What a ratio profile measures exactly

Ratio statistics and mathematical foundations

Count ordered pairs on their native support

For a positive readout multiset $v=(v_1,\ldots,v_n)$, let $E$ contain eligible ordered pairs $(i,j)$ with $i\ne j$ and set $N_R=|E|$. Kerr retains $0.05<v_i/v_j<5$. Hydrogen and spatial readouts have no finite upper cut. The fixed slot order is

$$(c_1,\ldots,c_{15})=(\varphi,\varphi^{-1},\varphi^2,\sqrt 2,\sqrt 3,\sqrt 5,\pi,\pi/2,\pi/10,e,e^{-1},\ln 2,j_{0,1},3/13,1/7). \tag{5}$$

Here $j_{0,1}$ is the first positive zero of $J_0$. The fine-structure constant is not one of the fifteen centers. At relative tolerance $\epsilon$,

$$h_j=\sum_{(a,b)\in E}\mathbf 1\{c_j(1-\epsilon)\le v_a/v_b\le c_j(1+\epsilon)\},\qquad p_j=h_j/N_R. \tag{6}$$ $$\begin{aligned} A_{\mathrm{SUM}}&=\frac{\sum_j h_j}{N_R},\\ A_{\mathrm{UNION}}&=\frac{1}{N_R}\sum_{(a,b)\in E}\mathbf 1\!\left\{v_a/v_b\in\bigcup_j[c_j(1-\epsilon),c_j(1+\epsilon)]\right\}. \end{aligned}\tag{7}$$

SUM counts overlap multiplicity. UNION counts a ratio once. The structural signature is $p=(p_1,\ldots,p_{15})$, not a vector of null-relative enrichments. Related centers, shared ratios and overlapping windows create dependence. Fifteen slots do not supply fifteen independent physical observations.

The retained tolerances are $\epsilon=0.03,0.01,0.005$. Primary abundance is 3% SUM for Kerr, 3% UNION for minimum-normalized hydrogen log1p, and 1% UNION for spatial graphs. Shape comparisons use the same tolerance across domains and retain all three widths. Spearman with average tied ranks is primary descriptive; Pearson is secondary. There is no independent-slot significance test. Scaling a nonzero profile by a positive scalar changes neither coefficient: $h/N_R$ and $h/\mathrm{SUM}$ yield the same within-profile shape correlation, with different abundance interpretations.

The exact log-autocorrelation identity

Write $u_i=\log v_i$, $\mu_n=n^{-1}\sum_i\delta_{u_i}$, and let $\check\mu_n$ be reflection about zero. The normalized unfiltered nonself measure is

$$\nu_n=\frac{1}{n(n-1)}\sum_{i\ne j}\delta_{u_i-u_j} =\frac{n}{n-1}\mu_n*\check\mu_n-\frac{1}{n-1}\delta_0. \tag{8}$$

Derivation. The convolution includes all $n^2$ ordered terms. Subtract exactly the $n$ diagonal self-pairs and divide the result by $n(n-1)$. Distinct equal-valued modes still contribute at zero after subtraction. Deduplication would change the measure.

$$I_j=[\log c_j+\log(1-\epsilon),\ \log c_j+\log(1+\epsilon)]. \tag{9}$$

With $B=\mathbb R$ in the uncut cases and $B=(\log 0.05,\log 5)$ for Kerr, the exact native profile is

$$p_j=\frac{\nu_n(I_j\cap B)}{\nu_n(B)},\qquad N_R=n(n-1)\nu_n(B). \tag{10}$$

Keep the closed matching intervals distinct from the open native cutoff. SUM adds window masses with multiplicity; UNION measures their union. Relative windows are not perfectly symmetric in log coordinates. The ordered measure $\nu_n$ is symmetric; an asymmetric Kerr cutoff can break reciprocal balance in the conditioned distribution.

The log spectrum means $\log v$, so in primary hydrogen it is $\log\log(1+\lambda/\lambda_{\min,+})$. Spatial mean normalization only translates it. For Fourier convention $\widehat\eta(k)=\int e^{ikt}\,\eta(dt)$, the unnormalized nonself difference measure gives

$$n(n-1)\widehat\nu_n(k)=\left|\sum_i e^{iku_i}\right|^2-n. \tag{11}$$

The squared modulus arises by multiplying the sum and its conjugate; the $-n$ removes diagonal terms. This is a form factor in native log coordinates. It is not an unfolded, ensemble-averaged GOE form factor. Equations (8)–(11) characterize exactly what was counted. An explanation must supply more than reconstruction from the same complete observed spectrum.