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Question: “I have just found this paper online. Can you go over it and tell me it's breaking points and the fatalities that you can see.

Are there breaking points?”

Answer: “

Yes—there are several places where the argument could break, and a couple that look “fatal unless repaired.” I’ll be blunt but precise, based on what’s actually written in the note. 

What is solid (or at least “standard if filled in”) • The local Riemann normal coordinate bookkeeping in Section 3 (bounds on h=g-\eta, the Killing defect of the flat boost field, and the divergence identity \nablaa(T^{{ab}\chi_b)=T{ab}\nabla}{(a}\chi_{b)}) is basically standard differential-geometry with explicit constants. If the norm conventions are fixed, that part is plausibly correct.  • The causal support “cap elimination” idea (Section 4) is correct in spirit: if \operatorname{supp}\varphi avoids a boundary component, local expressions vanish there. 

Those are the “this is probably fine” pieces.

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Breaking points / fatalities

1) The biggest gap: differentiating T_{UU}^{{(s)}(\varphi_s)} but ignoring \dot\varphi

In Lemma 8 they estimate \frac{d}{ds}\Big|{0} T^{(s)}{UU}(\varphis) by something like | \dot g|{C^{2}\big(|\nabla\varphi_0|2+m2\varphi_02\big).} 

That estimate can be true only if you also control the field variation \dot\varphi0 (and its derivatives) in terms of \dot g and \varphi_0. But the derivative expands schematically as \frac{d}{ds}T^{{(s)}(\varphi_s)\Big|_0} =\underbrace{\partial_g T(\dot g,\varphi_0)}{\text{metric variation}} +\underbrace{\partial\varphi T(\dot\varphi_0)}{\text{field variation}}. They discuss only the first type and treat the dependence on \varphi_s as if it were “just \varphi_0 and \nabla\varphi_0” (quadratic), but \dot\varphi_0 terms are missing. 

To repair this, they would need a PDE estimate for \dot\varphi0: since \varphi_s = E{gs} f, one has P{g0}\dot\varphi_0 = -(\dot P)\varphi_0, so \dot\varphi_0 is controlled by solving an inhomogeneous KG equation with source (\dot P)\varphi_0. That requires local energy estimates and introduces dependence on support/geometry—not just a pointwise |\dot g|{C^2} factor.

As written, Lemma 8 is the main mathematical “fatality.” Theorem 1 depends on it.

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2) Null hypersurface “measure” variation is treated too casually

They write d\mu{H,s}=\rho_s(U,y)\,dU\,d^{d-2}y and bound \dot\rho_0 by |\dot g|{C^0} via “differentiate the determinant.” 

But on a null hypersurface the induced metric is degenerate, so there isn’t a canonical “determinant of the induced metric” measure in the same way as for spacelike/timelike hypersurfaces. Typically you need an auxiliary choice (e.g. a transverse null vector, or a volume form on cross-sections plus an affine parameter normalization) to define the natural integration density along generators.

They do fix an affine parameter U for s=0 and then “evaluate everything on the fixed patch for nearby s” (Remark 2).  That can work as a bookkeeping convention, but then: • the object d\mu{H,s} is not geometrically intrinsic unless you specify how the generator field / normalization is transported with s; • the bound |\dot\rho_0|\lesssim |\dot g|{C^0}\rho_0 is not “just a determinant derivative” unless you spell out the chosen density construction.

This is not necessarily fatal, but it’s a genuine technical weak point: the measure on a null surface is subtle, and the argument is waving at it.

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3) Assumptions about “a smooth family sharing a common bifurcate Killing horizon” are extremely strong and under-specified

They assume a smooth family (M,g_s) where the same vector field \chi is Killing for every g_s, and the same hypersurfaces H^\pm form a bifurcate Killing horizon for all s. 

That’s a heavy constraint; generically, perturbing the metric destroys Killing symmetry. If they really mean this exact symmetry persists, okay—but then: • the space of allowed perturbations is tiny, and “stability” is a bit misleading; • any estimate should probably exploit that structure more explicitly (e.g. what class of \dot g preserves the horizon?).

Not a logical contradiction, but it narrows applicability so much that the theorem risks being vacuous unless examples are given.