Sharp null-energy lower bounds for Page-like entropy excursions in two-dimensional conformal field theory

Mohammed Daoud; Soufiane Karmoudi

REVIEW

Sharp null-energy lower bounds for Page-like entropy excursions in two-dimensional conformal field theory

Soufiane Karmoudi, Mohammed Daoud

High Energy Physics - Theory · Submitted 2 Oct 2026

Pith's verdict: Clean, honest variational result: sharp null-energy bounds for Page-like entropy excursions in 2D CFT, conditional on the primary QNEC, and the math holds up under independent checking.

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arXiv ID2610.03576 (version 2)

Pith IDNH3XOD65, Pith's permanent record of this paper and its review

CitationMohammed Daoud, Soufiane Karmoudi. "Sharp null-energy lower bounds for Page-like entropy excursions in two-dimensional conformal field theory." arXiv:2610.03576 (2026). https://pith.science/paper/NH3XOD65

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Editor's note

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This paper does what it says on the tin: sharp lower bounds on positive, negative, and net null energy for Page-like entropy excursions in 2D CFT, conditional on the primary QNEC (Eq 11). The derivation is the real content, and it survives checking. I traced the KKT step, p* = min{a psi, M} (44) to f_{alpha,R}(z) (49), the roots z+/- = R/(1 +/- q), the two-branch structure of g_alpha(R) (51), and the C^1 matching at R = 2. The net-energy coefficient 12 DeltaS^2/(pi c T) follows cleanly from a triangular linear profile, and the sinusoidal test ratio pi^2/8 reproduces. No fitted parameters, no invented entities; the constants are genuine infima over smooth conformal vacua (Sec VI), and the paper proves that rather than assuming it. The sum sqrt(DeltaS_i / c) < infinity condition in Sec VII is a nice consequence, stronger than finite total variation, and correctly flagged as physically idealized.

What is new: the sharp two-branch E+ bound with threshold (c/6) ln 2, the doubled E_net constant relative to [10], the saturating-sector E- bound, the optimal-profile classification (linear/logarithmic/mixed), and the Pareto frontier. Against the paper's own citations these are genuinely absent from the prior literature.

The soft spots, in proportion. First, the load-bearing premise is the primary QNEC, F >= F_S. The paper says this outright (b27, ref [9]): it holds and saturates in conformal vacua and holographic Banados states, and use elsewhere is an explicit assumption. If primary QNEC fails for thermal, quenched, or other physical states, the E+ and E_net bounds lose every non-saturating instance, and the E- bound is already restricted to the saturating sector by the authors' own logic (b71, b73). This is not a hidden flaw; it is an honestly stated condition, but it caps the physical reach, and a referee should say so plainly. Second, the factor-of-two comparison with [10] is asserted without reproducing the quoted value. [10] is published, so this is a ten-minute verification, not a substantive gap, but the paper should show it. Third, the reader flagged a single-sentence jump to the multi-segment bound (74); Appendix C actually contains the Cauchy-Schwarz and time-allocation argument, so I think that concern is largely answered in the text, though the per-segment additivity could be spelled out a bit more. Minor at worst.

Bottom line: this is a serious, well-executed variational calculation with honest scope conditions. It deserves a real referee. My request to the authors would be: verify the [10] comparison explicitly, and expand the multi-segment step. Anyone working on QNEC, energy inequalities, moving mirrors, or holographic 2D CFT gets value from this; I would bring it to reading group and I would cite it.

Referee report

Significance

If the primary QNEC (Eq. 11) holds in the intended physical classes, the results are sharp, parameter-free, and falsifiable: explicit minimizers are given for every branch, the constants are infima over smooth conformal vacua, and the sinusoidal test supplies concrete dimensionless ratios (π/2, π²/8) that can be checked against the bounds. The factor-of-two improvement over the Abdolrahimi-Page value [10] and the C¹-matching of the two E+ branches at R=2 are clean, testable claims. The variational core is transparent and was verified independently; the paper also contains an honest, clearly flagged conditional scope: all bounds live inside the primary-QNEC class, whose validity outside the saturating conformal-vacuum and Bañados sectors is explicitly acknowledged to be an assumption (Sec. II.B, ref [9]), and the E− bound is additionally restricted to the saturating sector by the reversal (57), which the authors state correctly. The main open question, as the paper itself admits, is whether any physically motivated non-saturating states exist in which (11) is satisfied; the value of the results for thermal, quenched, or other settings depends entirely on that question.

Major objections (3)

[Sec. VI.A (Eqs. 64–66)] The advertised sharpness of all constants as infima over smooth conformal vacua rests on the asserted convergence ∥D²ψε∥TV→∥D²ψ∗∥TV in (65), but this property is stated without proof, and it is not implied by the displayed weak-* convergence together with smoothness of the mollifiers. The strict-convergence implication needed for (66) is exactly the convergence of the Jordan masses, which requires the total-variation statement; please supply the argument (or a standard BV-approximation lemma) that the compression-then-mollification construction has converging total variation, and in particular that the boundary layers created by compressing v∗ into [2ε,τ−2ε] reproduce, rather than alter, the atomic masses of D²ψ∗ in the limit. Please also confirm explicitly that the smoothed approximants are admissible for the stated duration T (the compressed rise occupies τ−2ε, with the remainder filled by constant phases), since equality-tolerance in the duration constraint is part of what converts attainment in the relaxed class into an infimum in the smooth class.
[Sec. II.B, Eq. (11); Sec. IV.A] This is the decisive open condition for the applicability of (55), (56), and (82): the primary QNEC is established only in the saturating conformal-vacuum and vacuum-Bañados classes [7], and the authors rightly flag its use elsewhere as an explicit assumption (Eq. 11 and ref [9]). What the manuscript leaves unresolved is a statement of the present evidence and a concrete test of (11) in candidate non-saturating states (e.g., a thermal state or the quenched CFTs of ref [9]: one can in principle compute F−FS on the null-deformed interval and check nonnegativity), together with an explicit declaration of which advertised bounds survive if (11) is violated there. This matters because the observation in Sec. IV.A that the E+ minimum can be attained with Q>0 (non-saturating states with extra negative flux where FS<0) shows the attainability domain is wider than the saturating sector; the text should state that this widening is itself conditional on the same unverified (11). A paragraph in the Discussion addressing these points would make the reach of the paper match its abstract. The authors acknowledge this: “Primary QNEC and energy cost We assume that the state, region, regulator, and affine null deformation obey the primary two-dimensional QNEC, F(u)≥F S(u) = 1 2π [ S′′(u) + 6 cS′(u)2 ] .(11) This strengthened form is saturated in the conformal- vacuum sector discussed above and, in holographic CFTs, for states dual to vacuum Bañados geometries [7].”
[Appendix C (Eqs. 73–75) and Sec. III.B] The derivation of the multi-segment bound (74) applies the one-segment bound (52) to each monotone segment of an excursion whose interior segments share endpoints with their neighbors. In the relaxed class the flux measure contains atomic terms at the turnover points (Eq. 19), so the additivity of E± over segments needs an explicit bookkeeping statement: each shared atom must be assigned to exactly one segment, and the per-segment lower bounds (37)–(39) must use only that segment's own speed extrema so that the atom at a shared peak is counted exactly once (as 2M when rise and fall are symmetric). A few sentences in Appendix C would remove the only remaining ambiguity in the chain (52)→(74). Relatedly, in Sec. III.B the reduction argument explicitly controls Em_net and the maximal speed, whereas (52)–(54) bound the functional Cα = Enet + (1−α)E−; please spell out that the same per-segment E− bound transfers to arbitrary oscillatory profiles, or state that (54)–(56) are claims about the infimum over the relaxed class, which is what the reduction actually establishes.

Minor objections (7)

[Sec. VIII, Eq. (85)] The saturating flux in (85) carries an ℏc factor while (3), (8), and the energy formulas (88)–(93) do not; please fix the unit convention (state once that ℏ=c=1 is used throughout, or remove the stray ℏ) so the comparisons (90) and (93) are dimensionally transparent.
[Sec. IV.A] The sentence explaining that the absolute minimum of E+ can be attained by states that do not saturate the QNEC would be fully self-contained if it stated the explicit condition R0=0⇔[F]+=[FS]+, i.e., F=FS wherever FS>0 and F∈[FS,0] wherever FS<0.
[Appendix C] The first line of Appendix C reads 'From (52), on has first'; it should read 'one has first'.
[Eq. (61)] The symbol ū is used inside the display (61) but defined only in the surrounding sentence as ū=min{u,T−u}; moving the definition before the display would help the reader.
[Fig. 4] The caption says the dashed and dotted lines are the analytic limits π/2 and π²/8 but does not say which line is which; please label each curve with its limit.
[Sec. III.B, Eq. (28)] The effective crossing speed p(y) defined in (28) is distinct from the ray-tracing function p(u) of (2); the text notes this parenthetically, but renaming one of the two (for instance using w(y) for the crossing speed) would prevent reader confusion.
[Sec. IV.A, comparison with [10]] The factor-of-two comparison with the Enet value quoted in [10] would benefit from a short dictionary between the quantities used in [10] (entropy amplitude, duration, central charge, and the definition of net energy) and ΔS, T, c used here, so that the factor two is unambiguous to a reader who does not consult [10].

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The paper

The paper claims

For 2D CFT states satisfying a strengthened quantum null-energy condition (QNEC), the paper derives sharp lower bounds on the positive, net, and negative null energies required to traverse a Page-like entanglement-entropy excursion.

The reading

The paper asks what it minimally costs in null energy for a two-dimensional conformal field theory to traverse a prescribed Page-like excursion of entanglement entropy over a fixed affine time. For states obeying the primary quantum null energy condition — a strengthened inequality that saturates in conformal vacua and in holographic vacua — the answer is a sharp two-branch lower bound on positive null energy with a threshold at ΔS = (c/6)ln2, plus a sharp bound on net energy that is exactly twice the value quoted earlier by Abdolrahimi and Page. In the QNEC-saturating sector the paper also obtains a sharp lower bound on negative energy. The constants are shown to be infima over smooth conformal vacua, so they cannot be improved within that class, and the entropy profiles attaining each bound are identified explicitly. A final result is that an infinite sequence of small sub-excursions within finite duration requires the summability condition ∑√(ΔS_i/c) < ∞ for the total positive energy to stay finite, which is stricter than requiring finite total entropy variation.

What carries the argument

The load-bearing object is the Bianchi-Smerlak entropy-flux relation F_S(u) = (1/2π)[S''(u) + (6/c)S'²(u)], which fixes the null flux of a conformal vacuum in terms of the entanglement entropy and its derivatives. Substituting ψ = exp[(6/c)(S − S0)] rewrites it as F_S = K ψ''/ψ with K = c/12π, converting the energy cost into a one-dimensional variational problem over the positive profile ψ. The paper minimizes a cost function C_α = E+ − αE− over monotone rising-and-falling pulses, using the fact that a minimizing complete pulse is a single monotone rise and fall, and solves the constrained optimization (fixed duration, bounded slope M) through a KKT analysis that yields the bang-bang profile p*(ψ) = min{aψ, M}. The threshold ΔS = (c/6)ln2 (equivalently R = $e^{{6ΔS/c}}$ = 2) is where the active constraint regime switches, producing the two branches of the E+ bound. Sharpness is established by mollifying the relaxed BV optimizers into smooth conformal vacua with convergent second-distributional-derivative measures.

Extended reading notes

Core claim

The central claim is that the energy cost of a complete Page-like entropy excursion is fixed from below by the saturating-sector profile determined through the Bianchi-Smerlak entropy-flux relation. For positive null energy the sharp bound is piecewise: E+ ≥ (c/3πT)($e^{{6ΔS/c}}$ − 1) for ΔS ≤ (c/6)ln2 and E+ ≥ (c/3πT)(1 + 6ΔS/c − ln2)² above that threshold; the switch at ΔS = (c/6)ln2 reflects a change in the active constraint regime of the underlying variational problem. For net energy the bound Enet ≥ 12ΔS²/(πcT) doubles the coefficient quoted by Abdolrahimi and Page. In the QNEC-saturating sector, E− ≥ (c/3πT)(1 − $e^{{−6ΔS/c}}$). Because every state satisfying the primary QNEC dominates the saturating profile pointwise, the positive- and net-energy bounds transfer to the full QNEC class, while the negative-energy bound is shown not to transfer and is stated only for the saturating sector. Sharpness is established by smoothing the relaxed optimizers: as ε→0 the relaxed values are realized as limits of smooth conformal vacua, so the constants are infima over that class.

Load-bearing premise

The load-bearing premise is the primary quantum null energy condition, F(u) ≥ F_S(u) = (1/2π)[S''(u) + (6/c)S'²(u)]: if a physical state of interest does not obey this strengthened inequality, none of the three energy bounds applies, and the paper explicitly marks its use outside conformal vacua and holographic Bañados vacua as an assumption.

Editorial extensions

If this is right

Any Page-like entropy excursion in a primary-QNEC state of a 2D CFT carries positive null energy at least (c/3πT)(e^{6ΔS/c} − 1) for small heights and (c/3πT)(1 + 6ΔS/c − ln2)² for heights above ΔS = (c/6)ln2, so information-paradox-motivated protocols in this framework cannot evade a definite energy floor.
The net-energy bound Enet ≥ 12ΔS²/(πcT) is a factor of two stronger than the value quoted by Abdolrahimi and Page, tightening the semiclassical constraint on black-hole-radiation energy-entropy trade-offs.
The saturating-sector bound E− ≥ (c/3πT)(1 − e^{−6ΔS/c}) fixes the minimum negative-energy cost of a complete pulse in QNEC-saturating states, complementing quantum-interest and quantum-energy inequalities.
The summability condition ∑√(ΔS_i/c) < ∞ for finite positive energy under an infinite sequence of small sub-excursions is stronger than finite total entropy variation, so highly oscillatory entropy profiles with ∑ΔS_i < ∞ can still demand divergent positive energy.
The optimizing entropy profiles (linear, logarithmic, and mixed linear-logarithmic) are explicitly identified, giving concrete templates for constructing extremal moving-mirror and holographic pulses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the primary QNEC holds generically beyond the known sectors, the two-branch feature at ΔS = (c/6)ln2 should be observable as a sharp discontinuity in the energy cost of entanglement-entropy pulses in holographic computations for vacuum Bañados states; a direct test there would check whether the bounds tighten or are beaten.
The factor-of-two gap with the Abdolrahimi-Page value suggests the earlier bound is not sharp, and the relaxed-class variational machinery identifies where the slack enters; re-deriving that older value from the same formalism would confirm the improvement is genuine rather than artifact.
The √ΔS_i summation condition is reminiscent of compactness-type requirements and may transfer to other settings with an entropy-flux relation, such as moving-mirror models, where finiteness of energy for oscillatory profiles could be tested numerically.
Since a general conformal reparametrization changes positive and negative flux parts separately, a scaling-consistent version of these bounds uniform under time rescalings would require additional work beyond the affine-duration statement given here.

Original abstract

We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_+$ obeys a sharp lower bound with two branches separated by the threshold $\Delta S=(c/6)\ln 2$. We also derive a sharp lower bound on the net energy $E_{\rm net}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_-$. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize $E_+$ for general QNEC states and the one that minimizes $E_-$ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.

The compiled record

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-05: 1 request, 5,177 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.96 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory. For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_+$ obeys a sharp lower bound with two branches separated by the threshold $\Delta S=(c/6)\ln 2$. We also derive a sharp lower bound on the net energy $E_{\rm net}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_-$. The constants are sharp as infima over smooth conformal vacua. We determine the entropy profiles that minimize $E_+$ for general QNEC states and the one that minimizes $E_-$ in the saturating sector. Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We derive sharp lower bounds on the energy cost of Page-like entanglement entropy excursions in two-dimensional conformal field theory.	contribution 0.89	0.97	0.79	asserted 1.00
s1	For states satisfying the primary quantum null energy condition (QNEC), the positive null energy $E_+$ obeys a sharp lower bound with two branches separated by the threshold $\Delta S=(c/6)\ln 2$.	result 1.00	0.88	0.44	asserted 1.00
s2	We also derive a sharp lower bound on the net energy $E_{\rm net}$, with a coefficient larger by a factor of two than the value quoted by Abdolrahimi and Page [Phys. Rev. D 92, 083005 (2015)] and, in the QNEC-saturating sector, we obtain a sharp lower bound on the negative energy $E_-$.	result 0.95	0.94	0.21	asserted 1.00
s3	The constants are sharp as infima over smooth conformal vacua.	result 0.87	0.87	0.09	asserted 1.00
s4	We determine the entropy profiles that minimize $E_+$ for general QNEC states and the one that minimizes $E_-$ in the saturating sector.	result 0.61	0.96	0.10	asserted 1.00
s5	Finally, for an excursion containing an infinite sequence of small sub-excursions within a finite duration, we derive a necessary condition for the total positive energy to remain finite, which is stronger than requiring the entropy to have finite total variation.	result 0.94	0.97	0.09	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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arXiv listing abstract v1

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Dimensionless sharp positive-energy cost

**Figure 2.** FIG. 2. Optimal entropy profiles for the three variational regimes: the logarithmic profile for

**Figure 3.** FIG. 3. QNEC-saturating Pareto frontier for

**Figure 4.** FIG. 4. Positive-energy ratio of the smooth conformal pulse

Figures reproduced from arXiv:2610.03576 by the authors.

Discussion

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