IndisputableMonolith.Materials.Crystal_Structure2_FromConfigDim
Module linking RS configuration dimension to 3D crystal structure: it defines a domain cost from the RS cost functional, a positive canonical threshold, and an inhabited Bravais-3 certificate. Materials or lattice theorists cite Bravais3Cert and the threshold lemmas when connecting forced D=3 to admissible crystals. Argument is definitional plus nonnegativity/positivity lemmas on the cost side.
claimThe module defines a domain cost $C$ on configuration data, a canonical threshold $\theta>0$, and a certificate that admissible 3D Bravais lattices arise when configuration dimension is compatible with the RS forcing $D=3$ and the cost lies below $\theta$.
background
Recognition Science forces spatial dimension $D=3$ at T8 of the unified forcing chain (after the eight-tick octave at T7 and $\varphi$ at T6). Condensed-matter applications need that geometric fact packaged as a selectable crystal constraint.
This Materials module imports RS Constants (time quantum $\tau_0$) and Cost (the $J$-cost lineage from the Recognition Composition Law). It introduces a domain cost on configuration spaces, records its pointwise evaluation identity and nonnegativity, defines a canonical threshold and proves it positive, then packages a Bravais-3 certificate with an inhabited witness.
Local setting: configuration dimension plus cost threshold as the filter that retains ordinary 3D Bravais structure and excludes incompatible lattices.
proof idea
Definition module with short supporting lemmas. domainCost is defined from the imported Cost layer; domainCost_at_eq is an evaluation identity; domainCost_nonneg is nonnegativity. canonicalThreshold is defined and canonicalThreshold_pos proves positivity. Bravais3Cert is the certificate type; cert and cert_inhabited supply a concrete inhabited instance. No multi-step tactic developments beyond those positivity/nonnegativity facts.
why it matters in Recognition Science
Places crystal structure inside the RS materials layer by tying Bravais lattices to forced $D=3$ (T8) and the $J$-cost lineage. Export surface is Bravais3Cert together with the domain-cost and threshold lemmas. No downstream uses are recorded yet, so the module is a leaf certificate: later lattice, phonon, or condensed-matter developments can assume the inhabited 3D Bravais cert rather than re-deriving dimension compatibility.
scope and limits
- Does not classify all 14 Bravais types from first principles.
- Does not compute lattice constants, elastic moduli, or spectra.
- Does not treat quasicrystals or dimensions other than 3.
- Does not claim uniqueness of domainCost among all cost functionals.
- Does not connect to experimental crystallography databases.