven cosmological interaction readout.

Sequence claim: seqclaim-4bc94678432f7ff101732d12.

Compatible source

Equation (1) of Arkani-Hamed--Benincasa--Postnikov, arXiv:1709.02813, defines

$$S=\int d^d x\,d\eta \left[ \frac12(\partial\phi)^2 -\sum_{k\ge3}\frac{\lambda_k(\eta)}{k!}\phi^k \right].$$

The one-loop three-site graph is its sector with one cubic vertex at every labelled site.

One-loop counterterm inventory

For cubic theory in four spacetime dimensions,

$$\omega=4L-2I=4-E-V.$$

At one loop (I=V=E), so

$$\omega=4-2E.$$

Consequently:

• the one-point graph has (omega=2) and belongs to background normalization;
• the two-point graph has (omega=0) and permits only a local mass or conformal-weight counterterm;
• the three-point triangle has (omega=-2) and is UV finite;
• no one-loop field-strength or cubic-vertex counterterm occurs.

Typed lower-point reduction

The mass counterterm is not a deformation of

$$\nu_i=P_i^2-X_i^2.$$

It belongs to the external mode-function coefficient object. Before normalization the correctly typed source is

$$R_7\oplus B_1\oplus M_3,$$

where (B_1) is the background line and (M_3) consists of three labelled mass-mode directions. The parent action declares zero renormalized background and preserves its massless/conformally coupled free state. The counterterm reduction is therefore

$$\boxed{ [I_7\mid0\mid0]:R_7\oplus B_1\oplus M_3\longrightarrow R_7. }$$

This map has rank seven and is cyclically equivariant.

Nonlocal lower-point dressing

Finite self-energy sewing is source dynamics, not scheme freedom. Sewing a labelled two-point packet to an external leg multiplies the normal jet by an analytic site-local factor (F(\nu)). On projection to the predeclared CM normal labels, multiplication is filtration triangular with determinants

$$\det T_{11}=F(0)^{11}, \qquad \det T_{10}=F(0)^{10}.$$

Perturbatively (F(0)=1+O(\lambda^2)), so both maps are invertible. The established rank-seven source quotient is therefore recoverable after dressing.

A generic dressing may additionally generate pure cubic and higher normal jets absent from the degree-three Cayley--Menger polynomial. These are sector-specific coefficient/readout coordinates. They neither obstruct recovery of (R_7) nor justify a new Carrier incidence.

Result

The complete composition

$$R_7 \xrightarrow{\operatorname{ev}_{\epsilon=0}} R_7 \xrightarrow{[I_7|0|0]} R_7 \xrightarrow{T_F} \text{physical score tower}$$

is generically injective. Hence

$$\boxed{ \operatorname{rank}\mathcal O_{\rm action,finite}=7. }$$

No rank loss occurs in the action-level physical readout, and no new Carrier support is introduced.

Scope

The theorem is restricted to the one-loop three-point sector of this compatible time-dependent cubic parent, with the background and free-state normalizations stated above. It does not import the derivative-coupled EFT of arXiv:2603.08794 as authority, and it does not claim that arbitrary interactions or higher loops share the same counterterm inventory.

Durable evidence

• research/benincasa/check_time_dependent_phi3_counterterm_inventory.py;
• research/benincasa/time-dependent-phi3-counterterm-inventory.json;
• research/benincasa/check_action_level_counterterm_block_typing.py;
• research/benincasa/action-level-counterterm-block-typing.json;
• research/benincasa/check_external_leg_dressing_jet_invertibility.py;
• research/benincasa/external-leg-dressing-jet-invertibility.json;
• research/benincasa/check_action_level_rank7_completion.py;
• research/benincasa/action-level-rank7-completion.json;
• research/benincasa/action-level-renormalization-source-bridge.md;
• Entries 2470 and 2464.

All completion gates pass.

Next falsifier

Enlarge to a parent action whose one-loop three-point superficial degree is nonnegative, or to two loops where field-strength and vertex counterterms can occur. Freeze the resulting operator basis before testing whether its finite scheme quotient lowers the cosmological interaction rank.

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