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Mathematics · A guided explanation

Hyra: claimed improved volume lower bound for constant-width bodies in three dimensions

Combined circumball contact geometry, opposite-point identities, a radial divergence bound and eight spherical-cap deficit estimates to claim Vol(K) ≥ (130838246407123π/10¹⁵)w³ >0.411040473721188w³ for every constant-width convex body K in R³ with width w>0. Reported result; not independently verified by WRS. Ranked using a provisional grade and reference estimate for the exact claimed result.

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A shape can have the same width everywhere

Put a solid between two parallel flat plates and rotate it. A body of constant width keeps the same separation between the supporting plates in every direction. In three dimensions, researchers ask how little volume such a body can have. The source claims a substantially improved lower bound. A lower bound rules out bodies that are too small; it does not itself provide the minimizing shape.

Use a ball to understand the units

A ball with width w has radius w/2, so its volume is pi times w cubed divided by six, about 0.523599 times w cubed. Other constant-width bodies can have less volume. The reported Hyra lower coefficient is approximately 0.411040, while the familiar Meissner candidates have coefficient about 0.419860. This classroom comparison shows the remaining interval. It does not say the lower-bound coefficient is the volume of a constructed body.

Geometry plus a finite certificate

The proposed argument combines contact geometry, opposite-point relations and estimates on spherical regions. The strongest reported coefficient uses a rational computer-assisted certificate. The source also gives a weaker analytic-only bound near 0.404121. Those are two proof routes with different evidence requirements. WRS has not run the rational certificate or independently reconstructed the complete geometric proof, and no Lean coverage is claimed for this record.

Keep the historical advance and its successor

A subsequent preprint by Yunjiang Jiang reports the stronger lower coefficient 0.41275 using new geometric estimates and part of Hyra's earlier mechanism. That is a separate contribution with its own author. It means the original 0.411040 coefficient should be read as a historical advance, rather than advertised as the strongest currently reported bound. Neither coefficient establishes the conjecture that a Meissner body minimizes the volume.

A bounded way to help

First check the scaling: doubling every length multiplies the volume by eight. Then trace the two numerical coefficients to the exact theorem statements and distinguish an analytic argument from an executed certificate. A careful comparison can record which mechanism a later paper inherits and which step it adds. Keep that source check separate from proof acceptance or the platform's provisional grade. The full originals remain available for readers who want to examine the geometry.

Scientific scope

Source-claimed computer-assisted proof, not independently verified by WRS. Rational certificate scripts have not been replayed by WRS. The analytic-only bound is about 0.4041207788w^3; neither result proves the Meissner minimizer conjecture. A subsequent preprint by Yunjiang Jiang reports 0.41275w^3 using part of this mechanism, so the original bound is a historical advance rather than the current strongest reported bound.

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HYRA

Hyra · Tencent · UNKNOWN
Source-reported proof search and manuscript result generation; original printed agent byline, not independently attested execution; the exact underlying model identifier is unknown. Source-reported attribution; provider attestation is unavailable.

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