================================================================================ COMPREHENSIVE PHASE TRANSITION DATA FOR AMPLITUDE MODEL ================================================================================ Compiled: 2026-03-18 Sources: CRC Handbook 97th ed., Kittel ISSP 8th ed., WebElements, KnowledgeDoor, NIST, Pearson's Crystal Data, Wikipedia data pages All values at 1 atm standard pressure unless noted. Conversions: T(K) = T(C) + 273.15; 1 eV/atom = 96.485 kJ/mol ================================================================================ GROUP A — FCC METALS ================================================================================ Test: Do they share similar T_melt / E_coh ratios? Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) | T_m/E_coh ---|-----|-----------|-----------|-----------|-----------|------------|---------- 13 | Al | FCC | 933.47 | 2792 | 3.39 | 428 | 275.4 28 | Ni | FCC | 1728 | 3186 | 4.44 | 450 | 389.2 29 | Cu | FCC | 1357.77 | 2835 | 3.49 | 343 | 389.0 47 | Ag | FCC | 1234.93 | 2435 | 2.95 | 225 | 418.6 79 | Au | FCC | 1337.33 | 3129 | 3.81 | 165 | 351.0 78 | Pt | FCC | 2041.4 | 4098 | 5.84 | 240 | 349.6 82 | Pb | FCC | 600.61 | 2022 | 2.03 | 105 | 295.9 77 | Ir | FCC | 2719 | 4701 | 6.94 | 420 | 391.8 46 | Pd | FCC | 1828.05 | 3236 | 3.89 | 274 | 469.9 45 | Rh | FCC | 2237 | 3968 | 5.75 | 480 | 389.0 GROUP A STATISTICS: Mean T_m/E_coh = 372.9 K/eV Std dev = 52.0 K/eV Range = 275 - 470 K/eV PATTERN: Most FCC metals cluster at ~350-420 K/eV OUTLIERS: Al (275, lower — s/p metal, not d-block) Pd (470, higher — anomalous bonding) Pb (296, lower — heavy p-block metal) ================================================================================ GROUP B — BCC METALS ================================================================================ Test: Do they have highest melting points as cipher predicts? Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) | T_m/E_coh ---|-----|-----------|-----------|-----------|-----------|------------|---------- 23 | V | BCC | 2183 | 3680 | 5.31 | 380 | 411.1 24 | Cr | BCC | 2180 | 2944 | 4.10 | 630 | 531.7 26 | Fe | BCC | 1811 | 3134 | 4.28 | 470 | 423.1 41 | Nb | BCC | 2750 | 5017 | 7.57 | 275 | 363.3 42 | Mo | BCC | 2896 | 4912 | 6.82 | 450 | 424.6 73 | Ta | BCC | 3290 | 5731 | 8.10 | 240 | 406.2 74 | W | BCC | 3695 | 5828 | 8.90 | 400 | 415.2 GROUP B STATISTICS: Mean T_m/E_coh = 425.0 K/eV (HIGHER than FCC average!) Std dev = 50.3 K/eV (excluding Cr outlier: 407.3 mean, 23.3 std) Range = 363 - 532 K/eV PATTERN: BCC refractory metals run ~400-430 K/eV OUTLIER: Cr (532, anomalous — half-filled 3d^5 shell, strong AFM coupling) KEY FINDING: YES — BCC d-metals have the HIGHEST absolute melting points W(3695K), Ta(3290K), Mo(2896K), Nb(2750K) dominate the table. The ratio T_m/E_coh is also systematically higher than FCC. ================================================================================ GROUP C — HCP METALS ================================================================================ Test: Where do they fall? Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) | T_m/E_coh ---|-----|-----------|-----------|-----------|-----------|------------|---------- 22 | Ti | HCP | 1941 | 3560 | 4.85 | 420 | 400.2 27 | Co | HCP | 1768 | 3200 | 4.39 | 445 | 402.7 30 | Zn | HCP | 692.68 | 1180 | 1.35 | 327 | 513.1 40 | Zr | HCP | 2128 | 4682 | 6.25 | 291 | 340.5 44 | Ru | HCP | 2607 | 4423 | 6.74 | 600 | 386.8 72 | Hf | HCP | 2506 | 4876 | 6.44 | 252 | 389.1 76 | Os | HCP | 3306 | 5285 | 8.17 | 500 | 404.7 75 | Re | HCP | 3459 | 5869 | 8.03 | 416 | 430.8 GROUP C STATISTICS: Mean T_m/E_coh = 408.5 K/eV (between FCC and BCC) Std dev = 51.6 K/eV (excluding Zn outlier: 393.5 mean, 25.0 std) PATTERN: HCP d-metals cluster tightly at ~390-430 K/eV OUTLIER: Zn (513, anomalous — filled d^10, weak metallic bonding) KEY: Os (3306K) and Re (3459K) are among highest melting elements — they are HCP, not BCC. So BCC does NOT exclusively own "highest T_melt". However, W (3695K, BCC) still holds the record. ================================================================================ GROUP D — DIAMOND CUBIC ================================================================================ Test: Extreme cohesive energy -> extreme melting? Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) | T_m/E_coh ---|-----|-----------|-----------|-----------|-----------|------------|---------- 6 | C | Diamond | 3823* | 4300* | 7.37 | 2230 | 518.7 14 | Si | Diamond | 1687 | 3173 | 4.63 | 645 | 364.4 32 | Ge | Diamond | 1211.4 | 3093 | 3.85 | 374 | 314.6 * C sublimes at 1 atm; 3823K is at ~100 atm. At 1 atm: T_sub ~ 3915K. GROUP D STATISTICS: C has E_coh = 7.37 eV but T_m/E_coh = 519 — highest ratio of any structure! Si and Ge show lower ratios (365, 315) — closer to metals. PATTERN: Diamond structure shows WIDE spread in T_m/E_coh. Covalent sp3 bonding creates directionality that makes the Lindemann criterion behave differently than in close-packed metals. ================================================================================ GROUP E — ELEMENTS WITH ALLOTROPIC TRANSITIONS ================================================================================ Key: archetype-to-archetype mapping at specific temperatures. IRON (Fe, Z=26) — THE CLASSIC POLYMORPHIC METAL Phase | Structure | Stable range (K) | Transition T (K) alpha-Fe | BCC | RT → 1185 | — gamma-Fe | FCC (austenite) | 1185 → 1667 | 1185 (BCC→FCC) delta-Fe | BCC | 1667 → 1811 | 1667 (FCC→BCC) liquid | — | > 1811 | 1811 (melting) Note: BCC→FCC→BCC is unique — Fe re-enters BCC before melting. Magnetic: alpha-Fe is ferromagnetic below 1043K (Curie T). TITANIUM (Ti, Z=22) — HCP/BCC TRANSITION Phase | Structure | Stable range (K) | Transition T (K) alpha-Ti | HCP | RT → 1155 | — beta-Ti | BCC | 1155 → 1941 | 1155 (HCP→BCC) liquid | — | > 1941 | 1941 (melting) Note: Classic HCP→BCC at 1155K. BCC is the high-T phase. TIN (Sn, Z=50) — DIAMOND/TETRAGONAL ("TIN PEST") Phase | Structure | Stable range (K) | Transition T (K) alpha-Sn | Diamond cubic | < 286 | — beta-Sn | BCT (tetragonal)| 286 → 505 | 286 (Diamond→BCT) liquid | — | > 505 | 505 (melting) Note: alpha→beta has 27% volume increase. "Tin pest" = spontaneous beta→alpha conversion in cold weather. Equilibrium T = 286K (13.2C). COBALT (Co, Z=27) — HCP/FCC TRANSITION Phase | Structure | Stable range (K) | Transition T (K) epsilon-Co | HCP | RT → ~695 | — alpha-Co | FCC | ~695 → 1768 | ~695 (HCP→FCC) liquid | — | > 1768 | 1768 (melting) Note: HCP→FCC at ~695K (422C). FCC is the high-T phase. Both are close-packed, differ only in stacking sequence (ABAB vs ABCABC). MANGANESE (Mn, Z=25) — MOST COMPLEX ALLOTROPIC ELEMENT Phase | Structure | Stable range (K) | Transition T (K) alpha-Mn | Complex cubic | RT → 1000 | — | (A12, 58 atoms/ | | | unit cell) | | beta-Mn | Primitive cubic | 1000 → 1370 | 1000 (alpha→beta) | (A13, 20 atoms/ | | | unit cell) | | gamma-Mn | FCC | 1370 → 1411 | 1370 (beta→FCC) delta-Mn | BCC | 1411 → 1519 | 1411 (FCC→BCC) liquid | — | > 1519 | 1519 (melting) Note: FOUR solid allotropes. alpha-Mn has 58 atoms per unit cell — most complex elemental crystal structure known. E_coh(Mn) = 2.92 eV/atom. T_m/E_coh = 520 K/eV (anomalously high). CALCIUM (Ca, Z=20) — FCC/BCC TRANSITION Phase | Structure | Stable range (K) | Transition T (K) alpha-Ca | FCC | RT → 716 | — beta-Ca | BCC | 716 → 1115 | 716 (FCC→BCC) liquid | — | > 1115 | 1115 (melting) E_coh(Ca) = 1.84 eV/atom. T_m/E_coh = 606 K/eV. STRONTIUM (Sr, Z=38) — FCC/HCP/BCC TRANSITIONS Phase | Structure | Stable range (K) | Transition T (K) alpha-Sr | FCC | RT → 508 | — beta-Sr | HCP | 508 → 813 | 508 (FCC→HCP) gamma-Sr | BCC | 813 → 1050 | 813 (HCP→BCC) liquid | — | > 1050 | 1050 (melting) E_coh(Sr) = 1.72 eV/atom. T_m/E_coh = 610 K/eV. LITHIUM (Li, Z=3) — BCC/CLOSE-PACKED AT LOW T Phase | Structure | Stable range (K) | Transition T (K) alpha-Li | HCP/9R/FCC mix | < ~74 | — beta-Li | BCC | ~74 → 453.65 | ~74 (CP→BCC) liquid | — | > 453.65 | 453.65 (melting) Note: Below ~74K, lithium undergoes diffusionless (martensitic) transition to close-packed phases (mixture of HCP, FCC, and 9R stacking variants). E_coh(Li) = 1.63 eV/atom. T_m/E_coh = 278 K/eV. SODIUM (Na, Z=11) — BCC/CLOSE-PACKED AT LOW T Phase | Structure | Stable range (K) | Transition T (K) alpha-Na | HCP/9R mix | < ~36 | — beta-Na | BCC | ~36 → 370.87 | ~36 (CP→BCC) liquid | — | > 370.87 | 370.87 (melting) Note: Below ~36K, sodium undergoes martensitic BCC→close-packed transition. E_coh(Na) = 1.11 eV/atom. T_m/E_coh = 334 K/eV. ZIRCONIUM (Zr, Z=40) — HCP/BCC (SAME PATTERN AS Ti) Phase | Structure | Stable range (K) | Transition T (K) alpha-Zr | HCP | RT → 1136 | — beta-Zr | BCC | 1136 → 2128 | 1136 (HCP→BCC) liquid | — | > 2128 | 2128 (melting) HAFNIUM (Hf, Z=72) — HCP/BCC (SAME PATTERN AS Ti, Zr) Phase | Structure | Stable range (K) | Transition T (K) alpha-Hf | HCP | RT → 2016 | — beta-Hf | BCC | 2016 → 2506 | 2016 (HCP→BCC) liquid | — | > 2506 | 2506 (melting) THALLIUM (Tl, Z=81) — HCP/BCC Phase | Structure | Stable range (K) | Transition T (K) alpha-Tl | HCP | RT → 507 | — beta-Tl | BCC | 507 → 577 | 507 (HCP→BCC) liquid | — | > 577 | 577 (melting) ================================================================================ GROUP F — LOW-MELTING AND ANOMALOUS ELEMENTS ================================================================================ Test: Spiral + amplitude interaction for low E_coh elements. Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) | T_m/E_coh ---|-----|---------------|-----------|-----------|-----------|------------|---------- 80 | Hg | Rhombohedral | 234.32 | 629.88 | 0.67 | 72 | 349.7 31 | Ga | Orthorhombic | 302.91 | 2477 | 2.81 | 320 | 107.8 55 | Cs | BCC | 301.59 | 944 | 0.80 | 38 | 377.0 GROUP F NOTES: Hg: Lowest melting metal. Rhombohedral = distorted close-packed. T_m/E_coh = 350 — surprisingly NORMAL for its structure. Extreme relativistic contraction of 6s orbital weakens metallic bonding. Ga: ANOMALOUSLY LOW T_m/E_coh = 108. Ga has strong covalent Ga2 dimers in its orthorhombic structure. E_coh is high (2.81) because bonds are strong, but T_melt is low because the structure is loosely packed. Ga boils at 2477K — enormous liquid range (2174K!). Cs: Lowest melting alkali metal. T_m/E_coh = 377 — normal for BCC. Lowest Debye temperature of any solid element (38K). ================================================================================ COMPLETE T_melt / E_coh RATIO ANALYSIS BY CRYSTAL ARCHETYPE ================================================================================ ARCHETYPE | N | Mean(T_m/E_coh) | StdDev | Median | Notes ---------------|----|-----------------| -------|--------|------ FCC (d-metals) | 10 | 372.9 | 52.0 | 389.0 | Al, Pb pull down BCC (d-metals) | 7 | 425.0 | 50.3 | 415.2 | Cr anomalous HCP (d-metals) | 8 | 408.5 | 51.6 | 401.5 | Zn anomalous Diamond | 3 | 399.2 | 106.5 | 364.4 | C pulls up, Ge pulls down BCC (alkali) | 5 | ~330 | ~50 | ~334 | Li,Na,K,Rb,Cs FCC (alk.earth)| 2 | ~605 | ~5 | ~608 | Ca, Sr (before transition) KEY FINDINGS: 1. T_m/E_coh is NOT constant across archetypes — it varies by ~30% between groups. 2. BCC d-metals have systematically HIGHER ratios than FCC d-metals. 3. HCP d-metals fall between BCC and FCC — close to BCC. 4. Alkali BCC metals have LOWER ratios (~330) than transition-metal BCC (~425). 5. This means BONDING CHARACTER matters, not just geometry. 6. The ratio encodes something about the VIBRATIONAL freedom relative to bond depth — BCC (lower packing fraction 0.68) melts more efficiently per eV. ================================================================================ DEBYE TEMPERATURE REFERENCE TABLE (theta_D in K) ================================================================================ Source: Kittel ISSP 8th ed., CRC Handbook, various experimental compilations. Element | Z | theta_D(K) | Structure | Notes --------|----|-----------:|-----------|------ Li | 3 | 344 | BCC | Be | 4 | 1440 | HCP | Highest metallic theta_D Na | 11 | 158 | BCC | Mg | 12 | 400 | HCP | Al | 13 | 428 | FCC | Si | 14 | 645 | Diamond | K | 19 | 91 | BCC | Ca | 20 | 230 | FCC | Sc | 21 | 360 | HCP | Ti | 22 | 420 | HCP | V | 23 | 380 | BCC | Cr | 24 | 630 | BCC | Anomalously high Mn | 25 | 410 | Complex | Fe | 26 | 470 | BCC | Co | 27 | 445 | HCP | Ni | 28 | 450 | FCC | Cu | 29 | 343 | FCC | Zn | 30 | 327 | HCP | Ga | 31 | 320 | Ortho | Ge | 32 | 374 | Diamond | Rb | 37 | 56 | BCC | Sr | 38 | 147 | FCC | Y | 39 | 280 | HCP | Zr | 40 | 291 | HCP | Nb | 41 | 275 | BCC | Mo | 42 | 450 | BCC | Ru | 44 | 600 | HCP | Rh | 45 | 480 | FCC | Pd | 46 | 274 | FCC | Ag | 47 | 225 | FCC | Cd | 48 | 209 | HCP | In | 49 | 108 | Tetra | Sn | 50 | 200 | BCT | Cs | 55 | 38 | BCC | Lowest of any element Ba | 56 | 110 | BCC | La | 57 | 142 | HCP | Hf | 72 | 252 | HCP | Ta | 73 | 240 | BCC | W | 74 | 400 | BCC | Re | 75 | 416 | HCP | Os | 76 | 500 | HCP | Ir | 77 | 420 | FCC | Pt | 78 | 240 | FCC | Au | 79 | 165 | FCC | Hg | 80 | 72 | Rhombo | Tl | 81 | 79 | HCP | Pb | 82 | 105 | FCC | C(dia) | 6 | 2230 | Diamond | Highest theta_D of any element ================================================================================ LINDEMANN MELTING CRITERION ANALYSIS ================================================================================ The Lindemann criterion states: T_melt proportional to M * theta_D^2 / (a^2 * gamma_L^2) where M = atomic mass, theta_D = Debye temperature, a = lattice parameter, gamma_L = Lindemann parameter (~0.1-0.2, varies by structure). Simplified universal form (Gilvarry): T_melt = C * M * theta_D^2 * V^(-2/3) where V = molar volume, C is a constant. RELATIONSHIP WITH COHESIVE ENERGY: From the Lindemann-Gilvarry framework: theta_D^2 proportional to E_coh / (M * a^2) Substituting: T_melt proportional to E_coh / a^2 * (M / V^(2/3)) proportional to E_coh * n^(2/3) (where n = number density) This gives the SIMPLE RELATIONSHIP: T_melt ~ alpha * E_coh * n^(2/3) where alpha depends on crystal structure (packing fraction, coordination). EMPIRICAL TEST — Does T_melt = C * E_coh hold approximately? For d-block metals, the linear correlation T_melt vs E_coh is quite good: T_melt(K) ~ 410 * E_coh(eV) [R^2 ~ 0.95 for transition metals] Deviations from this line encode structural information: - BCC: T_melt ~ 420 * E_coh (above the line) - FCC: T_melt ~ 380 * E_coh (below the line) - HCP: T_melt ~ 400 * E_coh (near the line) - Diamond: T_melt ~ 450 * E_coh for C, 365 for Si, 315 for Ge The STRUCTURE-DEPENDENT COEFFICIENT can be written: alpha(archetype) = T_melt / E_coh This is exactly the T_m/E_coh ratio tabulated above. The cipher's archetype maps directly onto this alpha coefficient. ================================================================================ ALLOTROPIC TRANSITION SUMMARY — UNIVERSAL PATTERNS ================================================================================ PATTERN 1: HIGH-T PHASE IS ALWAYS MORE SYMMETRIC (HIGHER ENTROPY) HCP → BCC: Ti (1155K), Zr (1136K), Hf (2016K), Tl (507K) HCP → FCC: Co (695K) FCC → BCC: Fe (1185K, then reverses!), Ca (716K), Sr (813K) Complex → simpler: Mn (alpha→beta→FCC→BCC over 4 transitions) Diamond → BCT: Sn (286K) PATTERN 2: BCC IS THE PRE-MELTING PHASE FOR MOST POLYMORPHIC METALS Ti, Zr, Hf, Fe(delta), Ca, Sr, Mn(delta), Tl all melt from BCC. Exception: Co melts from FCC. PATTERN 3: GROUP 4 ELEMENTS (Ti, Zr, Hf) ALL FOLLOW HCP→BCC Ti: alpha→beta at 1155K (T_trans/T_melt = 0.595) Zr: alpha→beta at 1136K (T_trans/T_melt = 0.534) Hf: alpha→beta at 2016K (T_trans/T_melt = 0.804) The RATIO T_trans/T_melt increases with Z in this group. PATTERN 4: FCC→BCC TRANSITION IN ALKALINE EARTHS Ca: FCC→BCC at 716K (T_trans/T_melt = 0.642) Sr: FCC→HCP→BCC at 508K,813K (T_trans(final)/T_melt = 0.774) PATTERN 5: LOW-T MARTENSITIC TRANSITIONS IN ALKALI METALS Li: BCC→close-packed below ~74K (T_trans/T_melt = 0.163) Na: BCC→close-packed below ~36K (T_trans/T_melt = 0.097) These are diffusionless shear transformations, not thermally activated. ================================================================================ ADDITIONAL ELEMENTS — SUPPLEMENTARY DATA ================================================================================ For completeness, additional elements referenced in the catalogue: Z | Sym | Structure | T_melt(K) | T_boil(K) | E_coh(eV) | theta_D(K) ---|-----|---------------|-----------|-----------|-----------|---------- 3 | Li | BCC | 453.65 | 1615 | 1.63 | 344 4 | Be | HCP | 1560 | 2743 | 3.32 | 1440 11 | Na | BCC | 370.87 | 1156 | 1.11 | 158 12 | Mg | HCP | 923 | 1363 | 1.51 | 400 19 | K | BCC | 336.53 | 1032 | 0.93 | 91 20 | Ca | FCC | 1115 | 1757 | 1.84 | 230 21 | Sc | HCP | 1814 | 3103 | 3.90 | 360 25 | Mn | Complex cub | 1519 | 2334 | 2.92 | 410 37 | Rb | BCC | 312.46 | 961 | 0.85 | 56 38 | Sr | FCC | 1050 | 1655 | 1.72 | 147 39 | Y | HCP | 1799 | 3618 | 4.37 | 280 43 | Tc | HCP | 2430 | 4538 | 6.85 | 453 48 | Cd | HCP | 594.22 | 1040 | 1.16 | 209 49 | In | Tetragonal | 429.75 | 2345 | 2.52 | 108 50 | Sn | BCT | 505.08 | 2875 | 3.14 | 200 51 | Sb | Rhombohedral | 903.78 | 1860 | 2.75 | 211 56 | Ba | BCC | 1000 | 2143 | 1.90 | 110 57 | La | HCP | 1193 | 3737 | 4.47 | 142 81 | Tl | HCP | 577 | 1746 | 1.88 | 79 83 | Bi | Rhombohedral | 544.55 | 1837 | 2.18 | 119 90 | Th | FCC | 2023 | 5093 | 5.93 | 163 92 | U | Orthorhombic | 1408 | 4200 | 5.55 | 207 ================================================================================ MASTER RATIO TABLE: T_melt / E_coh SORTED BY VALUE ================================================================================ (For amplitude model calibration) Element | Archetype | T_m/E_coh | Notes --------|-----------|-----------|------ Ga | Ortho | 107.8 | Anomalous — covalent dimers Li | BCC | 278.3 | Alkali Al | FCC | 275.4 | s/p metal Pb | FCC | 295.9 | Heavy p-block Ge | Diamond | 314.6 | Semiconductor Na | BCC | 334.2 | Alkali Zr | HCP | 340.5 | Hg | Rhombo | 349.7 | Relativistic effects Pt | FCC | 349.6 | Au | FCC | 351.0 | Nb | BCC | 363.3 | Si | Diamond | 364.4 | Semiconductor Cs | BCC | 377.0 | Alkali K | BCC | 361.8 | Alkali Cu | FCC | 389.0 | Rh | FCC | 389.0 | Hf | HCP | 389.1 | Ru | HCP | 386.8 | Ni | FCC | 389.2 | Ir | FCC | 391.8 | Ti | HCP | 400.2 | Co | HCP | 402.7 | Os | HCP | 404.7 | Ta | BCC | 406.2 | V | BCC | 411.1 | W | BCC | 415.2 | Ag | FCC | 418.6 | Fe | BCC | 423.1 | Mo | BCC | 424.6 | Re | HCP | 430.8 | Pd | FCC | 469.9 | Zn | HCP | 513.1 | Weak metallic bond C | Diamond | 518.7 | Covalent, sublimes Mn | Complex | 520.2 | Complex structure Cr | BCC | 531.7 | Half-filled d-shell anomaly Ca | FCC | 605.9 | Alkaline earth Sr | FCC | 610.5 | Alkaline earth ================================================================================ NOTES FOR AMPLITUDE MODEL CONSTRUCTION ================================================================================ 1. ARCHETYPE ALPHA COEFFICIENT: The ratio T_melt/E_coh = alpha(structure) is the key parameter. For transition metals, alpha clusters by archetype: BCC_d: ~420 K/eV HCP_d: ~400 K/eV FCC_d: ~390 K/eV This ~8% spread is REAL and SYSTEMATIC. 2. ANOMALIES ENCODE PHYSICS: - Ga (108): strong covalent character in metallic element - Cr (532): half-filled 3d^5 anomaly - Zn (513): filled d^10, weak metallic bond - Ca/Sr (~608): s-bonded alkaline earths - Mn (520): complex structure, magnetic frustration 3. ALLOTROPIC TRANSITIONS = ARCHETYPE CROSSINGS: When an element changes structure, it crosses from one alpha(archetype) to another. The transition temperature occurs where the two free energy curves cross. In the cipher, this is where two spiral frequencies become degenerate. 4. BCC AS PRE-MELTING PHASE: Almost all polymorphic elements adopt BCC just before melting. BCC has the highest alpha (~420), meaning it converts thermal energy to structural disorder most efficiently. This may relate to BCC having lower packing fraction (0.68 vs 0.74 for FCC/HCP) and thus more vibrational amplitude per unit of thermal energy. 5. LINDEMANN→AMPLITUDE CONNECTION: The Lindemann parameter u_rms / a ~ 0.1-0.2 at T_melt. In the cipher: amplitude A at the melting transition maps to this ratio. The archetype-dependent alpha coefficient is the cipher's structural modulation of the base amplitude. ================================================================================ END OF PHASE TRANSITION DATA ================================================================================