Open questions and potential solutions

Finite measure and square integrability first make the flux integral finite. Expanding about the mean gives the exact identity $V=E-Q^2\mathrm{/}A$. Subtracting this identity from the energy budget separates the deficit into two nonnegative terms. The function $A\mapsto 4\pi -Q^2\mathrm{/}A$ is nondecreasing, so the area upper bound gives the stated parameter estimate. Integrating a pointwise indicator inequality gives the exceptional-set bound. A two-point measure space realizes equality and establishes abstract sharpness.

Use direct square completion, isolate the nonnegative energy slack, substitute the area upper bound monotonically, and establish sharpness by an explicit finite example. All proof inputs are mathematical; the independent horizon literature supplies context only.

1. For either $f=e$ or $f=b$, the pointwise inequality $2\mathrm{\mid }f\mathrm{\mid }\le 1+f^2$ and finite measure give $\int \mathrm{\mid }f\mathrm{\mid }d\mu \le (A+\int f^{2}d\mu )\mathrm{/}2<\mathrm{\infty }$. Thus $Q$ is finite. Absolute values, scalar multiples and sums are measurable by the closure supplier. Squaring a real measurable function preserves measurability directly: for $a\ge 0$, $\{f^2>a\}=\{\mathrm{\mid }f\mathrm{\mid }>\sqrt{a}\}$, and for $a<0$ the superlevel is $S$. Also $(e-Q\mathrm{/}A{)}^2\le 2e^2+2(Q\mathrm{/}A{)}^2$, so every integral used below is finite. [H-data, math:def-integrable-real-and-complex-functions-and-their-integrals, math:prop-closure-properties-of-measurable-functions-used-by-the-integral, math:prop-order-and-scalar-rules-for-the-nonnegative-integral, math:def-integral-of-a-nonnegative-simple-function, math:prop-the-nonnegative-integral-agrees-with-the-simple-integral, math:thm-linearity-of-the-lebesgue-integral-on-l-one]
2. Write $m=Q\mathrm{/}A$. Linearity and the constant-integral formula give $V=E-2mQ+m^{2}A=E-Q^2\mathrm{/}A$. Consequently $4\pi -Q^2\mathrm{/}A=V+(4\pi -E)$. When $E\le 4\pi$, both terms on the right are nonnegative. This proves the variance bound and $Q^2\le 4\pi A$. When only $E\le 4\pi +\eta$, the same exact identity gives $V\le 4\pi +\eta -Q^2\mathrm{/}A$. [step:1, H-bounds, math:thm-linearity-of-the-lebesgue-integral-on-l-one, math:prop-order-and-scalar-rules-for-the-nonnegative-integral, math:def-integral-of-a-nonnegative-simple-function, math:prop-the-nonnegative-integral-agrees-with-the-simple-integral]
3. Under the parameter assumptions, $Q^2=16{\pi }^2(1-q^2)\ge 0$. Since $0<A\le 4\pi +ϵ$, one has $Q^2\mathrm{/}A\ge Q^2\mathrm{/}(4\pi +ϵ)$. Therefore $4\pi -Q^2\mathrm{/}A\le 4\pi -16{\pi }^2(1-q^2)\mathrm{/}(4\pi +ϵ)=4\pi (ϵ+4\pi q^2)\mathrm{/}(4\pi +ϵ)$. Combine this with step 2; the relaxed energy budget adds exactly $\eta$. In particular, the variance tends to zero whenever $q,ϵ,\eta$ tend to zero under these hypotheses, even if the underlying measure spaces vary. [step:2, H-bounds]
4. Put $h=(e-Q\mathrm{/}A{)}^2+b^2$. For $t>0$, its measurable exceptional set $B_t=\{h>t^2\}$ satisfies $h\ge t^2{\mathbf{1}}_{B_t}$. Monotonicity and the simple-integral formula yield $t^2\mu (B_t)\le \int hd\mu =V$. If $4\pi -Q^2\mathrm{/}A=0$ under the exact energy budget, step 2 gives $V=0$. Applying the same estimate to $\{h>1\mathrm{/}n\}$ shows each has measure zero. Their countable union is $\{h>0\}$, so $h=0$ almost everywhere, giving the asserted equality case. [step:1, step:2, math:prop-order-and-scalar-rules-for-the-nonnegative-integral, math:def-integral-of-a-nonnegative-simple-function, math:prop-the-nonnegative-integral-agrees-with-the-simple-integral, math:thm-finite-and-countable-subadditivity-of-measures]
5. For any $A>0$ and real $Q$ with $D=4\pi -Q^2\mathrm{/}A\ge 0$, take $S=\{-1,1\}$ with all subsets measurable and mass $A\mathrm{/}2$ at each point. Set $b=0$ and $e(\pm 1)=Q\mathrm{/}A\pm \sqrt{D\mathrm{/}A}$. Finite sums give $\int ed\mu =Q$, $V=D$ and $E=Q^2\mathrm{/}A+D=4\pi$. Thus the general variance bound is attained. Taking $A=4\pi +ϵ$ and the specified $Q$ also attains the parameter bound. Replacing $D$ by $D+\eta$ attains the corresponding relaxed bounds. These examples establish sharpness only in the stated class of measure spaces. [H-bounds, math:def-measure-space, math:def-integral-of-a-nonnegative-simple-function, math:prop-the-nonnegative-integral-agrees-with-the-simple-integral, math:thm-linearity-of-the-lebesgue-integral-on-l-one]

• H-data: The space has finite positive mass $A$; $e,b$ are real measurable functions with finite squared integral $E$. Define $Q=\int e$ and $V=\int ((e-Q\mathrm{/}A{)}^2+b^2)$. These are mathematical hypotheses and inline definitions. Source: self-contained; Insight statement; proof step 1 establishes the required integrability.. Reading limits: Locally specified assumptions; no geometric realization or physical premise is assumed.
• H-bounds: The principal case assumes $E\le 4\pi$. The parameter specialization additionally assumes $q\in [0,1]$, $Q=4\pi \sqrt{1-q^2}$, $ϵ\ge 0$ and $A\le 4\pi +ϵ$. The separately stated error variant assumes $E\le 4\pi +\eta$, $\eta \ge 0$. Source: self-contained; Insight statement; proof steps 2–3.. Reading limits: Explicit conditional assumptions, not conclusions imported from the article.
• math:def-measure-space: A measure space consists of a set, a sigma-algebra and a measure on that sigma-algebra. Source: math:def-measure-space; Definition.. Reading limits: Read the complete supplier file and checked its frozen hash; did not independently audit its dependency closure or bibliography.
• math:def-integrable-real-and-complex-functions-and-their-integrals: A real measurable function is integrable exactly when its absolute-value integral is finite; its integral is defined through positive and negative parts. Source: math:def-integrable-real-and-complex-functions-and-their-integrals; Definition, real-valued portion.. Reading limits: Read the complete supplier file and checked its frozen hash; only the real-valued interface is used.
• math:prop-closure-properties-of-measurable-functions-used-by-the-integral: Finite real measurable functions remain measurable under sums, real scalar multiplication and absolute value. Source: math:prop-closure-properties-of-measurable-functions-used-by-the-integral; Statement clauses 1–3; Proof steps 1.3, 2.1 and 3.1.. Reading limits: Read the complete supplier file and checked its frozen hash. Measurability of squares is justified directly in proof step 1.
• math:thm-linearity-of-the-lebesgue-integral-on-l-one: Integrable functions form a vector space and their Lebesgue integral is linear. Source: math:thm-linearity-of-the-lebesgue-integral-on-l-one; Statement; Proof steps 1.1–2.1.. Reading limits: Read the complete supplier file and checked its frozen hash; its cited textbooks were not consulted.
• math:prop-order-and-scalar-rules-for-the-nonnegative-integral: The nonnegative integral is monotone and homogeneous for positive finite scalars; the zero function has integral zero. Source: math:prop-order-and-scalar-rules-for-the-nonnegative-integral; Statement clauses 1–2; Proof steps 1.1–2.1.. Reading limits: Read the complete supplier file and checked its frozen hash; no independent closure audit was performed.
• math:def-integral-of-a-nonnegative-simple-function: For pairwise disjoint measurable sets, the simple integral of ${\sum }_j{c}_j{\mathbf{1}}_{E_j}$ is ${\sum }_j{c}_j\mu (E_j)$. Source: math:def-integral-of-a-nonnegative-simple-function; Definition.. Reading limits: Read the complete supplier file and checked its frozen hash; use is restricted to finite-measure constants and indicators.
• math:prop-the-nonnegative-integral-agrees-with-the-simple-integral: The nonnegative Lebesgue integral of a nonnegative simple function equals its simple integral. Source: math:prop-the-nonnegative-integral-agrees-with-the-simple-integral; Statement; Proof steps 1.1–2.1.. Reading limits: Read the complete supplier file and checked its frozen hash; no reading credit is claimed for its underlying references.
• math:thm-finite-and-countable-subadditivity-of-measures: The measure of a countable union of measurable sets is at most the sum of their measures. Source: math:thm-finite-and-countable-subadditivity-of-measures; Statement; Proof steps 1.1–3.1.. Reading limits: Read the complete supplier file and checked its frozen hash; used only to conclude that a countable union of null sets is null.

Decision: accept

Independently checked all five steps. Finite measure and square integrability ensure the required integrability; expansion about the mean proves the exact decomposition. Nonnegative energy slack gives the deficit and area bounds, and substitution of the area upper bound gives the stated parameter estimate with the additive energy error. The indicator argument proves the exceptional-set estimate and almost-everywhere rigidity. The two-point examples attain the variance bounds. Acceptance covers this abstract corollary and the explicitly conditional geometric context; it establishes no horizon realization, metric stability or Kerr conclusion. The limited novelty comparison is appropriately qualified.

Dependency review: Read all eight immediate mathematical supplier files in full and checked their relevant interfaces and applications. Their measure-space, measurability, integrability, linearity, monotonicity, simple-integral and subadditivity contracts apply without additional geometric assumptions or sigma-finiteness. H-data and H-bounds are explicit mathematical hypotheses; square measurability is established locally. No inference depends on a physical claim, an open problem or the article's own conclusion. Independently checked Dain–Jaramillo–Reiris, section 2 equations (1)–(4), section 3 Definitions 3.1–3.2, Lemmas 3.3–3.4 and equations (30)–(34): their conventions and integrated bound support the contextual application under spacetime outer stability, the stated Einstein–Maxwell normalization, nonnegative cosmological constant and the dominant energy condition for other matter. That application is correctly identified as a conditional physics-theorem, with physical assumptions rather than observations. The original article serves only attribution and comparison.

Review reading limits: Freshly read the complete assigned draft, its five numbered steps and dependency records, and all eight listed local supplier files. Read the frozen article's extracted text on PDF pages 1–3, 6–7 and 26–27, and visually inspected retained images of pages 26–27, including the flux definitions, equation (4.4) and Corollary 4.1. Independently consulted the cited Dain–Jaramillo–Reiris PDF text at the passages specified in dependencies_review; its screenshot request failed, so no visual reading of that PDF is claimed. Did not audit deeper supplier closures, consult their textbook bibliographies, reproduce the author's frozen-hash checks, perform a broad priority search, read all 31 article pages or assess the article's global proof. No earlier reading credit or original-article assessment is inherited.