{"id":"54b4d3c8-c3ab-4344-9931-bad9f938d36f","arxiv_id":"2611.03217","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For unitary, bosonic 2D CFTs just above c=1, crossing symmetry excludes scalar operators with 0.17≲Δ≲1, yielding Δ_gap<0.17 at c=1.0005 and c=1.001.","lead":"The paper uses the numerical conformal bootstrap to show that, just above central charge c=1, a scalar operator whose self-OPE contains no lighter primary cannot have dimension between about 0.17 and 1. This forces the lightest primary of any compact unitary CFT at c=1.0005 or 1.001 to be lighter than 0.17, and ties the edge of the forbidden region to the known Z5 parafermion theory at c=8/7.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"Equation (5) requires the excluded set to be a full interval in Δ_φ, but only grids with spacing ≤0.07 were checked and no monotonicity theorem exists because the blocks depend on Δ_φ; one allowed window would falsify Δ_gap<0.17.","rationale":"The reader's CONDITIONAL verdict is the right one, and my stress-test does not move it. The single most load-bearing step is the inference from finitely many excluded points in Δ_φ to the statement that the whole range between D_Λ(c) and Δ_mod(c) is excluded, which is what converts the bootstrap exclusion into the headline bound Δ_gap < 0.167 at c = 1.0005. I checked whether any argument in the paper supplies interval-ness or monotonicity: it does not, and the data themselves show non-monotonicity — exclusion reappears as allowed above U_Λ(c) even though the constraint set (2) shrinks monotonically with Δ_φ, which means the Δ_φ-dependence of the Virasoro blocks, not the constraint set, controls feasibility. The c = 1.0001 and 1.00001 rows confirm that at fixed Λ the excluded set degrades to isolated points near c = 1, so c = 1.0005 — the anchor of the headline — is the least secure point at which a full interval is claimed. I gave the paper credit where the text supports it: the grid check is stated honestly in the Table 1 caption; the anchor exclusions at 0.1668, 0.1687 and 0.45 carry dense supplementary verification (768-bit precision, independent block tables, spins to 3000, twists to 1000, refined Kac-pole grids); the c_min → 8/7 claim is explicitly caveated as an upper limit ([b123]); and the external inputs I rely on — the modular bound [17], the fusion rules [43], the candidate spectra [18] — are published sources, not attacked here. I rate the ℓ > 40 twist-line sampling as a smaller risk than the reader's emphasis implies, because the anchors that set D_19 are exactly the points with the dense re-verification, and the remaining continuum gap between sampled twists is the accepted limitation of numerical bootstrap work. By contrast, a hole in the excluded interval does not become visible through any amount of verification at the endpoints; only a dense scan or a theorem would settle it. The concrete test I propose (step-0.005 scans with contiguity checks at Λ = 19 and Λ = 23) would either shrink the possible hole to 0.005–0.02 or expose an interruption; in the latter case, Eq. (5) would need re-scoping. Until such a scan is run, the crisp formulations 'we show' and 'forbids' overstate what a sampled certificate supports, and the conditionality in the reader's verdict is exactly appropriate.","tokens_in":17830,"tokens_out":28642,"duration_ms":187619,"concrete_test":"At c = 1.0005 and c = 1.001, re-run the Λ = 19 exclusion search over Δ_φ = 0.10 to 1.10 in steps of 0.005 (about 200 SDPB runs per c), using the recursion of Appendix A at 512-bit precision. For each Δ_φ, first evaluate the anchor functional (certified at Δ_φ = 0.1668 for c = 1.0005, 0.1687 for c = 1.001) on the constraint set (2) and on F_{0,0}(Δ_φ); run a fresh SDPB search only where that evaluation fails. Then test contiguity of the excluded set: no allowed point strictly between two excluded points, and endpoints matching Table 1. Repeat the same scan at Λ = 23 with spacing 0.02. If the excluded set is contiguous at both orders and its endpoints match, the interval claim behind Eq. (5) survives with residual windows ≤ 0.005; if any allowed point interrupts the interval, re-check that point at Λ = 27, and if it remains allowed, Eq.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The gap bound (5) depends on more than the exclusion of individual points: it requires the entire interval (D_Λ(c), Δ_mod(c)) to be excluded, because the lightest primary of any compact theory has Δ_gap ≤ Δ_mod(c) < 0.525 [17] and U_19(c) > Δ_mod(c). The only certificate for interval-wide exclusion is grids in Δ_φ of spacing at most 0.07 (Table 1 caption). That sampling cannot establish the interval claim. The excluded set at fixed Λ is the set of Δ_φ for which an SDPB functional exists, and no monotonicity theorem links neighboring Δ_φ: the Virasoro blocks and the identity term F_{0,0} in (3) depend on the external dimension Δ_φ, so the feasibility problem changes between grid points. The paper's own data show that exclusion is not monotone in Δ_φ: it fails above U_Λ(c) ≈ 0.999–1.037 although the constraint set (2) only shrinks as Δ_φ grows, so a smaller constraint set does not imply exclusion. Moreover, the c = 1.0001 and 1.00001 rows of Table 1 show exclusion at Λ = 19 becoming spotty (isolated points) as c → 1, and the headline anchor c = 1.0005 is the nearest tabulated c at which the full interval is claimed. A single allowed window of width < 0.07 inside (0.1668, Δ_mod(1.0005)) would put a compact theory's lightest primary above 0.167 and falsify the crisp statements 'we show' and 'forbids'. The authors state honestly that the interval was 'checked on grids of points', so the defect is a gap between numerical evidence and the conclusion-statement, not a hidden error. I rate the Appendix A sampling for ℓ > 40 (28 fixed-twist lines plus limits) a smaller risk, because the anchor exclusions that set D_19 received dedicated dense re-verification (spins to 3000, twists to 1000, refined grids near Kac poles) and positivity along each line is imposed exactly; the unresolved risk is the Δ_φ direction, which no amount of per-point verification can repair.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper applies the numerical conformal bootstrap to the four-point function of a real scalar Virasoro primary φ in unitary bosonic CFTs with c > 1, assuming only that no primary lighter than φ appears in the φ×φ OPE (Eq. (2)). At derivative order Λ, a point (c, Δ_φ) is called excluded when a linear functional on the truncated crossing equation is positive on all allowed exchanged primaries and on the identity term. The paper reports three main results. First, just above c = 1, an interval 0.17 ≲ Δ_φ ≲ 1 is excluded at Λ = 19, with endpoints D_Λ(c) < Δ_φ < U_Λ(c) tabulated for c from 1.0005 to 1.14. Second, combining the lower endpoint with the modular bound Δ_mod(c) < 0.525 yields Δ_gap < 0.167 at c = 1.0005 and Δ_gap < 0.169 at c = 1.001 (Eq. (5)), with similar bounds up to c = 1.10. Third, the upper boundary c_min(Δ_φ) has sharp minima at Δ_φ = 1/2 and 4/7 whose bisection upper limits approach c = 8/7 within 2×10^-7; the extremal functionals at those points have zeros matching the spectra of the Z5 parafermion and WB_2(5,7), which also saturate a second bound on the lightest heavier scalar in φ×φ. The N = 1 minimal model at c = 81/70 is found to lie just above the boundary on a short plateau.","tokens_in":18310,"tokens_out":18466,"duration_ms":106860,"significance":"If the interval exclusion is correct, the paper's gap bound is the strongest constraint to date on compact unitary CFTs in the 1 < c < 8/7 desert, and the c_min(Δ_φ) minima give a striking bootstrap signature of the Z5 parafermion and its orbifold, in the spirit of the three-dimensional Ising kink conjecture. The paper earns credit for several verifiable strengths: the excluded region is monotone in the derivative order Λ by construction; the numerics are careful (512/768-bit precision, q-truncation cross-checks to relative accuracy 10^-20, refined grids near Kac poles); the two anchor exclusions behind Eq. (5) were re-verified with an independent higher-order block table; and the comparison with the known theories at c = 8/7 is made after the fact, with no quantity fitted to them. The authors also state several limitations honestly, notably the bisection upper-limit caveat for c_min and the grid-based nature of the interval check. The principal weakness is that the categorical interval-level conclusions go beyond the certified numerical evidence.","major_comments":[{"comment":"The headline conclusions — 'crossing symmetry excludes the interval 0.17≲Δ_φ≲1' and 'Δ_gap < 0.167 at c = 1.0005' — require that every point of the interval (D_Λ(c), Δ_mod(c)) be excluded, because the lightest primary of a compact theory could sit at any dimension below Δ_mod(c). The evidence is a grid in Δ_φ of spacing at most 0.07 (Table 1 caption), with only a handful of interior points per row, and the paper states this honestly ('which we checked on grids of points at each of these values of c'). The unresolved step is that no monotonicity or openness argument transfers exclusion from grid points to the whole interval, and such an argument is not automatic: the Virasoro blocks and the identity term F_{0,0} in Eq. (3) depend on Δ_φ, and the paper's own data show the excluded set at fixed c is not monotone in Δ_φ — it is an interval, with allowed points both below D_Λ(c) and above U_Λ(c) ≈ 1, even though the constraint set (2) only shrinks as Δ_φ grows. An allowed window of width smaller than 0.07 inside (0.1668, Δ_mod(1.0005)) would put a compact theory's lightest primary above 0.167 and falsify Eq. (5). The authors should either (i) prove that a functional excluding Δ_φ = x excludes a neighborhood of controlled size (for example by interval or polynomial bounds in Δ_φ), (ii) scan densely enough to rule out windows of the relevant width and state the result with that qualification, or (iii) soften the conclusion statements in the abstract, Section 3, and Section 6 accordingly.","section":"§3, Eq. (5) and Table 1","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Positivity of the functional is certified only on a sample of the allowed (h, h̄) spectrum, not on all of it: for spins ℓ > 40 the conditions are imposed on 28 fixed-twist lines (with ℓ continuous) plus the large-spin and large-dimension limits, and near the Kac poles h_{r,s}(c) only on finite grids. The appendix presents these sampled conditions as the positivity conditions (4) without flagging the un-sampled twists as a technical assumption, and the conclusion is phrased categorically ('No unitary CFT then has a primary φ with these values of c and Δ_φ'). For the anchor exclusions used in Eq. (5) the extra checks are substantial (spins to 3000, twists to 1000, refined grids near Kac poles), but they remain finite samples; a primary with ℓ > 3000 or on an intermediate twist is never evaluated. This is standard practice in the numerical two-dimensional bootstrap, but in a paper whose abstract claims that crossing symmetry 'forbids' the interval, the step from sampled constraints to Eq. (4) should either be justified with denser or adaptive sampling plus a continuity statement, or be stated explicitly as an assumption on which the exclusions rest.","section":"Appendix A and Eq. (4)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"The statement that c_min lies 'within 2×10^-7 of c = 8/7' is an upper limit from bisection, as the body correctly explains (c = 8/7 itself is allowed, and the figure plots upper-limit arrows); please carry that qualification into the abstract and the first paragraph of Section 4, since the current phrasing could be read as a proven approach from below.","section":"Abstract and Fig. 2","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The test of the candidate at c = 1.05 is reported as consistent 'only if some of its dimensions lie outside their quoted uncertainties'; please state which dimensions and by how much, so that the test is reproducible rather than a binary verdict.","section":"Section 3, candidate spectra","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The manuscript does not state where the SDPB input files, functional certificates, or grid data can be obtained; given that the claims are numerical, an availability statement or a data release would substantially strengthen verifiability.","section":"Reproducibility","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"Reference [20] is a conference talk slide reporting preliminary results; since the manuscript compares its bound with those results ('whose minimum is about 0.2'), please cite the published or preprint version when it becomes available. The comparison is not load-bearing, so this does not affect the conclusions.","section":"References","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"For the editor: the manuscript's honest reporting of its own limitations (the grid-based interval check and the bisection upper limits) is a point in its favor, as is the transparent acknowledgment of the AI-assisted computation and the proper coordination with the companion paper [49]. The decisive issue is the gap between certified per-point exclusions and the interval-level claims in the abstract and Section 6; this is fixable within the paper's scope, and the c = 8/7 extremal-functional analysis is likely to survive the revision intact. If the authors choose option (iii) and soften the claims, the paper would still be a solid contribution, though a weaker one."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Short version: this is a serious numerical bootstrap paper. It finds a new excluded region just above c=1 and a sharp gap bound, and it points convincingly at the two known theories at c=8/7. The one thing to keep in mind before quoting the headline number is that the interval-wide exclusion behind (5) is verified on grids, not proved, and there is no monotonicity in Δ_φ that would make grid verification enough.\n\nWhat's new: the assumption (2) — no primary lighter than φ in φ×φ — is natural and every compact CFT's lightest primary satisfies it. Under that assumption, crossing symmetry excludes 0.17 ≲ Δ_φ ≲ 1 just above c=1, the compact boson realizing all these dimensions at c=1. Combining with the modular bootstrap gives Δ_gap < 0.17 at c=1.0005 and 1.001, a third of the c=1 value. Further out, c_min(Δ_φ) has sharp minima at 1/2 and 4/7 that approach 8/7 within 2×10^-7, matching scalar dimensions in the Z5 parafermion and WB2(5,7). The numerics look careful: 512-bit SDPB, q-truncation checks to 10^-20, dense re-verification of the anchor exclusions near Kac poles, and a sensible control test. The paper also tests the candidate spectra of Benjamin et al. and is explicit about what those tests can and cannot show.\n\nThe main soft spot is the interval claim. Equation (5) needs every Δ between D_Λ(c) and Δ_mod(c) to be excluded, not just the sampled points. The checks are grids with spacing at most 0.07, and the blocks depend on Δ_φ, so there is no monotonicity argument. The paper's own data show exclusion is not monotone in Δ_φ: it fails above U_Λ(c) about 1.0–1.06 even though the constraint set shrinks as Δ_φ grows. A single allowed window of width less than 0.07 would falsify the crisp statement. The authors honestly say the interval was 'checked on grids of points', so this is a presentation gap, not a hidden error — but the abstract's 'we show' is stronger than the evidence strictly licenses.\n\nMinor: positivity for ℓ>40 is sampled on 28 fixed-twist lines rather than proved over all twists. This is standard practice, the sensitive points got dedicated dense checks, so I don't treat it as a real risk. No code or data are shipped; independent reproduction requires reimplementation.\n\nWho this is for: bootstrap practitioners and anyone working on the c>1 desert. It deserves a serious referee. I'd send it to review and ask the referee to focus on whether the interval claim can be tightened — denser scans, a partial monotonicity argument, or softer wording — and to encourage shipping the code. The core observation is likely to survive.","headline":"Serious numerical bootstrap result: new excluded region just above c=1 and Δ_gap < 0.17, with the interval claim resting on grids rather than proof.","tokens_in":18986,"tokens_out":6629,"would_cite":true,"duration_ms":39135,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["81T40","81R10"],"pacs":["11.25.Hf"],"model":"deepseek/deepseek-v4-flash","headline":"Crossing symmetry bans operator dimensions $0.17$ to $1$ just above $c=1$; with the modular bootstrap this forces a compact unitary CFT's lightest primary below $0.17$.","keywords":["conformal bootstrap","two-dimensional CFTs","Virasoro algebra","crossing symmetry","modular bootstrap","operator spectrum gap","unitarity","compact boson"],"falsifier":"Run the functional that excludes $\\Delta_\\phi=0.1668$ at $c=1.0005$ (or $0.1687$ at $c=1.001$) over a dense scan of twists between the 28 sampled lines and at spacing $10^{-4}$ near the poles of the block recursion, up to the twist and spin ranges the paper checked; one negative value would refute the certificate and remove the bound. The decisive alternative is constructive: exhibit any compact unitary bosonic CFT at $c=1.001$ whose lightest primary has dimension in $0.17\\lesssim\\Delta\\lesssim1$.","tokens_in":19262,"feed_emoji":"⛔","tokens_out":25120,"duration_ms":134382,"temperature":0.7,"pith_summary":"Representation theory alone does not forbid a compact unitary two-dimensional CFT just above $c=1$, yet no such theory is known until $c=8/7$, so any obstruction must come from the remaining consistency conditions, crossing symmetry and modular invariance. The paper shows that, slightly above $c=1$, crossing symmetry together with unitarity excludes the whole interval $0.17\\lesssim\\Delta_\\phi\\lesssim1$ for a scalar Virasoro primary $\\phi$ whose self-OPE contains no lighter primary, even though the compact boson at exactly $c=1$ realizes every one of those dimensions. Combined with the modular-bootstrap bound on the lightest primary, the exclusion forces every compact unitary bosonic CFT at $c=1.0005$ or $c=1.001$ to contain a scalar lighter than about $0.17$. Farther from $c=1$, the boundary of the excluded region dips twice, at $\\Delta_\\phi=1/2$ and $4/7$, to within $2\\times10^{-7}$ of $c=8/7$, where two known theories have scalars of exactly those dimensions. The paper leaves open whether any theory occupies the still-allowed band of very light scalars below $0.17$.","feed_headline":"Crossing symmetry bans operator dimensions 0.17 to 1 above c=1","feed_subtitle":"Combined with modular invariance, this forces the lightest primary below 0.17 at c=1.0005 and c=1.001.","key_machinery":"The machine is the numerical conformal bootstrap run on Virasoro blocks, the holomorphic-times-antiholomorphic contributions of Virasoro-symmetry multiplets. At each candidate point $(c,\\Delta_\\phi)$ the paper asks whether a linear functional $\\alpha$ on the derivatives $\\partial_z^m\\partial_{\\bar z}^n$ (with $m<n$, $m+n$ odd, $m+n\\le\\Lambda$) of the crossing combination $F_h(z)F_{\\bar h}(\\bar z)-F_h(1-z)F_{\\bar h}(1-\\bar z)$ at $z=\\bar z=1/2$ can satisfy $\\alpha[F_{0,0}]>0$ while staying non-negative on every block with $\\Delta_O\\ge\\max(\\Delta_\\phi,\\ell_O)$; such an $\\alpha$ is a certificate that no unitary CFT has a primary $\\phi$ of that dimension. Zamolodchikov's recursion in the elliptic nome writes each Taylor coefficient as a polynomial in the conformal weight divided by a Kac-determinant denominator, and for $1<c<25$ that denominator is positive on $h\\ge0$, so the semidefinite program solver can search for $\\alpha$ directly. A certificate found at order $\\Lambda$ remains valid at every higher order, so the excluded region can only grow with $\\Lambda$, and the boundary $c_{\\rm min}(\\Delta_\\phi)$ is located by bisection. At the boundary the zeros of the extremal functional reproduce the dimensions of the known theories (twist $10/7$ and $16/7$ families for the $N=1$ model; $8/5$ and $12/7$ for the two $c=8/7$ theories), which is how the paper identifies the sharp minima with the $\\mathbb{Z}_5$ parafermion and its orbifold. The compact-boson fact that $\\cos(X/R)\\times\\cos(X/R)$ contains only $\\cos(2X/R)$ below the continuum is what lets every $\\Delta_\\phi\\le2$ satisfy the no-lighter-primary assumption at $c=1$.","core_discovery":"On its own terms, the paper's claim is a two-part statement about the $(c,\\Delta_\\phi)$ plane for a real scalar Virasoro primary $\\phi$ whose self-OPE contains no lighter primary. First, just above $c=1$ the crossing equation of $\\langle\\phi(0)\\phi(z,\\bar z)\\phi(1)\\phi(\\infty)\\rangle$ has no solution with non-negative squared OPE coefficients in the interval $0.17\\lesssim\\Delta_\\phi\\lesssim1$: a positive linear functional on the identity cancels the crossing of every allowed Virasoro block. At $c=1$ the same claim is false, because the compact boson realizes every $\\Delta_\\phi\\le2$. Second, combining the excluded interval with the modular-bootstrap bound that the lightest primary near $c=1$ is a scalar below about $1/2$ forces $\\Delta_{\\rm gap}<0.167$ at $c=1.0005$ and $\\Delta_{\\rm gap}<0.169$ at $c=1.001$ (inequality (5)), with the same argument excluding gaps in $0.3<\\Delta_{\\rm gap}<0.5$ up to $c=1.14$. Away from $c=1$, the boundary $c_{\\rm min}(\\Delta_\\phi)$ of the excluded region has two sharp minima near $\\Delta_\\phi=1/2$ and $4/7$ that approach $c=8/7$ within $2\\times10^{-7}$ at order $\\Lambda=31$; these are exactly the dimensions of the twist field of $\\mathrm{WB}_2(5,7)$ and the energy operator $\\epsilon_1$ of the $\\mathbb{Z}_5$ parafermion, the two known theories at that central charge.","pith_inferences":["My reading: the lower endpoint of the excluded interval at $c=1.0005$ falls from $0.229$ at $\\Lambda=11$ to $0.159$ at $\\Lambda=27$, so the $0.167$ and $0.169$ certificates are conservative; the true bound is plausibly smaller, and an analytic functional realizing it would be the natural limit of this construction.","My inference: the argument compresses the open existence question into the band $\\Delta_{\\rm gap}<0.17$ — at $c=1.001$ a compact unitary CFT either contains a very light scalar or does not exist, so constructing or excluding such a theory is now a sharply focused problem rather than a search over all gaps below $1/2$.","My testable suggestion: run the same crossing scan with positivity imposed on a dense twist grid (spacing $10^{-4}$) near the poles of the block recursion at $c\\approx1.156$; if the plateau just below the $N=1$ minimal model's central charge persists and its extremal zeros keep matching the twist-$10/7$ and twist-$16/7$ families, it would indicate a genuine family of crossing solutions there rathe"],"forward_implications":["At $c=1.0005$ and $c=1.001$, every compact unitary bosonic CFT must contain a scalar lighter than about $0.17$; no such theory can have a lightest primary anywhere in the range $0.17\\lesssim\\Delta\\lesssim1$.","Up to $c=1.14$, gaps in $0.3<\\Delta_{\\rm gap}<0.5$ are impossible, filling a window in which modular bootstrap alone had found no candidate spectra below $c=1.06$.","The bounds are monotone in the numerical order: a functional that excludes a point at order $\\Lambda$ works at every higher order, so the excluded interval grows and the $\\Delta_{\\rm gap}<0.17$ bound can only tighten as $\\Lambda$ increases.","If the excluded interval reaches $c=1$ as $\\Lambda\\to\\infty$, the space of allowed theories is discontinuous at $c=1$, since the compact boson realizes every $\\Delta_\\phi\\le2$ there.","The two theories at $c=8/7$ sit exactly at the sharp minima of the boundary and saturate the upper bound on the lightest scalar in $\\phi\\times\\phi$ heavier than $\\phi$ ($8/5$ for the twist field, $12/7$ for the parafermion energy operator), so crossing symmetry singles out their spectra at those points."],"supporting_citations":[{"why":"Supplies the $c=1$ contrast: the compact boson has a scalar $\\cos(X/R)$ of every dimension $\\le2$ that satisfies the no-lighter-primary assumption.","marker":"[12]"},{"why":"Supplies the modular-bootstrap upper bound on the lightest primary (below $0.525$ up to $c=1.14$) that turns the four-point exclusion into the gap bound.","marker":"[17]"},{"why":"Provides the candidate spectra at $c=1.05$–$1.13$ whose lightest scalars the paper tests against the crossing equation.","marker":"[18]"},{"why":"The origin of the numerical-bootstrap method, linear functionals on derivatives of the crossing equation, that the paper applies to Virasoro blocks.","marker":"[28]"},{"why":"The recursion representation of Virasoro blocks in the elliptic nome, which makes each Taylor coefficient a polynomial in the conformal weight, the form the semidefinite solver needs.","marker":"[34]"},{"why":"The semidefinite program solver used to search for the exclusion functionals at each candidate point.","marker":"[39]"},{"why":"Gives the $\\mathbb{Z}_5$ parafermion energy operator of dimension $4/7$ at $c=8/7$, one of the two minima of the exclusion boundary.","marker":"[6]"},{"why":"Supplies the fusion rules of the charge-conjugation orbifold used to verify that its twist field of dimension $1/2$ satisfies the no-lighter-primary assumption.","marker":"[43]"},{"why":"The formula used to compute the twist-field fusion rules from the modular $S$-matrix.","marker":"[45]"},{"why":"Shows that the $N=1$ minimal model's scalar $\\Phi_{13}$ does not appear in its own OPE, supplying the third known theory as a check point at $c=81/70$.","marker":"[47]"}],"fun_headline_variants":["Crossing symmetry rules out dimensions 0.17–1 just above c=1","Forbidden scalar dimensions 0.17–1 just above c=1","Bootstrap excludes scalar dimensions 0.17–1 above c=1","Numerical bootstrap leaves a gap below 0.17 at c=1.0005"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The load-bearing premise is that the crossing exclusions are valid everywhere, although positivity of the functional is checked only on a sample: for spins above 40 on 28 fixed-twist lines plus large-spin and large-dimension limits, near the poles of the block recursion on finite grids, and across the interval between the lowest excluded dimension and the modular bound on grids of spacing at most 0.07 — a single negative value between sampled points would invalidate the exclusion and with it the $\\Delta_{\\rm gap}<0.17$ bound.","fun_headline_variants_meta":{"raw":{"variants":["Crossing symmetry rules out dimensions 0.17–1 just above c=1","Forbidden scalar dimensions 0.17–1 just above c=1","Bootstrap excludes scalar dimensions 0.17–1 above c=1","Numerical bootstrap leaves a gap below 0.17 at c=1.0005"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000122,"raw_usage":{"cost":0.0001222858,"is_byok":false,"cost_details":{"upstream_inference_cost":0.0001222858,"upstream_inference_prompt_cost":3.60178e-05,"upstream_inference_completions_cost":8.6268e-05},"total_tokens":1709,"prompt_tokens":1235,"completion_tokens":474,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":1049,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":394}},"tokens_in":1235,"tokens_out":474,"cost_source":"openrouter_usage","duration_ms":5811,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":394,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-11-03T16:29:39.097381+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Run the functional that excludes $\\Delta_\\phi=0.1668$ at $c=1.0005$ (or $0.1687$ at $c=1.001$) over a dense scan of twists between the 28 sampled lines and at spacing $10^{-4}$ near the poles of the block recursion, up to the twist and spin ranges the paper checked; one negative value would refute the certificate and remove the bound. The decisive alternative is constructive: exhibit any compact unitary bosonic CFT at $c=1.001$ whose lightest primary has dimension in $0.17\\lesssim\\Delta\\lesssim1$.","supporting_citations":[{"cited_title":"Diakonov, V","cited_arxiv_id":null,"evidence_quote":"Supplies the $c=1$ contrast: the compact boson has a scalar $\\cos(X/R)$ of every dimension $\\le2$ that satisfies the no-lighter-primary assumption."},{"cited_title":"Modular Bootstrap Revisited","cited_arxiv_id":"1608.06241","evidence_quote":"Supplies the modular-bootstrap upper bound on the lightest primary (below $0.525$ up to $c=1.14$) that turns the four-point exclusion into the gap bound."},{"cited_title":"Descending into the Modular Bootstrap","cited_arxiv_id":"2604.01275","evidence_quote":"Provides the candidate spectra at $c=1.05$–$1.13$ whose lightest scalars the paper tests against the crossing equation."},{"cited_title":"Bounding scalar operator dimensions in 4D CFT","cited_arxiv_id":"0807.0004","evidence_quote":"The origin of the numerical-bootstrap method, linear functionals on derivatives of the crossing equation, that the paper applies to Virasoro blocks."},{"cited_title":"Conformal symmetry in two-dimensional space: Recursion representation of conformal block,","cited_arxiv_id":null,"evidence_quote":"The recursion representation of Virasoro blocks in the elliptic nome, which makes each Taylor coefficient a polynomial in the conformal weight, the form the semidefinite solver needs."},{"cited_title":"A Semidefinite Program Solver for the Conformal Bootstrap.JHEP, 06:174,","cited_arxiv_id":null,"evidence_quote":"The semidefinite program solver used to search for the exclusion functionals at each candidate point."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the $\\mathbb{Z}_5$ parafermion energy operator of dimension $4/7$ at $c=8/7$, one of the two minima of the exclusion boundary."},{"cited_title":"Fusion rules for $\\mathbb{Z}_{2}$-orbifolds of affine and parafermion vertex operator algebras","cited_arxiv_id":"1904.01798","evidence_quote":"Supplies the fusion rules of the charge-conjugation orbifold used to verify that its twist field of dimension $1/2$ satisfies the no-lighter-primary assumption."},{"cited_title":"Fusion Rules and Modular Transformations in 2D Conformal Field Theory","cited_arxiv_id":null,"evidence_quote":"The formula used to compute the twist-field fusion rules from the modular $S$-matrix."},{"cited_title":"Structure constants in the N=1 superoperator algebra","cited_arxiv_id":"hep-th/9109050","evidence_quote":"Shows that the $N=1$ minimal model's scalar $\\Phi_{13}$ does not appear in its own OPE, supplying the third known theory as a check point at $c=81/70$."}],"review_version":1}