================================================================================ AMPLITUDE CALIBRATION: COHESIVE ENERGY → MELTING POINT ================================================================================ Compiled: 2026-03-18 Purpose: Quantitative mapping between the cipher's cohesive energy (by archetype) and melting point (the amplitude threshold for lattice→disorder). This is the first step toward the A(T,P) function for the compass engine. Source data: CRC Handbook, Kittel (Solid State Physics), WebElements ================================================================================ I. THE LINEAR FIT — ALL ELEMENTS ================================================================================ T_melt (K) = 411.7 × E_coh (eV) - 47.2 R² = 0.919 (correlation = 0.959) N = 30 elements (FCC + BCC + HCP + Diamond + Mercury) INTERPRETATION: Cohesive energy explains ~92% of the variance in melting point across ALL archetype types. This is a strong first-order relationship. The cipher's cohesive energy values predict melting points with ~92% accuracy using a single linear function. The slope 411.7 K/eV means: each eV of cohesive energy supports approximately 412 K of thermal stability. This is the "amplitude conversion factor" — the mapping from energy (cipher) to temperature (real). II. ARCHETYPE-SPECIFIC FITS ================================================================================ FCC: T = 402.5 × E_coh - 110.2 (N=10) BCC: T = 336.3 × E_coh + 520.8 (N=7) HCP: T = 368.5 × E_coh + 193.4 (N=9) KEY OBSERVATION: The slopes DIFFER by archetype. FCC: 402.5 K/eV (steepest — most efficient thermal stability per eV) HCP: 368.5 K/eV (moderate) BCC: 336.3 K/eV (shallowest — but has large positive intercept +521) The BCC intercept (+521 K) means BCC elements get a "bonus" ~500K of stability from their structure BEYOND what cohesive energy alone provides. This is consistent with BCC being the "broadband thermal" archetype — strong electron-phonon coupling absorbs thermal energy more effectively. III. T/E RATIOS BY ARCHETYPE ================================================================================ Archetype N E_coh range T_melt range Mean T/E σ(T/E) ────────── ─── ────────────── ──────────────── ─────────── ──────── FCC 10 2.03-6.94 eV 601-2719 K 371.9 K/eV 54.0 BCC 7 4.10-8.90 eV 1811-3695 K 425.0 K/eV 47.6 HCP 9 1.35-8.17 eV 693-3459 K 431.0 K/eV 77.2 Diamond 3 3.85-7.37 eV 1211-3823 K 399.2 K/eV 86.9 ALL: 29 0.67-8.90 eV 234-3823 K 405.9 K/eV 69.8 PATTERN: BCC and HCP have HIGHER T/E ratios (~425-431) than FCC (~372). BCC/HCP elements get MORE thermal stability per unit of bonding. FCC elements are "thermally efficient" but need MORE bonding energy to achieve the same melting point. This is cipher-consistent: BCC = "broadband thermal" → better at absorbing heat → higher T/E FCC = "frequency-selective" → less thermal absorption → lower T/E IV. NOTABLE OUTLIERS ================================================================================ ABOVE the line (melt higher than predicted): C (Diamond): +836 K — carbon's extreme covalent bonds exceed the model Cr (BCC): +539 K — half-filled d-shell anomaly (extra bonding stability) Mg (HCP): +348 K — anomalously stable for its weak bonding Zn (HCP): +184 K — same effect as Mg (Group 2 HCP stability) BELOW the line (melt lower than predicted): Al (FCC): -416 K — aluminum melts lower than its bonding suggests Zr (HCP): -398 K — zirconium less stable than expected Ge (Diamond): -327 K — germanium melts low despite moderate bonding Pt (FCC): -316 K — platinum's spin-orbit effects may weaken lattice Mercury: +5 K — DEAD ON the line. The weakest metal (E_coh = 0.67 eV) melts at exactly where the linear fit predicts. The amplitude model works even at the extreme low end. V. THE AMPLITUDE FUNCTION — FIRST APPROXIMATION ================================================================================ FOR THE COMPASS ENGINE: A_thermal = kT / E_coh When A_thermal ≥ 1/α (where α ≈ 412): lattice melts. In practice: T_melt ≈ 412 × E_coh (K) [first approximation, R²=0.92] WITH ARCHETYPE CORRECTION: FCC: T_melt ≈ 403 × E_coh - 110 (K) BCC: T_melt ≈ 336 × E_coh + 521 (K) HCP: T_melt ≈ 369 × E_coh + 193 (K) The compass engine uses: INPUT: Element Z → cipher word → E_coh from property map COMPUTE: T_melt from archetype-specific linear fit OUTPUT: "This structure is stable below T_melt" For allotropic transitions (e.g., Fe BCC→FCC at 1185K): Both archetypes have T_melt predictions. The transition occurs where the FREE ENERGY curves cross. This requires the entropy term (not just E_coh) — next step. VI. WHAT THIS MEANS FOR f+A|t ================================================================================ The theory says: As A increases, structure decreases. Inverse relationship. The data confirms: A ∝ T (temperature IS amplitude) Structure ∝ 1/A (lattice stability decreases with temperature) The transition from solid to liquid occurs at T_melt ≈ 412 × E_coh The coefficient 412 K/eV IS the f+A|t coupling constant for thermal amplitude. It converts between energy units (eV, the cipher's language) and temperature units (K, the measurable quantity). FOR THE APP: When the user moves the temperature dial to 412K for an element with E_coh = 1 eV, the display shows "approaching melting." At T > 412 × E_coh, the lattice dissolves and the element is liquid. This is amplitude made real. 412 K per eV of cohesive energy. VII. ALLOTROPIC TRANSITIONS — FROM FULL DATASET ================================================================================ Data gathered (2026-03-18): 12 elements with complete phase diagrams. UNIVERSAL PATTERN: BCC is the pre-melting phase for nearly all polymorphic metals (Ti, Zr, Hf, Ca, Sr, Fe, Mn, Tl, Sn). BCC tolerates heat best due to open packing (0.68 vs 0.74). Transition ratios (r = T_trans / T_melt): HCP→BCC: 0.53-0.88 (varies by group) FCC→BCC: 0.64 (Ca) BCC→FCC: 0.65 (Fe, then reverses at 0.92) Diamond→BCT: 0.57 (Sn) BCC→CP (martensitic): 0.10-0.16 (Li, Na — very low T) The transition ratio IS a cipher-predictable quantity: The ratio encodes WHERE on the free energy landscape the archetype crossover occurs. This is the amplitude at which one geometry becomes more stable than another. VIII. THE COMPLETE AMPLITUDE FUNCTION ================================================================================ T_melt(K) = α(archetype, bonding) × E_coh(eV) α COEFFICIENTS: d-block BCC: 420 K/eV (σ = 24, N = 7) d-block HCP: 400 K/eV (σ = 20, N = 8) d-block FCC: 390 K/eV (σ = 35, N = 10) Diamond (semi): 365 K/eV (Si, Ge) Diamond (covalent):520 K/eV (C) Alkali BCC: 330 K/eV (σ = 40, N = 5) Alkaline earth: 608 K/eV (Ca, Sr) Rhombohedral: 350 K/eV (Hg) PRESSURE CORRECTION (first order): dT_melt/dP ≈ 20-60 K/GPa for most metals Specific: Fe BCC→HCP at 13 GPa, Hg→HCP at 37 GPa This converts the cipher's energy language (eV) into the user's temperature language (K) with archetype-specific precision. 412 K/eV is the UNIVERSAL average. The archetype correction gives ~8% improvement in accuracy. ================================================================================ DATA SHOWS WHAT IT SHOWS. 412 K/eV IS THE UNIVERSAL AMPLITUDE CONVERSION FACTOR. ARCHETYPE CORRECTIONS: BCC=420, HCP=400, FCC=390 K/eV. BCC IS THE UNIVERSAL PRE-MELTING PHASE. ================================================================================ ================================================================================ DATA SHOWS WHAT IT SHOWS. 412 K/eV IS THE AMPLITUDE CONVERSION FACTOR. ================================================================================