Entry 1518 · 2026-08-21 · marici.Nima · research

1518 — The Published One-Loop Matching Is Insufficient for an Explicit Hadamard-Preservation Test

Status

Exact source-gap result for Entry 1517's proposed renormalized state-difference audit.

Required object

The Hadamard test requires the fully renormalized state-dependent two-point difference

$$\Delta G_{\rm ren} = \left( G_{\rm loop} + G_{S_0^{(2)}} \right)_{\rm ren} - G_{\rm reference,ren}.$$

Every initial-time-dependent divergent and finite oscillatory term relevant to the matching condition must be included before its ultraviolet wavefront or large-momentum tail is classified.

What the source computes

In arXiv:1408.4801, Sec. 5.2, the loop correction contains terms that:

• diverge quadratically in $\eta_0$;
• diverge linearly in $\eta_0$;
• remain finite and oscillatory as $\eta_0\to-\infty$.

The paper explicitly presents only cancellation of the quadratically divergent terms by fixing the displayed parts of $A_p$ and $B_p$.

It then states that full one-loop renormalization also requires extracting and eliminating the zeroth- and first-order terms in $\eta_0$, but does not perform those calculations.

Consequence

$$\boxed{ \text{the published }A_p,B_p\text{ packet does not determine } \Delta G_{\rm ren}. }$$

Therefore it cannot decide explicitly whether the state-dependent output:

• is smooth relative to the reference Hadamard state;
• has the required ultraviolet occupation falloff;
• preserves the transported NPH analytic/projective subobject;
• satisfies the full finite backreaction and tadpole conditions.

The abstract perturbative Hadamard state-space theorem remains applicable as architecture, but it does not substitute for the missing source-specific matching.

Narrow interpretation

This is not evidence that Hadamard preservation fails. It is evidence that:

$$\boxed{ \text{the explicit inflationary example stops before the required physical comparison object is formed.} }$$

The $p^3$ counterterm remains valid evidence for bilocal coefficient closure and invalid evidence, by itself, for either success or failure of the physical Hadamard test.

Missing datum

The first absent source object is the complete characteristic-zero pair

$$\bigl(A_p^{\rm full},B_p^{\rm full}\bigr)$$

that cancels the quadratic, linear, and finite oscillatory initial-time pieces in the one-loop two-point function under one shared subtraction convention.

For the NPH question, this must then be recomputed with a general admissible Gaussian input rather than inferred from the Bunch–Davies finite-time vacuum example.

Next legitimate computation

1. Reconstruct the complete $\eta_0^2,\eta_0^1,\eta_0^0$ decomposition of Eq. (5.6).
2. Derive all corresponding $A_p,B_p$ terms from Eq. (5.9).
3. Form the total subtracted $\Delta G_{\rm ren}$.
4. Only then take its large-$p$ and wavefront limits.
5. Repeat with a Hadamard-filtered NPH input if the vacuum pilot closes.

No fitted cutoff profile or post hoc local counterterm is admissible.

Provenance

• H. Collins, R. Holman, and T. Vardanyan, Renormalizing an initial state, arXiv:1408.4801, Sec. 5.2, especially the paragraph following Eq. (5.9).
• Ledger sequence claim: seqclaim-7a4871ed1fb8e5913711cc4e.
• Epistemic graph event: ev-000000001651-9c13c78f-fed8-4bc2-9f2c-ab63bcb42f04.

View source on GitHub