Results

5 Results

We evaluated an ensemble of $N = 20,000$ paths, with path 0 set to the classical Hawking solution ( $k = 1$ only, $ϕ = 0$ ) and paths $1$ through $19,999$ drawn from the confound-controlled random ensemble described above.

Table 1: Representative values from the scalar-field minisuperspace ensemble ( $N = 20,000$ paths, $Kmax = 12$ , $Λ = 1$ , $κ = 0.15$ ). The Hawking instanton ( $k = 1$ only) attains $Cs = ω1 ∕Δω = 1.0$ , the theoretical minimum.

The Spearman rank correlation between $SE$ and $Cs$ across the full ensemble exceeds $ρ = 0.96$ . The partial Spearman correlation $ρ(C ,S | roughness) s E$ remains strongly positive with $p ≪ 0.01$ , confirming that the correlation is not driven by the common sensitivity to path roughness.

The classical Hawking instanton is the unambiguous joint minimum of both $Cs$ and $SE$ : it uses only the $k = 1$ mode for the scale factor and zero scalar field, attaining the lowest possible spectral complexity $Cs = 1$ . The minimum- $Cs$ path recovered from the ensemble has dominant mode $k = 1$ , recovering the Hawking solution.

Figure 1: Minimum-cost paths vs. Hawking solution. The scale factor (left) and scalar field (right) for the Hawking instanton (red), the minimum- $Cs$ path (blue dashed), and the minimum- $SE$ path (green dotted).

Figure 2: $Cs$ distribution (left), $SE$ distribution (centre), and the partial correlation residual scatter after removing the linear effect of roughness (right). The Hawking instanton (red vertical line) sits at the low end of both distributions.

Figure 3: Scatter of $SE$ vs. $Cs$ across all 20,000 paths (linear and log scales). The Hawking instanton (red star) is the joint minimum. Spearman $ρ$ and partial $ρ$ (controlling for roughness) are shown in the panel titles.

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Configuration	Euclidean Action $SE$	Spectral Complexity $Cs$

Classical (no-boundary, $k = 1$ )	7.2450	1.0
Min- $Cs$ path	$∼$ 7.2	1.0
Typical perturbed (median)	$∼$ 15–200	$∼$ 10–80