Contribution notes
One stand-alone note per contribution of the project, each with a PDF, written for a reader who wants the result and its proof or certificate without the project's history. The HTML versions here are rendered from the same Markdown as the PDFs. Where a later note supersedes part of an earlier one, the later note says so; the earlier notes are kept as they were issued.
| note | date | manifest | |
|---|---|---|---|
| Theorem H for frozen pyrochlore ice: proofs on the cells with a coprime pair of extents and on the cells with two chains in every pair of planes, a reduction to Theorem U alone on every cell, and the cubic cells of side one to eight Theorem H proved by hand on every cell with a coprime pair of extents and on the cells with two chains in every pair of planes; reduced to Theorem U alone on every cell (30 September; $K_u$ no longer a step); by certificates on the cubic cells of side one to eight and on four cells that are not cubic (1 October); the signed moments as currents along the chains; where the method stops at three chains; what is known about Theorem U | 2026-10-01 | 34 claims | |
| A sign-gauge obstruction to negative association in pyrochlore ice no sign gauge makes the bond correlations negatively associated: the three-bond certificate | 2026-09-06 | not yet | |
| Flux sectors of pyrochlore ice: log-concave counts, an octahedral winding support, and a certified failure of real-rootedness flux-sector counts are log-concave, the support is the even lattice in an octahedron, the marginals are not real-rooted at high degree | 2026-09-06 | not yet | |
| Frozen zero-flux states of pyrochlore ice are periodic at \(L = 2\) and \(L = 3\) every frozen zero-flux state of cubic L = 2 and L = 3 is periodic; L = 4 incomplete | 2026-09-06 | not yet | |
| How the work was done: the method of a language-model research programme on pyrochlore ice, and what it produced beyond its results how the work was done: pre-registration and stage gates, the status vocabulary, the two-method gate, the certification ladder, the division of labour, the record; what the method caught and missed; generalisable findings | 2026-09-06 | not yet | |
| The dynamical exponent of the single-hexagon-flip chain on pyrochlore spin ice the exponent, z = 2.004 ± 0.060 over thirteen sizes to 221,184 spins, the 1/L² correction, the fixed-size test | 2026-09-06 | not yet | |
| A pseudoscalar slow mode in pyrochlore ice dynamics: the emergent magnetic helicity at a thin cell the slow even mode at the thin cell is the emergent magnetic helicity; absent as a slow mode in cubic geometry | 2026-09-06 | not yet | |
| The giant-component conjecture for the hexagon-flip graph of pyrochlore ice the giant-component conjecture with exact censuses and 27,000 certified flip paths | 2026-09-06 | not yet | |
| Two frozen-state theorems for pyrochlore spin ice under hexagon flips frozen states exist at every size: Theorem 1 (the smallest cell is frozen) and Theorem 2 (the translation-frozen family, C(2c,c)² and the 16^L bound) | 2026-09-06 | not yet | |
| Frozen pyrochlore ice and the \(\mathbb{Z}_2\) curl: a reduction of the periodicity conjecture, with proofs in one dimension and two and certificates in three the periodicity conjecture reduced to a parity statement: Lemmas A and L, Theorem 2D and the profile lemma proved; Theorems H and P certified; slab sums | 2026-09-06 | not yet |