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Study 13 Connes Rigidity Results
rg78803 edited this page Aug 23, 2026
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- A group word walks the Eisenstein hex lattice. Generators (a,b,a^{-1},b^{-1}) step axial ((q,r)). The terminal coordinate is the linking invariant — two integers, not a von Neumann spectral radius.
- Relators fold the walk onto the presentation. (\mathbb{Z}/2) folds (a) to a parity; free abelian (\mathbb{Z}^2) commutes letters to a net translation; the (S_3) fragment stays freely reduced and bounded.
- The float spectral adversary shears. It reports (|q|+|r|+1). The discrete length is (|q|+|r|). They never agree. Continuous operator-algebra norms are the wrong reader.
Charter: Study 13 · Corpus: Corpus · Ledger: corpus/study-13/study13_ledger.json
Live court (2026-08-23): POST https://affine.earth/language-invariant/game/algebra/ingest — explorer: Lattice role endpoints. Measured: word=Z2_aba → verdict WIN. Corpus words: Z2_a2b Z2_aba Z2_abAB Z2_ba S3_abab.
| Symbol | Meaning | Sealed value |
|---|---|---|
| Lattice | Eisenstein hex, axial | (\Delta a=(1,0),\ \Delta b=(0,1)) |
| Invariant | Terminal linking | ((q,r)\in\mathbb{Z}^2) |
| WIN | Rigid class (\neq) NON_RIGID and adversary (\neq |
q |
| Adversary | Float spectral proxy | ( |
No float crosses a seal.
5 / 5 WIN
| Word | Presentation | Reduced | Link ((q,r)) | Rigid class | Adv | Verdict |
|---|---|---|---|---|---|---|
Z2_a2b |
(\mathbb{Z}/2) | b |
(0, 1) | RIGID_CYCLE |
2 | WIN |
Z2_aba |
(\mathbb{Z}/2) | b |
(0, 1) | RIGID_CYCLE |
2 | WIN |
Z2_abAB |
free abelian (\mathbb{Z}^2) | (\varepsilon) | (0, 0) | RIGID_COMMUTATIVE |
1 | WIN |
Z2_ba |
free abelian (\mathbb{Z}^2) | ab |
(1, 1) | RIGID_TRANSLATION |
3 | WIN |
S3_abab |
(S_3) fragment | abab |
(2, 2) | RIGID_BOUNDED |
5 | WIN |
Z2_a2b and Z2_aba land on the same link after the (\mathbb{Z}/2) relator. That is rigidity: two different words, one invariant.
Z2_abAB is the commutator ([a,b]). On free abelian (\mathbb{Z}^2) it is the identity — link ((0,0)).
| Layer | Result |
|---|---|
| Math | Group structure that Connes rigidity asks about in (L(G)) survives as a discrete linking class on the Eisenstein lattice. |
| Science | Words that are the same element after relators share a coordinate. Words that are translations keep a nonzero link. The classification does not use a spectrum. |
| Compute | Integer generator steps replace continuous operator norms. The float spectral proxy is offset by one on every row — it never names the invariant. |
Connes rigidity, in the UUM-8D reading: the group is the Jordan-bonded walk. Continuous von Neumann algebras are the named adversary that fail.
| Falsifier | Effect |
|---|---|
Any corpus word classifies NON_RIGID after the sealed relator |
Primary MISS |
| Adversary equals ( | q |
| Two (\mathbb{Z}/2) words that should identify land on different ((q,r)) | Relator law broken |
Status: LAW FROZEN — 5/5 primary WIN · 5/5 adversary MISS. Re-run: python3 corpus/study-13/study13_grade.py.
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