Crystal Regime (τ-Lattice)

The τ-crystal regime is the many-body matter phase characterized by μ ≈ 0 (locked) — a lattice-ordered configuration of T² defect bundles whose translational degree of freedom is frozen. Bravais lattices arise as T²-subgroups (IV.P254), Bloch's theorem on T² (IV.T237) governs band structure, and Penrose tilings (IV.R176) appear as quasi-crystal sub-cases. Formalized as CrystalRegimeCh53 in TauLib.

Categorical invariant. Many-body T² defect-bundle configuration with μ ≈ 0 (locked translational mode); periodic invariant under a T²-subgroup.

Primary registry anchor: IV.D235

Supporting items: IV.T237, IV.P254, IV.P233, IV.D391

1. I.K0 — Universe Postulate
2. IV.D12 — Particle Kind on T² fiber
3. IV.P229 — Ten regime instantiations of many-body matter
4. IV.P231 — Type-I/II classification from defect-tuple inequality
5. IV.D235 — Crystal regime: μ ≈ 0 (locked)
6. IV.P254 — Bravais lattices from T² subgroups
7. IV.T237 — Bloch's theorem on T² (band structure)
8. IV.P233 — Decidability of crystal and glass classification

Lean modules referenced: TauLib.BookIV.ManyBody.FluidRegimes

Calibration anchor: PG-P01-neutron

Calibration chain:

1. T²-subgroup gives lattice symmetry
2. Lattice constant a from minimal-bundle pair correlator
3. Bloch bands from Schrödinger on the T²-quotient
4. Atomic spacing recovered via m_n calibration cascade

Manuscript reference: manuscript-sources/book-04/part07/ch62-crystals-glass-phases.tex

Status: Formalized

Module: TauLib.BookIV.ManyBody.FluidRegimes

Lean kind: structure

Lean symbol: CrystalRegimeCh53

Related glossary entries

• PG-P01-neutron τ-Neutron
• PG-Q01-mass Mass
• PG-O07-finite-stage-strong-vacuum Finite-Stage Strong Vacuum

Referenced by

• PG-O09-macroscopic-compression Macroscopic Compression
• PG-O13-macro-tau-crystal Macro τ-Crystal