A continuous route
to close packing.

Move the lattice. Keep the bars.

A valid dense endpoint does not tell us how to get there. The construction below supplies the journey: the octahedron inside each cell stays fixed, while the vectors that repeat it through space change. The connecting bars swivel, keeping exactly the same length.

The older searches found sampled paths to both hcp and fcc, including one-octahedron cells. Their evidence remains VERIFIED NUMERICALLY. This is a separate fcc witness with an all-parameter argument. A certified continuous hcp path is still an open question.

One curve. An infinite packing.

Use units in which the six centres are P = {±e₁, ±e₂, ±e₃}, each ball has radius 1/√2, and each bar has length √2. The infinite packing consists of p + nL(s), for p ∈ P and integer row vectors n ∈ ℤ³.

    ⎡ 0 −1 −1 ⎤
C = ⎢ 1  0 −1 ⎥
    ⎣ 1 −1  0 ⎦

1. Preserve the bars exactly.

   The twelve octahedron edges do not move. Each of the three connecting bars has squared length a² + 2t² = 2. This identity holds throughout the interval.

2. Bound all distant images at once.

   The symmetric part of L has eigenvalues d, d + t, d − t, where d = 2 + a and d − t ≥ 1. Thus |nL| ≥ |n|. Every translation with |n|² ≥ 12 is safely too far away to overlap, even after allowing for the motif offsets.

3. Certify the remaining inequalities.

   171 translations and 19 distinct motif differences leave 3,248 non-self distance conditions. Clearing the positive denominator makes each a quartic. All 16,240 Bernstein coefficients are nonnegative in exact arithmetic over ℚ(√2). Since the Bernstein basis is nonnegative on [0, 1], every remaining distance is safe for the entire motion.

4. Identify the endpoint and the density.

   The endpoint has determinant 12; the six motif cosets fill a translate of the even-parity lattice D₃, which is fcc. A derivative argument shows the determinant decreases strictly along the motion. The ball volume stays constant, so the packing fraction increases strictly to π/√18.

Here PROVEN means a written argument with an exact finite computational component. It does not mean peer reviewed or formalized in a proof assistant. A reviewer still needs to inspect the analytic reduction and the arithmetic implementations.

The framework family is known.

The literature audit found a clear precedent: Borcea and Streinu, Periodic tilings and auxetic deployments (2021), Section 7, discuss regular cross-polytopes connected by bars in arbitrary dimensions. The earlier suggestion that this family had not been found in the literature has been withdrawn.

Their deployment example and this equal-bar packing problem use different proportions and periodic descriptions. A useful comparison must separate fixed-lattice Bloch modes, variable-lattice motions, rigid-body kinematics and finite-stiffness spring elasticity. They are different questions, even when a diagram looks similar.

Goodwin (2006) is a close precedent for linearly bridged rigid-unit frameworks. Li, Refson and Dove (2023) investigate actual hexaborides. The comparison note records which sources were read in full, by section, or only through abstracts. Search non-discovery does not establish novelty.

What changed in this research pass.

The assessment and priorities led to the literature comparison, the continuous-motion construction and a focused paper. The same audit also proposed a stronger attainment proof and proposed a repair for gaps in certificate coverage. These submissions are linked below; the research drafts have not been accepted by an outside reviewer.

All five PRs remain open at this October 2026 site update; follow each PR for its current review state. The proof branch depends on the proposed commitment fix, and the paper depends on the proof branch.

The next contribution is scrutiny.

This is meaningful, reproducible mathematical exploration. Its value lies in a construction others can inspect and reuse, explicit statements that can be challenged, and explanations that connect geometry to rigidity. The present record does not establish a major breakthrough, historical priority or a practical material application.

1. Specialist comparison. Check the precise formulas and equal-bar motion against periodic rigidity, auxetic deployment and crystallographic deformation literature.
2. Independent proof review. Review the infinite-image bound, enumeration coverage, exact arithmetic, endpoint cosets and density argument. The two implementations are internal AI checks, not independent human review.
3. Extend the geometry. Seek an all-parameter hcp witness, map other motion branches and determine what additional constraints prevent folding.
4. Test physical relevance separately. A realizable structure needs forces, bending stiffness, stability and fabrication tolerances beyond the ideal bar-and-ball model.