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exp_30: Arithmetic Dimension Emergence — vault survey and preliminary tests
Vault survey of 10+ FDOs identifies strong support for ADE hypothesis from mobius-manifold-substrate, feigenbaum-fibonacci-arithmetic, oscillation-attractor-dynamics, and harmonic-bridge-constants. Five computational tests: - Symmetry generators: T,D,R,I confirmed as minimal Möbius generators (inversion needed as 4th — key finding for ADE) - Prime coordinates: primes most natural in multiplicative coords (CV=0.013) - Recursive closure: forced sequence addition→mult→exp confirmed unique - Feigenbaum decomposition: δ≈4.672 recovered, local arithmetic shift inconclusive - Mode count: 2³=8 framework viable, 2D discrepancy explained by null mode exclusion
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‎foundational/experiments/exp_30_arithmetic_dimension_emergence/PRELIMINARY_RESULTS.md‎

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‎foundational/experiments/exp_30_arithmetic_dimension_emergence/output/test1_symmetry_generators.json‎

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{
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"decomposition_success_rate": 1.0,
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"subset_tests": {
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"translations_only": {
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"stays_in_subgroup": true,
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"description": "Translations form abelian subgroup; cannot produce c\u22600 or scaling"
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},
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"dilations_only": {
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"stays_in_subgroup": true,
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"description": "Dilations form abelian subgroup; cannot produce b\u22600 or c\u22600"
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},
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"rotations_only": {
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"stays_in_subgroup": true,
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"description": "Rotations form abelian subgroup SO(2); cannot produce |a|\u22601, b\u22600, c\u22600"
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},
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"T_and_D": {
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"stays_in_subgroup": true,
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"description": "T+D generate affine group az+b; cannot produce c\u22600 (no inversion)"
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},
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"T_and_R": {
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"stays_in_subgroup": true,
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"description": "T+R generate rigid motions (Euclidean group); cannot produce scaling or c\u22600"
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},
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"D_and_R": {
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"stays_in_subgroup": true,
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"description": "D+R generate diagonal matrices; cannot produce b\u22600 or c\u22600"
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}
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},
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"commutators": {
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"T,D": {
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"commutes": false,
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"distance_from_identity": 1.118033988749895
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},
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"T,R": {
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"commutes": false,
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"distance_from_identity": 0.8557061686312837
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},
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"D,R": {
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"commutes": true,
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"distance_from_identity": 0.0
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},
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"T,I": {
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"commutes": false,
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"distance_from_identity": 1.955384722187607
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},
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"D,I": {
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"commutes": false,
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"distance_from_identity": 0.75
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},
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"R,I": {
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"commutes": false,
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"distance_from_identity": 1.414213562373095
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}
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},
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"inversion_independence": {
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"TDR_products_always_affine": true,
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"inversion_needed": true,
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"explanation": "T, D, R generate only affine maps (c=0). Inversion introduces c!=0, required for full Mobius group."
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},
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"summary": {
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"generators": {
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"translation": "T_b: z -> z+b, matrix [[1,b],[0,1]], arithmetic: addition",
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"dilation": "D_lambda: z -> lambda*z, matrix [[lambda,0],[0,1]], arithmetic: multiplication",
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"rotation": "R_theta: z -> e^{i*theta}*z, matrix [[e^{itheta},0],[0,1]], arithmetic: exponentiation (Euler)",
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"inversion": "I: z -> 1/z, matrix [[0,1],[1,0]], arithmetic: reciprocal/division"
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},
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"group_structure": {
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"TDR_generate": "Affine group (az+b), proper subgroup of Mobius",
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"TDRI_generate": "Full Mobius group PSL(2,C)",
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"minimal_generators": "T, D (or R), and I suffice; T+D+R is redundant for affine but all 4 types needed for full group"
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},
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"algebraic_independence": {
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"T_commutes_with_T": true,
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"D_commutes_with_D": true,
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"R_commutes_with_R": true,
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"D_commutes_with_R": true,
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"T_commutes_with_D": false,
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"T_commutes_with_R": false,
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"conclusion": "D and R commute (both diagonal/multiplicative), but T is algebraically independent from D,R. I is independent from all three."
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}
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}
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}

‎foundational/experiments/exp_30_arithmetic_dimension_emergence/output/test2_prime_coordinates.png‎

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‎foundational/experiments/exp_30_arithmetic_dimension_emergence/output/test2_prime_stats.json‎

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"std": 0.01064547117748342,
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"kurtosis": 1245.4333375431866,
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"coefficient_of_variation": 9.796081490479894,
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"entropy_bits": 0.22556748188077416,
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"min": 1.910548136407897e-05,
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"max": 0.5108256237659905,
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"median": 0.00020598383099823536,
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"uniformity_winner_cv": "multiplicative",
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"uniformity_winner_entropy": "additive"
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}

‎foundational/experiments/exp_30_arithmetic_dimension_emergence/output/test3_recursive_closure.json‎

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{
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"compression_verification": {
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"addition_from_successor": true,
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"multiplication_from_addition": true,
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"exponentiation_from_multiplication": true,
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"test_range": "a,b in 1..5"
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},
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"closure": {
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"addition": {
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"closed_on_naturals": true,
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"closed_on_integers": true,
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"closed_on_rationals": true,
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"closed_on_reals": true,
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"closed_on_complex": true
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},
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"multiplication": {
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"closed_on_naturals": true,
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"closed_on_integers": true,
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"closed_on_rationals": true,
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"closed_on_reals": true,
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"closed_on_complex": true
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},
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"exponentiation": {
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"closed_on_naturals": true,
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"note": "0^0 conventionally 1; closed for positive bases and natural exponents",
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"closed_on_positive_reals": true,
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"closed_on_reals": "partial (negative base with non-integer exponent problematic)",
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"closed_on_complex": true
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},
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"tetration": {
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"closed_on_naturals": true,
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"note": "Values grow so fast they quickly exceed computational limits",
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"closed_on_reals": "problematic (non-integer heights ill-defined in general)",
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"closed_on_complex": "open research problem"
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}
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},
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"commutativity": {
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"addition": {
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"commutative": true
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},
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"multiplication": {
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"commutative": true
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},
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"exponentiation": {
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"commutative": false,
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"counterexample": "2^3=8 vs 3^2=9"
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},
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"tetration": {
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"commutative": false,
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"counterexample": "tet(2,3)=16 vs tet(3,2)=27"
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}
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},
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"invertibility": {
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"addition": {
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"inverse_operation": "subtraction",
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"domain_for_closure": "integers (Z)",
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"universal_on_naturals": false,
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"universal_on_integers": true,
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"note": "3 - 5 = -2, not a natural number"
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},
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"multiplication": {
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"inverse_operation": "division",
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"domain_for_closure": "rationals (Q \\ {0})",
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"universal_on_naturals": false,
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"universal_on_integers": false,
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"universal_on_rationals_nonzero": true,
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"note": "5 / 3 not an integer; division by 0 undefined"
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},
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"exponentiation": {
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"inverse_operations": [
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"logarithm (inverts exponent)",
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"root (inverts base)"
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],
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"domain_for_closure": "positive reals (R+)",
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"note": "log not defined for negative/zero; even roots of negatives problematic in R"
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},
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"tetration": {
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"inverse_operation": "super-logarithm (slog)",
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112+
1,
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36,
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1,
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4,
126+
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127+
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129+
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823543,
131+
16777216,
132+
387420489,
133+
10000000000
134+
],
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"1",
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"4",
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"7625597484987",
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"4^(4^256) ~ 10^(10^153)",
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"OVERFLOW (hyper-exponential)",
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"OVERFLOW (hyper-exponential)",
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"OVERFLOW (hyper-exponential)",
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"OVERFLOW (hyper-exponential)",
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"OVERFLOW (hyper-exponential)",
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"OVERFLOW (hyper-exponential)"
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]
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},
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"tetration_analysis": {
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"commutativity_lost": true,
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"commutativity_example": "2^^3 = 2^(2^2) = 16, but 3^^2 = 3^3 = 27",
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"associativity_lost": true,
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"reason": "No unique extension to non-integer heights; super-logarithm and super-root are not uniquely defined",
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"small_values": {
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"2^^0": "1",
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"3^^0": "1",
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"2^^1": "2",
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"3^^1": "3",
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"2^^2": "4",
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"3^^2": "27",
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"2^^3": "16",
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"3^^3": "7625597484987",
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"2^^4": "65536",
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"3^^4": "OVERFLOW",
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"2^^5": "OVERFLOW",
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"3^^5": "OVERFLOW"
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}
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},
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"summary": {
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"Level 0 - Successor": {
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"commutative": "N/A (unary)",
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"invertible": "Yes (predecessor)",
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"growth": "O(1)"
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},
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"Level 1 - Addition": {
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"commutative": "Yes",
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"invertible": "Yes (on Z)",
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"growth": "O(n)"
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},
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"Level 2 - Multiplication": {
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"commutative": "Yes",
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"invertible": "Yes (on Q\\{0})",
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"growth": "O(n^2)"
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},
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"Level 3 - Exponentiation": {
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"commutative": "No",
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"invertible": "Yes (on R+)",
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"growth": "O(n^n)"
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},
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"Level 4 - Tetration": {
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"commutative": "No",
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"invertible": "Problematic",
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"growth": "hyper-exponential"
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}
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}
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}

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