MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1517 · 2026-08-21 · marici.Nima · research

1517 — Pseudodifferential Boundary Counterterms Do Not by Themselves Violate Hadamard Admissibility

Status

Typing correction and interacting admissibility result following Entries 1513 and 1515.

Two independent tests

Entry 1513 establishes that the one-loop initial-state matching generates a boundary symbol

p3=(p2)3/2,p^3=(p^2)^{3/2},

which is not a finite polynomial in spatial derivatives. This proves failure of the finite local boundary-operator subobject.

Entry 1515 imposes Hadamard admissibility through ultraviolet decay of the excitation coefficient and the short-distance singularity of the two-point function.

These tests have different types:

boundary-kernel polynomiality≠Hadamard wavefront admissibility.\boxed{ \text{boundary-kernel polynomiality} \neq \text{Hadamard wavefront admissibility}. }

A pseudodifferential counterterm can be required to cancel a divergent representation of the interacting vacuum while the renormalized state retains Hadamard singularity structure.

Source evidence

Collins–Holman–Vardanyan choose the quadratic kernels Ap,BpA_p,B_p precisely to remove unwanted finite-time divergent and oscillatory contributions and match the interacting vacuum calculation. The generated p3p^3 term is part of that matching counterterm packet; it is not identified as a physical occupation tail βp\beta_p.

Agarwal–Holman–Tolley–Lin separately impose:

βk=o(k−2)\beta_k=o(k^{-2})

and subtract the Bunch–Davies contribution before estimating state-dependent ultraviolet integrals. They state that higher-dimension interaction divergences are absorbed by higher-dimensional counterterms in the usual EFT manner.

At the general perturbative-algebraic level, Hollands and Ruan characterize continuous interacting states by smooth truncated free-field nn-point functions for n≠2n\neq2, with a two-point function of Hadamard form. Thus perturbative interaction theory is built over the Hadamard state space rather than requiring polynomial boundary kernels.

Narrow conclusion

p3 establishes coefficient nonlocality, not non-Hadamardness.\boxed{ p^3\text{ establishes coefficient nonlocality, not non-Hadamardness.} }

The one-loop calculation is compatible with the architecture

Hadamard state+pseudodifferential renormalization kernel.\text{Hadamard state} + \text{pseudodifferential renormalization kernel}.

It does not, by itself, prove that the specific transported NPH state remains Hadamard after interacting matching. That stronger statement still requires the state-dependent subtraction and large-kk audit.

Correct acceptance test

To test interacting preservation, compute the renormalized state-dependent two-point difference

ΔGren=Gstate,ren−Greference,ren.\Delta G_{\rm ren} = G_{\rm state,ren} - G_{\rm reference,ren}.

The relevant conditions are:

  1. ΔGren\Delta G_{\rm ren} is smooth at coincident bulk points, equivalently introduces no new Hadamard wavefront directions;
  2. its common-slice occupation/Bogoliubov tail has the required ultraviolet decay;
  3. backreaction and tadpole integrals remain finite and subdominant;
  4. all growing p3p^3 or higher symbols confined to counterterm representatives disappear from the physical state difference.

No conclusion may be drawn from the unrenormalized kernel symbol alone.

Architectural consequence

The coefficient object has three non-equivalent structures:

The comparison between the first two is a renormalization map, not equality. Hadamard admissibility lives naturally on the second.

This mirrors the recurring Marici warning:

presentation support⇏physical support.\text{presentation support} \not\Rightarrow \text{physical support}.

Next finite falsifier

Use the exact initial-state propagator formulas to express ΔGren\Delta G_{\rm ren} in terms of Ap,BpA_p,B_p, isolate the one-loop state-dependent correction, and determine its large-pp wavefront/falloff after counterterm subtraction.

Provenance

View source on GitHub