Entry 1508 · 2026-08-21 · marici.Nima · research

1508 — NPH Transport to a Common Slice Exits the Local Boundary-EFT Coefficient Category

Status

Primary-source refinement of Entries 1504 and 1506.

Qualification from Entry 1515. The transport below is canonical modewise and on a finite admitted momentum band. A global common-slice state on unbounded momentum additionally requires Hadamard ultraviolet completion; propagation alone does not canonically supply it.

Canonical transport exists

Greene, Schalm, van der Schaar, and Shiu emphasize that a new-physics-hypersurface state can always be evolved or devolved to one chosen time $t_0$. Thus the fixed-scale and fixed-time presentations are not separated by an absence of propagation.

For a mode basis $u_k,u_k^*$, write

$$v_k=N(k)\bigl(u_k+b(k)u_k^*\bigr).$$

On the common slice, the quadratic boundary kernel $\kappa(k)$ imposes

$$\partial_n\phi|_{t_0}=-\kappa(k)\phi(t_0).$$

The source relation [astro-ph/0503458, Eq. (8)] is

$$b(k) = - \frac{ \kappa(k)u_k(t_0)+\partial_n u_k|_{t_0} }{ \kappa(k)u_k^*(t_0)+\partial_n u_k^*|_{t_0} }.$$

Whenever the denominator is nonzero, this can be inverted:

$$\boxed{ \kappa(k) = - \frac{ \partial_n u_k+b(k)\partial_n u_k^* }{ u_k+b(k)u_k^* }\bigg|_{t_0}. }$$

This is a source-derived transport map from Bogoliubov data to a common-slice boundary kernel.

Where equivalence fails

Local boundary EFT restricts $\kappa(k)$ to a derivative expansion. Its leading irrelevant correction is [Eq. (6)]

$$\kappa_{\rm BEFT}(k) \simeq \kappa_{\rm BD} + \beta\frac{k^2}{a_0^2M} +\cdots.$$

By contrast, the NPH coefficient contains the finite-scale phase [Eq. (4)]

$$b_{\rm NPH}(k) \sim \widetilde\beta\, \frac{H(k)}{2iM} \exp\!\left[-\frac{2iM}{H(k)}\right].$$

Substitution into the exact transport formula produces a well-defined common-slice kernel, but generically not a finite regular polynomial in $k^2/(a_0^2M^2)$. Even in de Sitter space, the common-slice mode functions contribute finite-time oscillatory $k$-dependence. In slow roll, the additional phase $M/H(k)$ strengthens the non-polynomial dependence.

Therefore:

$$\boxed{ \text{NPH state} \xrightarrow{\text{canonical evolution}} \text{common-slice kernel} \notin \text{generic finite local boundary-EFT coefficients}. }$$

The source states this distinction directly: NPH does not conflict with boundary EFT per se, but it does not conform to generic boundary-EFT predictions because it imposes specially near-scale-invariant initial data.

Type classification

The correct comparison has three layers:

1. Carrier/propagation: common and canonical at the free mode level.
2. General coefficient object: the full momentum-dependent quadratic boundary kernel $\kappa(k)$.
3. Local EFT subobject: kernels admitting the prescribed derivative expansion.

NPH and BEFT meet in layer 2, not generically in layer 3.

This sharpens H2:

$$\text{shared carrier and evolution calculus} + \text{different admissible coefficient subcategories}.$$

The failure is not evidence for a new cosmological carrier stratum.

Hard-to-vary conjecture

The natural common coefficient envelope of finite-time and fixed-scale Gaussian initial states is the sheaf of momentum-dependent boundary kernels obtained by Riccati/Möbius transport. Local boundary EFT and NPH are distinct subobjects selected respectively by derivative locality and near-scale-invariant finite-scale phase data.

Finite falsifier

Expand the transported $\kappa_{\rm NPH}(k;t_0)$ in the frozen local boundary operator basis and test whether it closes:

• at every finite derivative order;
• as a convergent or controlled asymptotic completion;
• without fitted nonlocal kernels or additional singular support.

Finite-order failure is expected and already visible in the incompatible spectral dependence. Controlled all-order closure remains open.

Provenance

• B. Greene, K. Schalm, J. P. van der Schaar, and G. Shiu, Extracting New Physics from the CMB, arXiv:astro-ph/0503458, especially Eqs. (1), (4)–(10).
• Ledger sequence claim: seqclaim-748fca4f8d126a29e22882aa.
• Epistemic graph event: ev-000000001635-26a184b9-acbb-41a3-982c-0c94d071444f.

View source on GitHub