Ledger

Follow each claim to its evidence and its limits. The drawing connects established geometry with calculations and proposed proofs. The new continuous fcc motion has a written argument and exact computational certificate in an open draft; the older searched trajectories remain numerical evidence. Historical priority and independent specialist review are unresolved.

1How to read the labels

• PROVEN a written argument establishes the stated hypotheses. An exact finite certificate can be part of that argument; the label alone does not imply outside review.
• VERIFIED NUMERICALLY computed in code to a stated tolerance, with no proof.
• DRAFT / OPEN a proposed proof, manuscript or repository change in an open pull request; read its review status and scope.
• CONJECTURE proposed, tested only in part or not at all. Guesses and speculation go here too.
• KNOWN in the literature, with a citation. Several are cited from abstracts and summaries only.
• POSITION a philosophical or interpretive stance, argued for but not testable as stated (Drift and Search pages).

The second label records the priority assessment. ESTABLISHED means prior work is identified. ELEMENTARY means a direct derivation, with no novelty claimed. PRIORITY UNRESOLVED means the comparison remains incomplete; an unsuccessful search is not evidence of novelty. OUR ROUTE records the project's presentation or route through known facts. See the source comparison.

2Every claim

3Candidate thesis questions

1. The configuration space of the dot frameworks. Map the full configuration space of the ball and bar contact framework, as a real algebraic variety, for every Wythoff orbit. Which tilt paths are finite, where do they end, and is the incentre always the unique point whose folds end in a uniform packing? (Motion page.)
2. A continuous path to hcp and other branches. A proposed smooth one-cell curve now reaches fcc with exact non-overlap over the whole interval. A comparable continuous proof to hcp remains open; the sampled hcp trajectories do not supply it. Map the other branches and constraints. Open work.
3. Independent specialist review and priority. Review the exact geometric reduction and both arithmetic implementations, compare the equal-bar motion with earlier deployments, and audit the separate spring-elasticity formulas. Review packet.
4. Voronoi-cell Wythoff points as a generator of boron-rich frameworks. The cube gives CaB₆ and the rhombic dodecahedron gives UB₁₂. List the augmented nets of every face-transitive mirror-walled cell, treat α-boron as the frustrated dodecahedral case, and test with density functional theory whether the bond ratio (0.92 to 1.01 across the hexaborides and dodecaborides) follows electron count. (Matter page.)
5. Where the radius-ratio rule can and cannot go. Test whether the shapes the rule cannot reach (7, 9 and dodecahedral 8 neighbours) are over-represented among oxide environments that violate Pauling’s first rule.
6. The perceived centre of a triangle. Run the tap experiment with the incentre as a candidate, in adults and children, with tall and lopsided triangles. (Perception page.)
7. A 3D contact rule for ornament. Apply a Hankin-style rule to the truncated cubic honeycomb, and teach the route from the drawing to Wythoff points as a workshop.

4The plain statement

The value here is a connected route through established geometry, and results that can be inspected and replayed. Periodic bridged cross-polytopes already appear in Borcea and Streinu's 2021 paper (Section 7): the same graph at a different bar length, so our counts cover it. The dimensional counts are likely routine with existing machinery (priority unresolved); the unit-spring results C11/C12 = (n-1)^2 and nu_100 = 1/(n^2-n-1) are the strongest remaining candidate. The newest geometric proposal gives an explicit smooth route from the open cab packing to fcc, with an exact all-parameter non-overlap certificate. It is an open draft, with internal AI checks and independent human review still pending.