module module high

`IndisputableMonolith.Chemistry.CrystalStructure`

CrystalStructure module classifies periodic atomic arrangements in three dimensions under Recognition Science rules. It supplies definitions for coordination numbers, packing efficiencies, and relations such as BCC to the eight-tick octave and HCP ratios near phi. Condensed-matter researchers cite it when connecting lattice geometry to the phi-ladder and RCL. The module is a collection of definitions and short lemmas with no central theorem.

claimCrystal structures $S$ equipped with coordination number $z(S)$ and packing efficiency $\eta(S)$ in $D=3$ space, satisfying relations derived from the Recognition Composition Law and the eight-tick periodicity.

background

The module sits in the Chemistry domain and imports the RS time quantum $\tau_0 = 1$ tick from Constants. It introduces Structure as the type of periodic unit cells, coordination as nearest-neighbor count, and packingEfficiency as occupied volume fraction. Sibling declarations include bcc_is_8_tick, close_packed_coordination, bcc_packing_lt_fcc, fcc_hcp_same_packing, idealHCPRatio, hcp_ratio_near_phi, and energyScale, all expressed in RS-native units.

proof idea

this is a definition module, no proofs

why it matters in Recognition Science

The module supplies the structural primitives required by the downstream CrystalSymmetry derivation (CM-003), which obtains the 32 point groups and 7 crystal systems from periodic filling of 3D space. It thereby connects the T0-T8 forcing chain and D=3 result to concrete lattice types.

scope and limits

Does not derive the 32 crystallographic point groups.
Does not compute exact numerical packing fractions beyond the listed approximations.
Does not treat non-periodic or quasicrystalline arrangements.
Does not connect structures to mass formulas or the alpha band.

used by (1)

From the project-wide theorem graph. These declarations reference this one in their body.

IndisputableMonolith.Chemistry.CrystalSymmetry module_import

depends on (1)

Lean names referenced from this declaration's body.

IndisputableMonolith.Constants module_import

declarations in this module (21)

inductive Structure L41
def coordination L48
def packingEfficiency L54
def packingEfficiencyApprox L60
theorem bcc_is_8_tick L66
theorem close_packed_coordination L69
theorem bcc_packing_lt_fcc L73
theorem fcc_hcp_same_packing L78
def idealHCPRatio L83
theorem ideal_hcp_ratio_value L86
theorem hcp_ratio_near_phi L104
def energyScale L124
theorem close_packed_lower_energy L130
def eightTickCoherence L140
theorem bcc_max_8tick_coherence L146
def stabilityScore L153
theorem stability_tradeoff L159
def prefersBCC L174
def prefersFCC L177
def prefersHCP L180
theorem alkali_prefer_bcc L183