Two live instruments, one hard boundary
The Middle Layer Breaks the Bound
Build the middle layer of a hypercube and watch distinct contributions collide with exact multiplicity. Then operate the later power-loss exponent to see what Cairo's 2025 preprints prove, what the browser merely computes, and why fixed spheres and paraboloids remain open.
A line-control estimate expected the total Fourier energy to stay within one scale-independent bound. Hannah Cairo's February 2025 preprint constructs a logarithmic loss. The finite mechanism at its centre can be operated exactly: take every bit string in the middle layer, switch one absent bit on, and count how many routes reach each point in the next layer.
Layer one: exact finite skeleton
Make the pileup
The browser enumerates every reachable bit string and every one-bit contribution. Select any output to expose all of its parents.
Waiting for the live enumeration.
Selected output
Cairo's Lemma 3.5 supplies a specially embedded configuration in which no plane meets more than this many radius R-1 balls. The number is computed from the chosen d. This toy does not construct or prove that embedding.
Exact here: the middle-layer counts, every route, every multiplicity, and the collision ratio. Imported from the paper: the analytic estimates, the choice N comparable to log R, and the plane-incidence configuration.
Layer two: the stronger dismissal answered
A logarithm is not a power
For every fixed positive epsilon, log R eventually grows more slowly than Repsilon. Cairo's first result therefore defeats the global constant bound, but does not by itself defeat the local Repsilon version. The December 2025 Cairo and Zhang preprint supplies a separate power-loss theorem on dense families of Ck perturbations.
Waiting for the live formula evaluation.
Waiting for the live formula evaluation.
The selected epsilon is at least beta, so beta minus epsilon is not a positive power. The formula is still evaluated, but this setting does not display a growing lower factor.
The check
These lines are regenerated whenever a control moves. They expose what this page derives and what it only receives from a theorem.
Finite count, recomputed
Power bench, recomputed
The theorem-sourced claim: Cairo's v2 preprint proves that for every compact C2 hypersurface in Rd not contained in a hyperplane, and for sufficiently large R, there are a function and a nonnegative weight whose weighted extension energy inside the radius-R ball is at least a hidden constant times log R times the function's squared norm and the largest line integral of the weight. That logarithmic factor contradicts the scale-independent global estimate.
Conventions, free choices, approximations, and uncertainty
- The page fixes the middle weight at floor(N/2), lists bits from coordinate N - 1 down to coordinate zero, orders masks numerically, and counts only switches from zero to one. The sliders' ranges and defaults are editorial choices.
- All finite counts use exact integer arithmetic within JavaScript's safe integer range. N stops at eighteen for responsive enumeration, not because the identity stops there.
- The displayed logarithm is the natural logarithm. Another fixed base changes only a constant in an asymptotic comparison. The R slider is logarithmic and sets R to ten raised to the displayed slider exponent.
- Power and logarithm displays use browser floating-point arithmetic. They illustrate the formulas at a chosen finite scale. They are not evidence for an asymptotic theorem.
- Cairo's analytic inequalities contain hidden constants. N comparable to log R is not equality, and the results apply for sufficiently large R. The finite cube verifies neither threshold nor hidden constant.
- The plane-incidence cap is computed from 2d-1, but the existence of the special embedding is imported from Lemma 3.5. An arbitrary placement of points need not obey it.
- Cairo and Zhang's power theorem concerns dense families obtainable by arbitrarily small Ck perturbation. Dense does not mean every surface. Many strictly convex examples does not mean every strictly convex surface.
- The exponent endpoint is not claimed. When k equals two, beta equals the general C2 upper exponent gamma, so the two bounds match only up to the endpoint.
Source literals and review status
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PreprintHannah Cairo, A Counterexample to the Mizohata-Takeuchi Conjecture, arXiv:2502.06137v2. Version 1 was submitted 10 February 2025 at 03:55:11 UTC. Version 2 was submitted 12 March 2025 at 13:44:52 UTC.
Used for Theorem 1.2, the middle-layer construction in Section 3.2, and the incidence cap in Lemma 3.5.
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PreprintHannah Cairo and Ruixiang Zhang, Power loss for the Mizohata-Takeuchi conjecture on Ck convex hypersurfaces, arXiv:2512.08064v1, submitted 8 December 2025 at 21:56:05 UTC.
Used for Theorem 1.3, the exponent, the perturbation scope, the C2 comparison, and the fixed-surface open problem.
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PreprintInbo Gottlieb Fenves, Cusp Excursions, Lattice Points on Manifolds, and the Mizohata-Takeuchi Conjecture, arXiv:2606.27020v1, submitted 25 June 2026 at 13:32:43 UTC.
Independent follow-up used for the current landscape: a new proof with explicit error terms and a generic Ck result.
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AnnouncementUC Berkeley Analysis and PDE Seminar, Hannah Cairo (UC Berkeley), posted 7 February 2025 for a 10 February 2025 talk.
Independent institutional confirmation of the talk title, date, and logarithmic counterexample description. It is not peer review.
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ProfileUniversity of Maryland CMNS, This Teen Solved a 40-Year-Old Math Mystery. Now She's Seeking a Ph.D. at UMD., 8 September 2025.
Biographical context only: the profile says Cairo was seventeen during the work and eighteen when beginning doctoral study. No exact birth date is inferred here.
Status check: the three mathematical works above are cited as arXiv preprints, not peer-reviewed articles. The source search was repeated on 29 July 2026 and did not locate a journal version for any of them.
Still open
The local Mizohata-Takeuchi conjecture for particular familiar hypersurfaces such as the sphere and paraboloid remains unresolved in the cited literature. Cairo and Zhang perturb surfaces, and Fenves gives a new proof and a genericity theorem, but neither method settles those fixed examples. The endpoint power exponents are also not claimed.