Entry 1513 · 2026-08-21 · marici.Nima · research

1513 — One-Loop Initial-State Renormalization Is Bilocal and Leaves the Finite Local Subobject

Status

Primary-source interaction test for Entry 1511's two-filtered coefficient architecture.

Frozen source map

Collins, Holman, and Vardanyan cancel the initial-time-dependent part of the one-loop two-point function by adjoining the quadratic initial action [arXiv:1408.4801, Eqs. (5.8)–(5.9)]

$$\begin{aligned} S_0^{(2)} =\frac12\int d^3x\,d^3y\, \bigl[ &\zeta_+(x)A(x-y)\zeta_+(y) -\zeta_-(x)A^*(x-y)\zeta_-(y)\\ &+2i\zeta_+(x)B(x-y)\zeta_-(y) \bigr]. \end{aligned}$$

The target is explicitly a pair of bilocal kernels $A(x-y)$ and $B(x-y)$, not a predeclared finite list of local boundary operators.

For the quadratically divergent matching, the source obtains

$$\operatorname{Re}A_p=0$$

and

$$\boxed{ \operatorname{Im}A_p = \frac{(3\epsilon+2\delta)^2}{32}\, p^3\, \operatorname{Inf} \left( I_4-J_1+2J_0+4J_2 \right). }$$

It also fixes $B_p$ independently from the divergent part of $J_0$.

Locality classification

For rotationally invariant local quadratic operators on the three-dimensional initial slice, a finite spatial-derivative action has a Fourier symbol polynomial in

$$p^2.$$

The generated symbol

$$p^3=(p^2)^{3/2}$$

is not such a polynomial. It is the symbol of a pseudodifferential/fractional operator.

Therefore:

$$\boxed{ \text{finite local spatial-derivative boundary kernels} \quad\text{are not closed under the displayed one-loop initial-state map}. }$$

The bilocal/pseudodifferential coefficient envelope is not optional decoration: the frozen interacting calculation already requires it.

What this does and does not establish

Established:

• the one-loop map preserves the broad translation-invariant bilocal kernel category;
• it generates independent statistical kernels $\operatorname{Im}A_p$ and $B_p$;
• its displayed divergent part exits the finite local $p^2$-polynomial subobject.

Not established:

• preservation of the specific NPH phase subobject;
• preservation of Entry 1511's analytic germ at every momentum;
• positivity of the finite renormalized density matrix;
• a complete all-orders interacting kernel atlas.

The source calculation uses the finite-time interacting vacuum matching condition, not an NPH state. It therefore types the target category without proving NPH stability.

Architectural update

The smallest source-supported quadratic coefficient category is now

$$\boxed{ \text{translation-invariant doubled bilocal kernels} \supset \text{projective Gaussian solution lines}. }$$

Within it:

• local boundary EFT is a restricted finite-jet subobject;
• transported NPH is an oscillatory infinite-jet subobject;
• loop renormalization generates fractional/pseudodifferential symbols.

This is stronger support for H2: the carrier remains the same initial boundary, while interaction enlarges and mixes coefficient layers.

Next falsifier

Insert a general Gaussian kernel $A_p,B_p$—including the transported NPH form—into the one-loop matching calculation and determine whether the renormalization map preserves the analytic/projective kernel atlas.

The first finite gates are:

1. closure under deck/Hermiticity constraints;
2. absence of new momentum support beyond source thresholds;
3. preservation or controlled enlargement of the analytic germ;
4. compatibility of the generated $p^3$ symbol with the adiabatic $H/M$ filtration.

Provenance

• H. Collins, R. Holman, and T. Vardanyan, Renormalizing an initial state, arXiv:1408.4801, Sec. 5.2, especially Eqs. (5.7)–(5.9).
• Ledger sequence claim: seqclaim-3b8c82e882d2209ad3a50826.
• Epistemic graph event: ev-000000001643-e8461a92-4ad2-4162-8967-cbd44eb49e29.

View source on GitHub