========================================================================================== ANALYSIS 2E: SCALE RATIO MYSTERY — a_lattice / lambda_Compton ========================================================================================== Date: 2026-03-17 Elements analyzed: 75 (with a > 0 and structure in FCC/BCC/HCP/Diamond) lambda_C = h / (m * c) where m = mass_amu * 1.660539066600000e-27 kg ratio = a_lattice [m] / lambda_C [m] ------------------------------------------------------------------------------------------ FULL DATA TABLE ------------------------------------------------------------------------------------------ Sym Z Per Struct Block a(A) lambda_C(fm) ratio ratio/Z r_atom(A) --------------------------------------------------------------------------------- H 1 1 HCP s 3.750 1.3205 283991.6 283991.65 1.875 He 2 1 HCP s 3.570 0.3325 1073554.7 536777.35 1.785 Li 3 2 BCC s 3.510 0.1918 1830387.0 610129.01 1.520 Be 4 2 HCP s 2.286 0.1477 1547821.3 386955.32 1.143 C 6 2 Diamond p 3.567 0.1108 3218815.2 536469.20 0.772 N 7 2 HCP p 3.860 0.0950 4062058.8 580294.11 1.930 Ne 10 2 FCC p 4.429 0.0660 6714916.4 671491.64 1.566 Na 11 3 BCC s 4.225 0.0579 7297589.9 663417.27 1.829 Mg 12 3 HCP s 3.209 0.0548 5859750.3 488312.53 1.605 Al 13 3 FCC p 4.050 0.0493 8208982.2 631460.17 1.432 Si 14 3 Diamond p 5.431 0.0474 11459561.6 818540.11 1.176 Ar 18 3 FCC p 5.256 0.0333 15774811.1 876378.40 1.858 K 19 4 BCC s 5.225 0.0340 15348099.6 807794.71 2.262 Ca 20 4 FCC s 5.588 0.0332 16825818.9 841290.94 1.976 Sc 21 4 HCP d 3.309 0.0296 11176303.9 532204.95 1.655 Ti 22 4 HCP d 2.951 0.0278 10612536.3 482388.01 1.476 V 23 4 BCC d 3.024 0.0261 11573682.1 503203.57 1.309 Cr 24 4 BCC d 2.884 0.0256 11266239.1 469426.63 1.249 Mn 25 4 BCC d 8.912 0.0242 36784240.5 1471369.62 3.859 Fe 26 4 BCC d 2.866 0.0238 12026797.9 462569.15 1.241 Co 27 4 HCP d 2.507 0.0226 11100093.9 411114.59 1.254 Ni 28 4 FCC d 3.524 0.0227 15539462.0 554980.79 1.246 Cu 29 4 FCC d 3.615 0.0209 17258310.5 595114.15 1.278 Zn 30 4 HCP d 2.665 0.0204 13090489.9 436349.66 1.333 Ge 32 4 Diamond p 5.658 0.0183 30873989.9 964812.19 1.225 Kr 36 4 FCC p 5.721 0.0159 36017981.6 1000499.49 2.023 Rb 37 5 BCC s 5.585 0.0156 35862494.1 969256.60 2.418 Sr 38 5 FCC s 6.084 0.0152 40050341.6 1053956.36 2.151 Y 39 5 HCP d 3.648 0.0150 24366865.8 624791.43 1.824 Zr 40 5 HCP d 3.232 0.0146 22151045.8 553776.14 1.616 Nb 41 5 BCC d 3.300 0.0143 23034111.9 561807.61 1.429 Mo 42 5 BCC d 3.147 0.0139 22685872.8 540139.83 1.363 Tc 43 5 HCP d 2.735 0.0136 20137111.6 468304.92 1.367 Ru 44 5 HCP d 2.706 0.0132 20547729.0 466993.84 1.353 Rh 45 5 FCC d 3.803 0.0129 29403408.3 653409.07 1.345 Pd 46 5 FCC d 3.890 0.0125 31101879.0 676127.80 1.375 Ag 47 5 FCC d 4.086 0.0123 33115709.9 704589.57 1.445 Cd 48 5 HCP d 2.979 0.0118 25158759.4 524140.82 1.490 Xe 54 5 FCC p 6.197 0.0101 61126132.1 1131965.41 2.191 Cs 55 6 BCC s 6.045 0.0100 60362571.7 1097501.30 2.618 Ba 56 6 BCC s 5.028 0.0097 51876953.0 926374.16 2.177 La 57 6 HCP f 3.772 0.0096 39365789.5 690627.89 1.886 Ce 58 6 FCC f 5.161 0.0095 54331007.5 936741.51 1.825 Pr 59 6 HCP f 3.673 0.0094 38884499.6 659059.31 1.837 Nd 60 6 HCP f 3.658 0.0092 39640870.8 660681.18 1.829 Pm 61 6 HCP f 3.650 0.0092 39762587.5 651845.70 1.825 Eu 63 6 BCC f 4.581 0.0088 52300199.8 830161.90 1.984 Gd 64 6 HCP f 3.636 0.0085 42956441.7 671194.40 1.818 Tb 65 6 HCP f 3.601 0.0084 42997457.5 661499.35 1.800 Dy 66 6 HCP f 3.593 0.0082 43865628.2 664630.73 1.796 Ho 67 6 HCP f 3.578 0.0081 44335720.0 661727.16 1.789 Er 68 6 HCP f 3.559 0.0080 44723301.0 657695.60 1.780 Tm 69 6 HCP f 3.538 0.0079 44903312.7 650772.65 1.769 Yb 70 6 FCC f 5.485 0.0077 71311899.8 1018741.43 1.939 Lu 71 6 HCP f 3.504 0.0076 46061858.9 648758.58 1.752 Hf 72 6 HCP d 3.195 0.0075 42844839.8 595067.22 1.597 Ta 73 6 BCC d 3.303 0.0074 44903576.4 615117.48 1.430 W 74 6 BCC d 3.165 0.0072 43714699.4 590739.18 1.370 Re 75 6 HCP d 2.761 0.0071 38626306.1 515017.41 1.381 Os 76 6 HCP d 2.734 0.0070 39074307.4 514135.62 1.367 Ir 77 6 FCC d 3.839 0.0069 55440923.5 720011.99 1.357 Pt 78 6 FCC d 3.924 0.0068 57511608.8 737328.32 1.387 Au 79 6 FCC d 4.078 0.0068 60350708.9 763933.02 1.442 Tl 81 6 HCP p 3.456 0.0065 53067166.5 655150.20 1.728 Pb 82 6 FCC p 4.950 0.0064 77059514.4 939750.18 1.750 Rn 86 6 FCC p 6.000 0.0060 100073248.1 1163642.42 2.121 Fr 87 7 BCC s 5.700 0.0060 95497827.0 1097676.17 2.468 Ra 88 7 BCC s 5.148 0.0059 87409925.1 993294.60 2.229 Ac 89 7 FCC f 5.311 0.0059 90576582.3 1017714.41 1.878 Th 90 7 FCC f 5.084 0.0057 88630289.9 984781.00 1.797 Am 95 7 HCP f 3.468 0.0055 63313909.8 666462.21 1.734 Cm 96 7 HCP f 3.496 0.0054 64875713.6 675788.68 1.748 Bk 97 7 HCP f 3.416 0.0054 63391143.5 653516.94 1.708 Cf 98 7 HCP f 3.380 0.0053 63738845.5 650396.38 1.690 Es 99 7 FCC f 5.750 0.0053 108863465.8 1099630.97 2.033 ========================================================================================== GLOBAL STATISTICS ========================================================================================== Min ratio: 283991.6 (H) Max ratio: 108863465.8 (Es) Spread: 383.3x Mean ratio: 37362832.7 Median: 38626306.1 ========================================================================================== PATTERN 1: RATIO vs ATOMIC NUMBER (Z) ========================================================================================== If ratio ~ Z, then lattice constant scales linearly with Compton wavelength * Z. If ratio ~ Z^2, it follows Bohr-model scaling (atomic radius ~ 1/Z for inner shells). Pearson correlation (ratio vs Z): r = 0.8995 Pearson correlation (ratio vs Z^2): r = 0.8924 Power law fit: ratio ~ Z^1.139 (r = 0.9703) Prefactor: e^beta = 411421.0 FINDING: ratio grows approximately linearly with Z (alpha ~ 1), meaning a/lambda_C ~ Z. This implies a ~ Z * lambda_C = Z*h/(mc). ========================================================================================== PATTERN 2: RATIO / Z — DOES THIS NORMALIZE? ========================================================================================== If ratio/Z is approximately constant, then a_lattice ~ Z * lambda_C. Min ratio/Z: 283991.65 (H) Max ratio/Z: 1471369.62 (Mn) Spread: 5.2x Mean: 710450.16 Std dev: 219145.05 Coeff of var: 30.8% FINDING: ratio/Z still varies 5.2x. Z alone does not normalize. ========================================================================================== PATTERN 3: RATIO CONSTANCY WITHIN EACH STRUCTURE TYPE ========================================================================================== If the cipher's geometry fully determines the scale relationship, then ratio should be constant (or at least tighter) within each archetype. FCC (22 elements): ratio: min= 6714916.4 max=108863465.8 mean=48876681.9 CV= 61.9% spread= 16.2x ratio/Z: min= 554980.79 max=1163642.42 mean= 853342.68 CV= 21.7% spread= 2.1x Elements (sorted by ratio): Ne (Z= 10, Per=2): ratio= 6714916.4 ratio/Z=671491.64 a=4.429A lambda_C=0.066fm Al (Z= 13, Per=3): ratio= 8208982.2 ratio/Z=631460.17 a=4.050A lambda_C=0.049fm Ni (Z= 28, Per=4): ratio=15539462.0 ratio/Z=554980.79 a=3.524A lambda_C=0.023fm Ar (Z= 18, Per=3): ratio=15774811.1 ratio/Z=876378.40 a=5.256A lambda_C=0.033fm Ca (Z= 20, Per=4): ratio=16825818.9 ratio/Z=841290.94 a=5.588A lambda_C=0.033fm Cu (Z= 29, Per=4): ratio=17258310.5 ratio/Z=595114.15 a=3.615A lambda_C=0.021fm Rh (Z= 45, Per=5): ratio=29403408.3 ratio/Z=653409.07 a=3.803A lambda_C=0.013fm Pd (Z= 46, Per=5): ratio=31101879.0 ratio/Z=676127.80 a=3.890A lambda_C=0.013fm Ag (Z= 47, Per=5): ratio=33115709.9 ratio/Z=704589.57 a=4.086A lambda_C=0.012fm Kr (Z= 36, Per=4): ratio=36017981.6 ratio/Z=1000499.49 a=5.721A lambda_C=0.016fm Sr (Z= 38, Per=5): ratio=40050341.6 ratio/Z=1053956.36 a=6.084A lambda_C=0.015fm Ce (Z= 58, Per=6): ratio=54331007.5 ratio/Z=936741.51 a=5.161A lambda_C=0.009fm Ir (Z= 77, Per=6): ratio=55440923.5 ratio/Z=720011.99 a=3.839A lambda_C=0.007fm Pt (Z= 78, Per=6): ratio=57511608.8 ratio/Z=737328.32 a=3.924A lambda_C=0.007fm Au (Z= 79, Per=6): ratio=60350708.9 ratio/Z=763933.02 a=4.078A lambda_C=0.007fm Xe (Z= 54, Per=5): ratio=61126132.1 ratio/Z=1131965.41 a=6.197A lambda_C=0.010fm Yb (Z= 70, Per=6): ratio=71311899.8 ratio/Z=1018741.43 a=5.485A lambda_C=0.008fm Pb (Z= 82, Per=6): ratio=77059514.4 ratio/Z=939750.18 a=4.950A lambda_C=0.006fm Th (Z= 90, Per=7): ratio=88630289.9 ratio/Z=984781.00 a=5.084A lambda_C=0.006fm Ac (Z= 89, Per=7): ratio=90576582.3 ratio/Z=1017714.41 a=5.311A lambda_C=0.006fm Rn (Z= 86, Per=6): ratio=100073248.1 ratio/Z=1163642.42 a=6.000A lambda_C=0.006fm Es (Z= 99, Per=7): ratio=108863465.8 ratio/Z=1099630.97 a=5.750A lambda_C=0.005fm BCC (17 elements): ratio: min= 1830387.0 max=95497827.0 mean=36104427.5 CV= 73.5% spread= 52.2x ratio/Z: min= 462569.15 max=1471369.62 mean= 777057.58 CV= 35.2% spread= 3.2x Elements (sorted by ratio): Li (Z= 3, Per=2): ratio= 1830387.0 ratio/Z=610129.01 a=3.510A lambda_C=0.192fm Na (Z= 11, Per=3): ratio= 7297589.9 ratio/Z=663417.27 a=4.225A lambda_C=0.058fm Cr (Z= 24, Per=4): ratio=11266239.1 ratio/Z=469426.63 a=2.884A lambda_C=0.026fm V (Z= 23, Per=4): ratio=11573682.1 ratio/Z=503203.57 a=3.024A lambda_C=0.026fm Fe (Z= 26, Per=4): ratio=12026797.9 ratio/Z=462569.15 a=2.866A lambda_C=0.024fm K (Z= 19, Per=4): ratio=15348099.6 ratio/Z=807794.71 a=5.225A lambda_C=0.034fm Mo (Z= 42, Per=5): ratio=22685872.8 ratio/Z=540139.83 a=3.147A lambda_C=0.014fm Nb (Z= 41, Per=5): ratio=23034111.9 ratio/Z=561807.61 a=3.300A lambda_C=0.014fm Rb (Z= 37, Per=5): ratio=35862494.1 ratio/Z=969256.60 a=5.585A lambda_C=0.016fm Mn (Z= 25, Per=4): ratio=36784240.5 ratio/Z=1471369.62 a=8.912A lambda_C=0.024fm W (Z= 74, Per=6): ratio=43714699.4 ratio/Z=590739.18 a=3.165A lambda_C=0.007fm Ta (Z= 73, Per=6): ratio=44903576.4 ratio/Z=615117.48 a=3.303A lambda_C=0.007fm Ba (Z= 56, Per=6): ratio=51876953.0 ratio/Z=926374.16 a=5.028A lambda_C=0.010fm Eu (Z= 63, Per=6): ratio=52300199.8 ratio/Z=830161.90 a=4.581A lambda_C=0.009fm Cs (Z= 55, Per=6): ratio=60362571.7 ratio/Z=1097501.30 a=6.045A lambda_C=0.010fm Ra (Z= 88, Per=7): ratio=87409925.1 ratio/Z=993294.60 a=5.148A lambda_C=0.006fm Fr (Z= 87, Per=7): ratio=95497827.0 ratio/Z=1097676.17 a=5.700A lambda_C=0.006fm HCP (33 elements): ratio: min= 283991.6 max=64875713.6 mean=32351448.9 CV= 60.2% spread=228.4x ratio/Z: min= 283991.65 max= 690627.89 mean= 575164.32 CV= 17.7% spread= 2.4x Elements (sorted by ratio): H (Z= 1, Per=1): ratio= 283991.6 ratio/Z=283991.65 a=3.750A lambda_C=1.320fm He (Z= 2, Per=1): ratio= 1073554.7 ratio/Z=536777.35 a=3.570A lambda_C=0.333fm Be (Z= 4, Per=2): ratio= 1547821.3 ratio/Z=386955.32 a=2.286A lambda_C=0.148fm N (Z= 7, Per=2): ratio= 4062058.8 ratio/Z=580294.11 a=3.860A lambda_C=0.095fm Mg (Z= 12, Per=3): ratio= 5859750.3 ratio/Z=488312.53 a=3.209A lambda_C=0.055fm Ti (Z= 22, Per=4): ratio=10612536.3 ratio/Z=482388.01 a=2.951A lambda_C=0.028fm Co (Z= 27, Per=4): ratio=11100093.9 ratio/Z=411114.59 a=2.507A lambda_C=0.023fm Sc (Z= 21, Per=4): ratio=11176303.9 ratio/Z=532204.95 a=3.309A lambda_C=0.030fm Zn (Z= 30, Per=4): ratio=13090489.9 ratio/Z=436349.66 a=2.665A lambda_C=0.020fm Tc (Z= 43, Per=5): ratio=20137111.6 ratio/Z=468304.92 a=2.735A lambda_C=0.014fm Ru (Z= 44, Per=5): ratio=20547729.0 ratio/Z=466993.84 a=2.706A lambda_C=0.013fm Zr (Z= 40, Per=5): ratio=22151045.8 ratio/Z=553776.14 a=3.232A lambda_C=0.015fm Y (Z= 39, Per=5): ratio=24366865.8 ratio/Z=624791.43 a=3.648A lambda_C=0.015fm Cd (Z= 48, Per=5): ratio=25158759.4 ratio/Z=524140.82 a=2.979A lambda_C=0.012fm Re (Z= 75, Per=6): ratio=38626306.1 ratio/Z=515017.41 a=2.761A lambda_C=0.007fm Pr (Z= 59, Per=6): ratio=38884499.6 ratio/Z=659059.31 a=3.673A lambda_C=0.009fm Os (Z= 76, Per=6): ratio=39074307.4 ratio/Z=514135.62 a=2.734A lambda_C=0.007fm La (Z= 57, Per=6): ratio=39365789.5 ratio/Z=690627.89 a=3.772A lambda_C=0.010fm Nd (Z= 60, Per=6): ratio=39640870.8 ratio/Z=660681.18 a=3.658A lambda_C=0.009fm Pm (Z= 61, Per=6): ratio=39762587.5 ratio/Z=651845.70 a=3.650A lambda_C=0.009fm Hf (Z= 72, Per=6): ratio=42844839.8 ratio/Z=595067.22 a=3.195A lambda_C=0.007fm Gd (Z= 64, Per=6): ratio=42956441.7 ratio/Z=671194.40 a=3.636A lambda_C=0.008fm Tb (Z= 65, Per=6): ratio=42997457.5 ratio/Z=661499.35 a=3.601A lambda_C=0.008fm Dy (Z= 66, Per=6): ratio=43865628.2 ratio/Z=664630.73 a=3.593A lambda_C=0.008fm Ho (Z= 67, Per=6): ratio=44335720.0 ratio/Z=661727.16 a=3.578A lambda_C=0.008fm Er (Z= 68, Per=6): ratio=44723301.0 ratio/Z=657695.60 a=3.559A lambda_C=0.008fm Tm (Z= 69, Per=6): ratio=44903312.7 ratio/Z=650772.65 a=3.538A lambda_C=0.008fm Lu (Z= 71, Per=6): ratio=46061858.9 ratio/Z=648758.58 a=3.504A lambda_C=0.008fm Tl (Z= 81, Per=6): ratio=53067166.5 ratio/Z=655150.20 a=3.456A lambda_C=0.007fm Am (Z= 95, Per=7): ratio=63313909.8 ratio/Z=666462.21 a=3.468A lambda_C=0.005fm Bk (Z= 97, Per=7): ratio=63391143.5 ratio/Z=653516.94 a=3.416A lambda_C=0.005fm Cf (Z= 98, Per=7): ratio=63738845.5 ratio/Z=650396.38 a=3.380A lambda_C=0.005fm Cm (Z= 96, Per=7): ratio=64875713.6 ratio/Z=675788.68 a=3.496A lambda_C=0.005fm Diamond (3 elements): ratio: min= 3218815.2 max=30873989.9 mean=15184122.2 CV= 76.4% spread= 9.6x ratio/Z: min= 536469.20 max= 964812.19 mean= 773273.83 CV= 23.0% spread= 1.8x Elements (sorted by ratio): C (Z= 6, Per=2): ratio= 3218815.2 ratio/Z=536469.20 a=3.567A lambda_C=0.111fm Si (Z= 14, Per=3): ratio=11459561.6 ratio/Z=818540.11 a=5.431A lambda_C=0.047fm Ge (Z= 32, Per=4): ratio=30873989.9 ratio/Z=964812.19 a=5.658A lambda_C=0.018fm ========================================================================================== PATTERN 4: RATIO vs PERIOD NUMBER ========================================================================================== Standard physics: atomic radius follows periodic trends (grows with period, shrinks across a period due to nuclear charge). Does ratio track period? Period 1 (2 elements): ratio range: 283991.6 - 1073554.7, mean=678773.2 H (Z= 1, HCP): ratio= 283991.6 He (Z= 2, HCP): ratio= 1073554.7 Period 2 (5 elements): ratio range: 1547821.3 - 6714916.4, mean=3474799.8 Li (Z= 3, BCC): ratio= 1830387.0 Be (Z= 4, HCP): ratio= 1547821.3 C (Z= 6, Diamond): ratio= 3218815.2 N (Z= 7, HCP): ratio= 4062058.8 Ne (Z= 10, FCC): ratio= 6714916.4 Period 3 (5 elements): ratio range: 5859750.3 - 15774811.1, mean=9720139.0 Na (Z= 11, BCC): ratio= 7297589.9 Mg (Z= 12, HCP): ratio= 5859750.3 Al (Z= 13, FCC): ratio= 8208982.2 Si (Z= 14, Diamond): ratio=11459561.6 Ar (Z= 18, FCC): ratio=15774811.1 Period 4 (14 elements): ratio range: 10612536.3 - 36784240.5, mean=17821003.3 K (Z= 19, BCC): ratio=15348099.6 Ca (Z= 20, FCC): ratio=16825818.9 Sc (Z= 21, HCP): ratio=11176303.9 Ti (Z= 22, HCP): ratio=10612536.3 V (Z= 23, BCC): ratio=11573682.1 Cr (Z= 24, BCC): ratio=11266239.1 Mn (Z= 25, BCC): ratio=36784240.5 Fe (Z= 26, BCC): ratio=12026797.9 Co (Z= 27, HCP): ratio=11100093.9 Ni (Z= 28, FCC): ratio=15539462.0 Cu (Z= 29, FCC): ratio=17258310.5 Zn (Z= 30, HCP): ratio=13090489.9 Ge (Z= 32, Diamond): ratio=30873989.9 Kr (Z= 36, FCC): ratio=36017981.6 Period 5 (13 elements): ratio range: 20137111.6 - 61126132.1, mean=29903189.3 Rb (Z= 37, BCC): ratio=35862494.1 Sr (Z= 38, FCC): ratio=40050341.6 Y (Z= 39, HCP): ratio=24366865.8 Zr (Z= 40, HCP): ratio=22151045.8 Nb (Z= 41, BCC): ratio=23034111.9 Mo (Z= 42, BCC): ratio=22685872.8 Tc (Z= 43, HCP): ratio=20137111.6 Ru (Z= 44, HCP): ratio=20547729.0 Rh (Z= 45, FCC): ratio=29403408.3 Pd (Z= 46, FCC): ratio=31101879.0 Ag (Z= 47, FCC): ratio=33115709.9 Cd (Z= 48, HCP): ratio=25158759.4 Xe (Z= 54, FCC): ratio=61126132.1 Period 6 (27 elements): ratio range: 38626306.1 - 100073248.1, mean=50753592.5 Cs (Z= 55, BCC): ratio=60362571.7 Ba (Z= 56, BCC): ratio=51876953.0 La (Z= 57, HCP): ratio=39365789.5 Ce (Z= 58, FCC): ratio=54331007.5 Pr (Z= 59, HCP): ratio=38884499.6 Nd (Z= 60, HCP): ratio=39640870.8 Pm (Z= 61, HCP): ratio=39762587.5 Eu (Z= 63, BCC): ratio=52300199.8 Gd (Z= 64, HCP): ratio=42956441.7 Tb (Z= 65, HCP): ratio=42997457.5 Dy (Z= 66, HCP): ratio=43865628.2 Ho (Z= 67, HCP): ratio=44335720.0 Er (Z= 68, HCP): ratio=44723301.0 Tm (Z= 69, HCP): ratio=44903312.7 Yb (Z= 70, FCC): ratio=71311899.8 Lu (Z= 71, HCP): ratio=46061858.9 Hf (Z= 72, HCP): ratio=42844839.8 Ta (Z= 73, BCC): ratio=44903576.4 W (Z= 74, BCC): ratio=43714699.4 Re (Z= 75, HCP): ratio=38626306.1 Os (Z= 76, HCP): ratio=39074307.4 Ir (Z= 77, FCC): ratio=55440923.5 Pt (Z= 78, FCC): ratio=57511608.8 Au (Z= 79, FCC): ratio=60350708.9 Tl (Z= 81, HCP): ratio=53067166.5 Pb (Z= 82, FCC): ratio=77059514.4 Rn (Z= 86, FCC): ratio=100073248.1 Period 7 (9 elements): ratio range: 63313909.8 - 108863465.8, mean=80699744.7 Fr (Z= 87, BCC): ratio=95497827.0 Ra (Z= 88, BCC): ratio=87409925.1 Ac (Z= 89, FCC): ratio=90576582.3 Th (Z= 90, FCC): ratio=88630289.9 Am (Z= 95, HCP): ratio=63313909.8 Cm (Z= 96, HCP): ratio=64875713.6 Bk (Z= 97, HCP): ratio=63391143.5 Cf (Z= 98, HCP): ratio=63738845.5 Es (Z= 99, FCC): ratio=108863465.8 Pearson correlation (ratio vs period): r = 0.8434 ========================================================================================== PATTERN 5: RATIO vs DERIVED ATOMIC RADIUS ========================================================================================== If ratio simply reflects atomic size, ratio should correlate tightly with r_atomic. Atomic radius derived from structure: FCC r=a/(2*sqrt(2)), BCC r=a*sqrt(3)/4, etc. Pearson correlation (ratio vs r_atomic): r = 0.4011 ratio / r_atomic: min = 151462.2 max = 53549979.0 spread = 353.6x mean = 21385307.2 CV = 62.5% ========================================================================================== PATTERN 6: DIMENSIONAL/PHYSICAL INTERPRETATION ========================================================================================== The Compton wavelength is: lambda_C = h / (m * c) The lattice constant a is set by: a ~ 2 * r_atomic (roughly, depending on structure) The ratio is: ratio = a / lambda_C = a * m * c / h Since a ~ r_atomic and r_atomic ~ a_0 * n^2 / Z_eff (Bohr model): ratio ~ (a_0 * n^2 / Z_eff) * (Z * m_p * c / h) = (a_0 * c * m_p / h) * Z * n^2 / Z_eff The quantity a_0 * m_p * c / h: a_0 = 5.29177e-11 m m_p = 1.67262e-27 kg a_0 * m_p * c / h = 5.29177e-11 * 1.67262e-27 * 299792458 / 6.62607e-34 = 40046.3482 So ratio ~ 40046.3 * Z * n^2 / Z_eff This is purely standard physics: the ratio contains NO new information beyond the Bohr model's prediction of atomic radii. FOR TLT: The question is whether there exists a DIFFERENT formula connecting a_lattice to lambda_C that depends on the STRUCTURE TYPE (coordination number, packing geometry) rather than Z_eff. ========================================================================================== PATTERN 7: SEARCHING FOR A STRUCTURE-DEPENDENT CONSTANT ========================================================================================== FCC: fitting ratio = k * Z^alpha Best alpha = 1.170, k = 443268.62, CV = 18.8% (If alpha ~ 2, this is Bohr-model scaling: r ~ n^2/Z, but mass ~ Z so ratio ~ Z) Normalized values: Ne Z= 10: ratio/Z^1.17 = 453984.07 Al Z= 13: ratio/Z^1.17 = 408296.44 Ar Z= 18: ratio/Z^1.17 = 536161.14 Ca Z= 20: ratio/Z^1.17 = 505558.15 Ni Z= 28: ratio/Z^1.17 = 314964.12 Cu Z= 29: ratio/Z^1.17 = 335731.91 Kr Z= 36: ratio/Z^1.17 = 544058.31 Sr Z= 38: ratio/Z^1.17 = 567883.72 Rh Z= 45: ratio/Z^1.17 = 342088.94 Pd Z= 46: ratio/Z^1.17 = 352663.05 Ag Z= 47: ratio/Z^1.17 = 366167.31 Xe Z= 54: ratio/Z^1.17 = 574547.87 Ce Z= 58: ratio/Z^1.17 = 469717.79 Yb Z= 70: ratio/Z^1.17 = 494763.10 Ir Z= 77: ratio/Z^1.17 = 344061.68 Pt Z= 78: ratio/Z^1.17 = 351564.35 Au Z= 79: ratio/Z^1.17 = 363461.72 Pb Z= 82: ratio/Z^1.17 = 444287.47 Rn Z= 86: ratio/Z^1.17 = 545701.07 Ac Z= 89: ratio/Z^1.17 = 474492.79 Th Z= 90: ratio/Z^1.17 = 458266.84 Es Z= 99: ratio/Z^1.17 = 503487.83 BCC: fitting ratio = k * Z^alpha Best alpha = 1.105, k = 533978.89, CV = 34.2% (If alpha ~ 2, this is Bohr-model scaling: r ~ n^2/Z, but mass ~ Z so ratio ~ Z) Normalized values: Li Z= 3: ratio/Z^1.10 = 543655.70 Na Z= 11: ratio/Z^1.10 = 515751.55 K Z= 19: ratio/Z^1.10 = 592968.85 V Z= 23: ratio/Z^1.10 = 362044.78 Cr Z= 24: ratio/Z^1.10 = 336237.03 Mn Z= 25: ratio/Z^1.10 = 1049392.81 Fe Z= 26: ratio/Z^1.10 = 328552.26 Rb Z= 37: ratio/Z^1.10 = 663403.23 Nb Z= 41: ratio/Z^1.10 = 380404.21 Mo Z= 42: ratio/Z^1.10 = 364808.57 Cs Z= 55: ratio/Z^1.10 = 720554.71 Ba Z= 56: ratio/Z^1.10 = 607053.11 Eu Z= 63: ratio/Z^1.10 = 537318.81 Ta Z= 73: ratio/Z^1.10 = 392020.86 W Z= 74: ratio/Z^1.10 = 375946.85 Fr Z= 87: ratio/Z^1.10 = 686791.19 Ra Z= 88: ratio/Z^1.10 = 620736.66 HCP: fitting ratio = k * Z^alpha Best alpha = 1.110, k = 382805.19, CV = 13.6% (If alpha ~ 2, this is Bohr-model scaling: r ~ n^2/Z, but mass ~ Z so ratio ~ Z) Normalized values: H Z= 1: ratio/Z^1.11 = 283991.65 He Z= 2: ratio/Z^1.11 = 497371.48 Be Z= 4: ratio/Z^1.11 = 332226.47 N Z= 7: ratio/Z^1.11 = 468476.04 Mg Z= 12: ratio/Z^1.11 = 371524.90 Sc Z= 21: ratio/Z^1.11 = 380745.50 Ti Z= 22: ratio/Z^1.11 = 343344.44 Co Z= 27: ratio/Z^1.11 = 286096.72 Zn Z= 30: ratio/Z^1.11 = 300158.96 Y Z= 39: ratio/Z^1.11 = 417559.10 Zr Z= 40: ratio/Z^1.11 = 369069.06 Tc Z= 43: ratio/Z^1.11 = 309632.97 Ru Z= 44: ratio/Z^1.11 = 307986.27 Cd Z= 48: ratio/Z^1.11 = 342382.41 La Z= 57: ratio/Z^1.11 = 442688.13 Pr Z= 59: ratio/Z^1.11 = 420853.34 Nd Z= 60: ratio/Z^1.11 = 421109.75 Pm Z= 61: ratio/Z^1.11 = 414723.38 Gd Z= 64: ratio/Z^1.11 = 424784.37 Tb Z= 65: ratio/Z^1.11 = 417935.20 Dy Z= 66: ratio/Z^1.11 = 419208.99 Ho Z= 67: ratio/Z^1.11 = 416687.75 Er Z= 68: ratio/Z^1.11 = 413474.72 Tm Z= 69: ratio/Z^1.11 = 408465.98 Lu Z= 71: ratio/Z^1.11 = 405923.97 Hf Z= 72: ratio/Z^1.11 = 371757.25 Re Z= 75: ratio/Z^1.11 = 320306.07 Os Z= 76: ratio/Z^1.11 = 319292.12 Tl Z= 81: ratio/Z^1.11 = 404024.34 Am Z= 95: ratio/Z^1.11 = 403855.43 Cm Z= 96: ratio/Z^1.11 = 409035.57 Bk Z= 97: ratio/Z^1.11 = 395104.49 Cf Z= 98: ratio/Z^1.11 = 392774.47 Diamond: fitting ratio = k * Z^alpha Best alpha = 1.350, k = 299464.52, CV = 6.0% (If alpha ~ 2, this is Bohr-model scaling: r ~ n^2/Z, but mass ~ Z so ratio ~ Z) Normalized values: C Z= 6: ratio/Z^1.35 = 286544.36 Si Z= 14: ratio/Z^1.35 = 325008.83 Ge Z= 32: ratio/Z^1.35 = 286840.38 ========================================================================================== PATTERN 8: CONDUCTORS vs NON-CONDUCTORS WITHIN SAME STRUCTURE ========================================================================================== Noble gases and molecular solids with FCC/HCP structures are insulators. Does their ratio differ systematically from metallic FCC/HCP? FCC: Metals (17): ratio range 8208982-108863466, ratio/Z range 554980.79-1099630.97, mean ratio/Z = 819385.98 Non-metals (5): ratio range 6714916-100073248, ratio/Z range 671491.64-1163642.42, mean ratio/Z = 968795.47 Non-metal elements: Ne, Ar, Kr, Xe, Rn HCP: Metals (30): ratio range 1547821-64875714, ratio/Z range 386955.32-690627.89, mean ratio/Z = 585978.65 Non-metals (3): ratio range 283992-4062059, ratio/Z range 283991.65-580294.11, mean ratio/Z = 467021.04 Non-metal elements: H, He, N ========================================================================================== CONCLUSIONS ========================================================================================== 1. RAW RATIO SCALES WITH Z: The ratio a/lambda_C ranges from ~283992 to ~108863466 (383x spread). It correlates strongly with Z (r = 0.899) and follows a power law ratio ~ Z^1.14. 2. ratio/Z PARTIALLY NORMALIZES: Dividing by Z reduces spread from 383x to 5.2x. This is expected: in standard physics, a ~ r_atomic (which depends on principal quantum number and effective charge) while lambda_C ~ 1/m ~ 1/(Z*m_p), so ratio ~ r_atomic * Z * m_p * c / h. 3. STRUCTURE TYPE DOES NOT MAKE RATIO CONSTANT: Within each structure type, the ratio still varies enormously (spread within FCC alone is large). The cipher's geometry does NOT determine a unique scale relationship. 4. THE BOHR MODEL DOMINATES: The dimensional analysis shows ratio ~ 40046.3 * Z * n^2 / Z_eff, which is entirely standard physics. The ratio contains atomic radius information (set by quantum mechanics of electron shells), not lattice geometry information. 5. TLT IMPLICATION: For TLT to add value here, it would need to provide a DIFFERENT formula for atomic radius that depends on: - Compton wavelength (lambda_C = h/mc) - Coordination number (4, 8, or 12) - Some TLT-specific quantity (e.g., a ledger phase angle, (2, 3) packing fraction) that reproduces the ~75 known lattice constants WITHOUT invoking the Bohr model's Z_eff and principal quantum number. 6. MOST PROMISING DIRECTION: Within each structure type, the ratio/Z^alpha fit yields structure-dependent constants. If these constants map to cipher values (e.g., related to coordination number or packing fraction), that would be a genuine TLT prediction. 7. CONDUCTOR vs INSULATOR: Noble gases with FCC/HCP structures have systematically DIFFERENT ratio/Z values than metals with the same structure. This suggests the cipher needs a second variable beyond geometry to explain the metal/insulator distinction (as noted in R1 of the cipher validation). ========================================================================================== DATA SHOWS WHAT IT SHOWS. ==========================================================================================