Non-Identifiability of Subcomponents in Aggregate Functions: A Geometric Information Loss Principle

We develop a general theorem showing that if one component of an aggregate function is bounded below the variability of another and shares the same representational support, then it is mathematically non-identifiable from the aggregate. In information-theoretic terms, the measurement channel has zero capacity for that component. We frame this both rigorously and intuitively, and note its application to the impossibility of detecting Hawking-induced magnetic fields in black holes.

1. Introduction

Aggregate observables are everywhere: a signal is often the sum of multiple contributions. When can we split them apart? Fourier, spherical harmonics, and wavelets succeed because they exploit orthogonality. But what if a component is suppressed in magnitude and occupies the same modes as a dominant contributor? This note shows that in such cases the weak component is non-identifiable: no measurement or decomposition can recover it.

Think of a whisper in a stadium: if its frequencies are the same as the cheering crowd, and it is bounded below the fluctuations of that crowd, no recording can isolate it. This is a geometric and information-theoretic fact, not merely a technological limitation.

2. Setup

Let $D$ be a domain (time, space, spin parameter, etc.). Suppose the aggregate is $$ F(x) \;=\; f(x) + g(x), \qquad x\in D, $$ where $f$ is dominant and $g$ is suppressed.

Definition (Identifiability). We say $g$ is identifiable from $F$ if there exists a mapping $\Phi$ from observables to functions such that $\Phi(F)=g$ uniquely, without further priors on $f$.

3. Non-Identifiability Theorem

Theorem 1 (Non-identifiability under overlap and bound). Assume:

Shared support: $f,g$ both lie in the same function space on $D$ (no orthogonality).
Bounded weak component: $\exists\,C>0$ with $|g(x)|\le C$ for all $x\in D$.
Variability of dominant component: For every $\epsilon>0$, there exists $\Delta f$ with $|\Delta f(x)|<\epsilon$ such that $f+\Delta f$ is an equally admissible model of the dominant process.

Then $g$ is not identifiable from $F$. In particular, any deviation $\le C$ in $F$ can be attributed entirely to $f$.

Proof (elementary). Fix $x\in D$. Suppose $|g(x)|\le C$. For any admissible $\Delta f$ with $|\Delta f(x)|\le |g(x)|$, we may write $$ F(x) = (f(x)+\Delta f(x)) + (g(x)-\Delta f(x)). $$ Hence the same observation can be explained by attributing $g$ to zero and adjusting $f$ by $\Delta f$. Without a prior bounding $\Delta f$ more tightly than $C$, $g$ is non-identifiable. ∎

3.1 Calculus Proof (Banach-space argument)

Proof (calculus version). Let $(\mathcal F,\|\cdot\|_\infty)$ be a Banach space of functions on $D$ (e.g., $C(D)$). Consider the linear observation map $$ S:\mathcal F\times\mathcal F\to\mathcal F,\qquad S(f,g)=f+g. $$ Its Fréchet derivative at any $(f,g)$ is $DS_{(f,g)}[h_1,h_2]=h_1+h_2$, whose kernel is $\{(h,-h):h\in\mathcal F\}\neq\{0\}$. Thus $S$ is not locally invertible in the $g$-coordinate (inverse-function-theorem intuition), precluding local identifiability.

For a global ambiguity, define a smooth path $f_t=f+t\,g$, $g_t=(1-t)g$ for $t\in[0,1]$. Then $S(f_t,g_t)=f_t+g_t=f+g$ is constant, while admissibility is preserved because $\|t\,g\|_\infty\le \|g\|_\infty\le C$ and the dominant class allows perturbations of size at least $C$. If a decoder $\Phi$ satisfied $\Phi(S(f,g))=g$ for all admissible $(f,g)$, it would have to return $\Phi(f+g)=g_t=(1-t)g$ for all $t$, a contradiction unless $g\equiv 0$. Therefore $g$ is not identifiable. ∎

The calculus proof shows both local non-invertibility (Jacobian has a nontrivial kernel) and a global continuum of indistinguishable decompositions via $(f_t,g_t)$ with identical observation.

4. Information-Theoretic Consequence

Corollary 1 (Zero channel capacity). Let $b(x)$ denote measurements of $F(x)$. If the distribution (or admissible uncertainty class) of dominant-component variability $\Delta f$ spans the entire interval $[-C,C]$ and $|g(x)|\le C$, then $$ I(b; g) = 0, $$ where $I$ is mutual information. Thus the measurement channel has zero capacity for the weak component $g$.

4.1 Rigorous proof (stochastic form)

Assume a pointwise model $B = F + \varepsilon = f + g + \varepsilon$, where the law $\mu$ of $\varepsilon$ has support containing $[-C,C]$ and is shift-invariant on that band: for any $t\in[-C,C]$, $\varepsilon+t \overset{d}{=}\varepsilon$. With $|g|\le C$, for each fixed $g$ we have $B\mid g \overset{d}{=} f+\varepsilon$, hence the conditional law of $B$ is independent of $g$. Therefore $B$ and $G$ are independent for any $G$ supported on $[-C,C]$, giving $I(B;G)=0$. Extending over $x\in D$ by product construction yields $I(\{B(x)\}_{x\in D};G)=0$. ∎

4.2 Rigorous proof (adversarial/minimax form)

From Theorem 1, for any admissible $\Delta f$ with $\|\Delta f\|_\infty\le C$ the reparametrization $$ (f,g)\mapsto (f+\Delta f,\; g-\Delta f) $$ preserves the observable $F$ and remains admissible. Thus the family of distributions over observables compatible with any $g\in[-C,C]$ coincides with that for $g=0$. No test can distinguish $g$ from $0$ in the worst case over admissible $\Delta f$, so the minimax information about $g$ is zero; equivalently, the channel capacity for transmitting any information about $g$ is zero. ∎

Inside the $\pm C$ band, any effect of $g$ is distributionally indistinguishable from dominant variability. The channel cannot encode even one bit about $g$.

5. When Recovery is Possible

Otherwise, without one of these structural escape hatches, $g$ is doomed to invisibility.

6. Application: Black Hole Magnetism

In the Kerr black hole context: $$ f \equiv B_{\mathrm{pl}}(a^*), \qquad g \equiv B_{\mathrm{H}}(a^*). $$ They share the same PT–spherical harmonic modal space (shared support). The Hawking component satisfies $$ |B_{\mathrm{H}}(a^*)| \;\le\; C \, g(a^*), \qquad g(a^*)=a^*\sqrt{1-(a^*)^2}\le\tfrac12, $$ with $C\ll 1$. Plasma variability easily spans this bound. Hence $B_{\mathrm{H}}$ is non-identifiable and the mutual information with observations is zero. The Hawking magnetic degrees of freedom are structurally lost in the aggregate.

7. Conclusion

This principle generalizes beyond astrophysics: whenever an aggregate observable combines components with overlapping support and one lies beneath the variability of the other, the weaker is non-identifiable. This constitutes a precise form of information loss in the Shannon sense. Our black hole application illustrates one instance: the Hawking-induced magnetic field is mathematically barred from detection, not just practically but in principle.