bcc_max_8tick_coherence
BCC crystal structure receives the highest 8-tick coherence value of 1.0 while FCC and HCP receive 2/3. Materials physicists modeling metal stability under Recognition Science cite this to explain BCC preference in alkali metals and high-temperature iron. The proof is a term-mode reduction that unfolds the coherence definition and verifies the numeric inequalities by normalization.
claim$C(BCC) > C(FCC) ∧ C(BCC) > C(HCP)$, where $C$ maps BCC to 1 and both FCC and HCP to $2/3$.
background
The CrystalStructure module derives lattice types from Recognition Science ledger periodicity. The eightTickCoherence function assigns scores based on coordination match to the 8-tick octave: BCC scores 1.0 for exact alignment with coordination 8, while FCC and HCP score 2/3 because their 12-fold coordination yields an 8/12 ratio. This rests on the tick as the fundamental time quantum and the eight-tick period as the basic evolution cycle. Upstream results include the structure of nuclear densities and photon fluxes from NucleosynthesisTiers, which places physical quantities on discrete φ-tiers, and the LedgerFactorization structure that calibrates the J-cost function underlying the ledger.
proof idea
The proof applies simp to substitute the numeric values from the eightTickCoherence definition, then uses constructor to split the conjunction and norm_num to confirm both inequalities 1 > 2/3 hold.
why it matters in Recognition Science
This result confirms the module prediction that BCC coordination equals 8 ticks and is favored when 8-tick coherence dominates. It directly supports the framework landmark T7 eight-tick octave and the energy-ordering statements in the crystal stability section. No downstream theorems are recorded, leaving open how the coherence scores combine with packing efficiency to produce quantitative cohesive energies.
scope and limits
- Does not calculate lattice energies or cohesive energies.
- Does not model temperature or pressure effects on phase stability.
- Does not compare coherence scores to experimental diffraction data.
- Does not treat defects, impurities, or non-ideal lattices.
formal statement (Lean)
146theorem bcc_max_8tick_coherence :
147 eightTickCoherence .BCC > eightTickCoherence .FCC ∧
148 eightTickCoherence .BCC > eightTickCoherence .HCP := by
proof body
Term-mode proof.
149 simp only [eightTickCoherence]
150 constructor <;> norm_num
151
152/-- Stability is a balance of packing and 8-tick coherence. -/