DOI: 10.5281/zenodo.18157426
title: HDBLAST v27: Kernel-family thresholds + kernel-free robustness certificate for PTA spectral knees

description:  

Release focus (v27 / KernelFamily): This release packages kernel-family closed forms, interpolated threshold tables (N=10 and N=30), kernel-free neff→evidence scaling notes, and the runnable “killswitch/scorecard” pipeline for stress-testing PTA spectral knee claims.

Status: methods + tools release. Physics “Big Bang spark / higher-dimensional” notes are exploratory and included as testable hypothesis scaffolding, not as established conclusions.

# HDBLAST v27 — Kernel-family covariance killswitch + n_eff universality (draft)

This release extends the original **HDBLAST covariance “killswitch”** beyond a single squared‑exponential (SE) correlation model and frames the effect of correlated spectra in terms of an **effective degrees‑of‑freedom** statistic:

\[n_\mathrm{eff}(R) \equiv \frac{(\mathrm{tr}\,R)^2}{\mathrm{tr}(R^2)}\,,\]for an N×N correlation matrix R (so tr R = N).

## What’s new (core idea)

### 1) Kernel family → shared “information budget” via n_eff

Different stationary kernels can share the **same adjacent‑bin correlation** \(\rho_\mathrm{adj}\equiv r_1\) yet induce very different long‑lag structure \(r_k\), and therefore different \(n_\mathrm{eff}\).  We provide closed‑forms / numerics for the mapping

\[\rho_\mathrm{adj} \longmapsto n_\mathrm{eff}\quad\text{(for fixed N)}\]

across a small, interpretable kernel family:

- **SQEXP (SE)**: \(r_k = \rho_\mathrm{adj}^{k^2}\)- **AR(1) / Matérn ν=1/2**: \(r_k = \rho_\mathrm{adj}^{k}\)- **Matérn ν=3/2**- **Matérn ν=5/2**- (Optional) **Banded lag‑1** (Toeplitz with r₁=ρ, r_{k>1}=0) — PSD only in a limited range.

### 2) A unified “killswitch threshold” in kernel space

The original v26 “killswitch” used a reference threshold \(\rho_\mathrm{adj}^\*\approx 0.775\) (for N=10 under SQEXP) as the level of adjacent‑bin correlation required to push the covariance‑frontier evidence from **knee**‑favoring to **SMBHB**‑favoring.

For a *fixed* reference \(\rho_\mathrm{adj}^\*=0.775\) (SQEXP, N=10) the implied effective DOF is

- **Reference:** \(n_\mathrm{eff}^\* \approx 4.419\) (N=10).

We then compute, for each kernel family member, the **critical adjacent correlation** \(\rho_\mathrm{crit}\) that yields this same \(n_\mathrm{eff}^\*\) at N=10:

| Kernel | r_k form | ρ_crit (N=10, n_eff*=4.419) ||---|---|---:|| SQEXP (SE) | ρ^{k²} | **0.775** || AR(1) / Matérn ν=1/2 | ρ^{k} | **0.659** || Matérn ν=3/2 | (1+ak) e^{-ak} (with a set by r1=ρ) | **0.726** || Matérn ν=5/2 | (1+ak+(ak)²/3) e^{-ak} | **0.749** |

**Interpretation:** AR(1) has the **slowest tail** for a given \(\rho_\mathrm{adj}\), so it reaches the same smoothing (same \(n_\mathrm{eff}\)) with **smaller** adjacent correlation.  Therefore AR(1) is the most conservative “danger kernel” in this family.

A simple conservative rule for N=10:

> If you can upper‑bound \(\rho_\mathrm{adj,max} < 0.659\), then **no kernel in the family** can reach \(n_\mathrm{eff}\le n_\mathrm{eff}^\*\) and the covariance‑artifact explanation for the knee is ruled out at this level of conservatism.

### 3) Kernel‑free path (copula‑free bounds → kernel feasibility)

A key next step (prototype scripts included) is to compute **pairwise correlation bounds** that require **no covariance matrix** and **no copula model**, only the 1D marginals (e.g., KDE grids) for each bin.  These bounds can be collapsed into per‑lag feasible intervals \([\underline r_k, \overline r_k]\). A kernel with correlations \(r_k(\theta)\) is feasible only if

\[\underline r_k \le r_k(\theta) \le \overline r_k\quad \forall k=1,\dots,N-1.\]

This yields a **data‑driven maximum feasible** adjacent correlation \(\rho_\mathrm{adj,max}\) (per kernel), which can then be compared to the \(\rho_\mathrm{crit}\) table above.

## Included artifacts

- Threshold tables for N=10 (and example scaling to larger N)- Closed‑form / Lambert‑W notes for Matérn half‑integer inversion- Prototype scripts:  - KDE‑grid copula‑free bounds + max feasible ρ_adj (per kernel family)  - Scorecard utility to turn ρ_adj,max into Pass/Fail/Inconclusive

## How this moves the project forward

This reframes the “killswitch” from **one kernel** into a **kernel‑family** and (ultimately) **kernel‑free** robustness certificate:

1) Convert “how correlated could the bins be?” → a bound on \(\rho_\mathrm{adj,max}\).  2) Compare \(\rho_\mathrm{adj,max}\) to the conservative AR(1) threshold (and/or kernel‑specific thresholds).  3) If \(\rho_\mathrm{adj,max}\) stays below threshold → the knee is robust to covariance artifacts under broad assumptions.

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*Draft prepared 2026-01-03. Numbers above come from the included threshold tables (N=10, interpolation anchored at ρ_adj*=0.775 under SQEXP).*