A theorem you can corner

Half the Cube Plus One

Choose just over half the corners of a hypercube. The page builds every one-bit edge, finds the forced high-degree corner, constructs Huang's signed matrix, checks its spectrum, then turns a painted Boolean truth table into exact sensitivity and polynomial degree.

At exactly half, choose every even-parity corner and no two chosen corners touch. Add one odd corner. A degree witness must appear. Try to hide it by choosing a different half-plus-one set.

Layer one

Choose the corners

Cube dimension n: waiting

Preparing the cube.

The adjacent binary button grid carries the same information as the diagram.

induced maximum degree waiting

signed principal eigenvalue estimate waiting

theorem threshold waiting

The guarantee activates strictly above half.

A₁ = [[0,1],[1,0]]; Aₙ = [[Aₙ₋₁,I],[I,−Aₙ₋₁]]

Exact matrix certificate waiting

Computed full spectrum waiting

Principal matrix waiting

Numerical residual waiting

Cyan cells are +1, rose cells are −1, and dark cells are zero. This is the complete recursively signed matrix, not the ordinary adjacency matrix.

For every selected corner, the browser flips each bit, tests whether the resulting corner is also selected, and counts the induced neighbours. The gold corner has the largest count found. Huang states the half-plus-one case. For any larger selection, discard chosen corners until half plus one remain, apply the theorem there, then restore them. Restoring corners cannot lower an induced degree. The resulting exact guarantee is Δ(H) ≥ ceil(√n).

Layer two, first instrument

One witness is not the whole structure

The later strengthening forces two selected witnesses of opposite parity. Each has at least n other selected corners at Hamming distance one or two. The center is not counted.

Even-parity witness waiting

Odd-parity witness waiting

Radius counts will follow the current cube selection.

waiting

Layer two, second instrument

Paint a Boolean function

Input bits: waiting

Each button is one truth-table row. Press it to switch the output between 0 and 1.

Exact zero-input sensitivity waiting

Exact one-input sensitivity waiting

Real multilinear degree waiting

Exhaustive block sensitivity waiting

Preparing the exact polynomial.

waiting

Every local sensitivity

The check

Every load-bearing value below is recomputed from the current controls. Integer counts and matrix identities are exact. Eigenvalues and eigenvectors use a floating-point cyclic Jacobi solver and are labelled as estimates. The theorem does not depend on rounded output.

Live cube audit

waiting

Live Boolean audit

waiting

Conventions, free choices, approximations

Vertices are integers in ascending order, displayed as n-bit binary strings. Bit 0 is the rightmost displayed bit. Two vertices are adjacent exactly when their XOR has one set bit.
The signing is Huang's recursive signing in that vertex order. Other switch-equivalent signings are possible. The signs change the spectrum, not which cube edges exist.
The display cap n ≤ 6 and truth-table cap n ≤ 5 are interface choices made for instant exhaustive recomputation. The cited theorems have no such cap.
The radius count is |N(v)| + |N²(v)| from the 2020 theorem: selected vertices at distance exactly one or exactly two. It excludes v itself.
The unique polynomial is multilinear over the real numbers. The page obtains integer coefficients by Möbius inversion. This is not algebraic normal form over F₂.
A cyclic Jacobi eigensolver stops after a computed convergence test or a fixed sweep cap. The displayed tolerance is waiting. The residual is ||A_Hv − λv||₂.
The random preset uses the browser's Math.random(). It is a free, non-reproducible display choice and never enters the proof.
The publication-age scope note is being computed.

What is proved, and what remains open

Huang proved polynomial relatedness, including the universal bound bs(f) ≤ s(f)⁴. The optimal universal relation between block sensitivity and sensitivity remains unresolved. The checked 2023 peer-reviewed source says the exact relationship remains open even for transitive functions. It records a quadratic lower-side gap while Huang's argument supplies the quartic upper bound. This status is time-sensitive and was rechecked on 29 July 2026; no claim is made that a search can rule out an unindexed later result.

Primary sources

Hao Huang, “Induced subgraphs of hypercubes and a proof of the Sensitivity Conjecture”, Annals of Mathematics 190(3), 949-955, published online 28 October 2019. DOI:10.4007/annals.2019.190.3.6. peer-reviewed
Hao Huang, arXiv:1907.00847, submitted 1 July 2019. This is the preprint record for the result. preprint
F. R. K. Chung, Zoltán Füredi, R. L. Graham, and P. Seymour, “On induced subgraphs of the cube”, Journal of Combinatorial Theory, Series A 49(1), 180-187, September 1988. DOI:10.1016/0097-3165(88)90034-9. Their construction establishes sharpness. peer-reviewed
Sophie Laplante, Reza Naserasr, and Anupa Sunny, “Sensitivity Lower Bounds from Linear Dependencies”, MFCS 2020, LIPIcs 170, Article 62, published 18 August 2020. DOI:10.4230/LIPIcs.MFCS.2020.62. This is the source for the opposite-parity radius witnesses and deg(f) ≤ s₀(f)s₁(f). peer-reviewed proceedings
Sophie Laplante, Reza Naserasr, Anupa Sunny, and Zhouningxin Wang, “Sensitivity Conjecture and Signed Hypercubes”, ACM Transactions on Computation Theory 18(1), Article 6, 1-25, published online 24 February 2026. DOI:10.1145/3777401. peer-reviewed
Siddhesh Chaubal and Anna Gál, “Tight bounds on sensitivity and block sensitivity of some classes of transitive functions”, Theoretical Computer Science 946, 113687, 10 February 2023. DOI:10.1016/j.tcs.2022.12.037. This is the checked source for the open-status statement. peer-reviewed

Offline differential verifier: node research/half-the-cube-plus-one/verify-half-the-cube-plus-one.mjs