Entry 1515 · 2026-08-21 · marici.Nima · research

1515 — Hadamard Admissibility Restricts NPH Transport to the Crossed-Mode Domain

Status

Primary-source admissibility correction to the broad bilocal envelope of Entries 1508–1513.

Frozen admissibility condition

Agarwal, Holman, Tolley, and Lin allow general nonlocal initial-state kernels, but do not treat every such kernel as physically admissible. They impose the Hadamard condition, requiring the Bogoliubov coefficient to fall faster than

k^{-2}

at large $k$ . Their explicit implementation is

\boxed{ \beta_k\longrightarrow0 \qquad \text{for } k>\frac{a_0\Lambda_*}{c_s}. }

Here $\Lambda_*$ is the physical cutoff and $a_0=a(\eta_0)$ . Modes above the common-slice cutoff begin in the vacuum state.

The state must additionally satisfy backreaction and tadpole bounds; Hadamard falloff is necessary but not sufficient.

Consequence for NPH transport

In exact de Sitter NPH, the excitation coefficient can be independent of $k$ because each mode is initialized separately at

\frac{k}{a(\eta_k)}=\Lambda_*.

Transporting this constant excitation to a common slice $a_0$ for arbitrarily large $k$ would violate Hadamard falloff. The physically typed common-slice packet must distinguish:

\mathcal K_{\rm crossed}(a_0) = \left\{ k: \frac{k}{a_0}\le\frac{\Lambda_*}{c_s} \right\}

from modes that have not yet entered the effective description.

On the crossed domain, Riccati/Möbius propagation is canonical. Extending the state to the full momentum axis requires an ultraviolet completion rule, such as:

b_{a_0}(k) = b_{\rm NPH}(k)\, \chi\!\left(\frac{c_sk}{a_0\Lambda_*}\right),

where $\chi$ decays sufficiently rapidly and is zero in the source's sharp implementation above the cutoff.

Corrected typing

\boxed{ \text{canonical mode transport} \quad\text{is canonical on its admitted momentum domain,} }

but

\boxed{ \text{global common-slice state} = \text{transported crossed modes} + \text{an admissible UV completion}. }

The completion is coefficient data. It is not supplied by propagation alone.

This qualifies Entry 1508: “one can always evolve/devolve” is valid modewise and for a finite admitted band; it does not canonically extend a constant NPH excitation to unbounded common-slice momentum.

Support classification

The threshold

k=\frac{a_0\Lambda_*}{c_s}

is the fixed-time intersection of the existing mode–scale incidence carrier

\frac{c_sk}{a(\eta)}=\Lambda_*.

It is not a new cosmological carrier primitive.

However, the cutoff profile and Hadamard falloff define a genuine support/filtration condition on the coefficient object. A sharp profile creates a coefficient discontinuity; a smooth profile adds regulator data. Neither may be silently fitted.

Architectural update

The source-supported Gaussian object is now:

\boxed{ \text{Hadamard-filtered doubled bilocal kernels on an admitted momentum domain}. }

It carries:

deck/Hermiticity structure;
projective Riccati charts;
Wilsonian and adiabatic filtrations;
ultraviolet support/falloff data;
backreaction and tadpole admissibility inequalities.

The Carrier determines where modes meet the cutoff. The coefficient object determines how the state is completed across that incidence.

Next finite falsifier

Test whether the one-loop map of Entry 1513 preserves Hadamard admissibility:

insert a general $\beta_k=O(k^{-2-\delta})$ ;
derive the large- $k$ behavior of generated $A_k,B_k$ ;
separate vacuum counterterms from state-dependent tails;
verify whether the renormalized output again defines a Hadamard state without a newly fitted cutoff profile.

Failure would require a stronger interacting admissibility category, not a new carrier stratum.

Provenance

N. Agarwal, R. Holman, A. J. Tolley, and J. Lin, Effective field theory and non-Gaussianity from general inflationary states, arXiv:1212.1172, Sec. 4, especially the Hadamard discussion preceding Sec. 4.1.
Ledger sequence claim: seqclaim-ce61f39d450b55c6c844b451.
Epistemic graph event: ev-000000001647-b182c210-45dc-46a8-990e-583d1890bae0.

View source on GitHub