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| Format: | Recurso digital |
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| Udgivet: |
Zenodo
2026
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| Online adgang: | https://doi.org/10.5281/zenodo.18437705 |
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Indholdsfortegnelse:
- <p>this book connects the Brahim Field Equations (single-body dynamics) to the N-body problem.<br>│ <br>│ Chapter 1: The Classical N-Body Problem <br>│ <br>│ - What is the N-body problem (Newton, gravity, pair interactions) <br>│ - Why it's unsolvable in general (Poincaré, chaos, no closed-form for N>=3) <br>│ - Known special solutions (Lagrange equilateral, Euler collinear, figure-8) <br>│ - Pair distances: C(N,2) = N(N-1)/2 for N bodies <br>│ - The standard approach: F = -dV/dr, integrate equations of motion <br>│ - Why this book takes a different approach <br>│ <br>│ Chapter 2: The Brahim Field Equations (Recap) <br>│ <br>│ - D(x) = -ln(x)/ln(phi), the dimension function <br>│ - phi^D(x) = 1/x (the algebraic identity) <br>│ - E(x) = 2pi (energy conservation, proven) <br>│ - The three field equations: evolution, conservation, wormhole <br>│ - x* = 0.7104... (the observer/fixed point) <br>│ - Key result: E = 2pi is flat — dE/dPsi = 0 everywhere <br>│ <br>│ Chapter 3: The Flat Potential <br>│ <br>│ PROOF 1: f = 0 <br>│ - In classical mechanics: F = -dV/dr. If V = constant, F = 0. <br>│ - In Brahim: E(Psi) = 2pi for ALL Psi. Therefore dE/dPsi = 0. <br>│ - For N bodies: no coupling force can be extracted from a constant potential <br>│ - Each body evolves independently: dPsi_i/dt = -(1/Psi_i)(Psi_i - x*) <br>│ - Theorem: The coupling function f between any pair of bodies is zero. <br>│ - Parallel to QM: the vector potential A appears in the Schrödinger equation directly; similarly D(Psi) appears in the Brahim evolution equation directly. Dynamics from field │<br>│ structure, not from potential gradients. <br>│ - Gauge freedom: E + C = 2pi + C, physics unchanged. <br>│ <br>│ Chapter 4: The Brahim Numbers <br>│ <br>│ - The sequence [27, 42, 60, 75, 97, 117, 139, 154, 172, 187] <br>│ - Mirror symmetry: B_n + B_{11-n} = 214 <br>│ - Center axis: 107 = 4*B_1 - 1 <br>│ - Sum: 1070 = 5 * 214 = 10 * 107 <br>│ - Phi-adic expansion: k - 1 = (phi-1)/32 + sum c_n * phi^(-B_n) <br>│ - All 10 coefficients (include c_117 = -11/65 and c_139 = 48/61) <br>│ - Convergence: 33 decimal digits <br>│ - The MirrorOperator M(x) = 214 - x as involution <br>│ - The MirrorProduct: |B_n> ◇ |M(B_n)> = |214> <br>│ <br>│ Chapter 5: The Generating Triangle <br>│ <br>│ PROOF 3: Unique 3-body geometry <br>│ - {B_2, B_4, B_5} = {42, 75, 97} <br>│ - B_2 + B_4 + B_5 = 214 (the sum constant) <br>│ - Exhaustive search: the UNIQUE triple from 120 possible C(10,3) triples <br>│ - Pairwise sums produce Brahim numbers: <br>│ - 42 + 75 = 117 = B_6 = M(B_5) <br>│ - 42 + 97 = 139 = B_7 = M(B_4) <br>│ - 75 + 97 = 172 = B_9 = M(B_2) <br>│ - Each pairwise sum is the mirror of the remaining vertex <br>│ - Self-generating: the triangle produces its own mirror partners <br>│ - Primitive vs composite: 7 primitive Brahim numbers, 3 composites <br>│ - 4 independent generators = 5 lower halves - 1 triangle constraint <br>│ - Conservation law: r_12 * r_13 * r_23 = phi^(-214) (verified exact to 100 digits) <br>│ - Meaning: for 3 bodies, the product of pair distances is fixed <br>│ <br>│ Chapter 6: The N-Body Scale Hierarchy <br>│ <br>│ PROOF 2 + PROOF 4: 369 scales, levels, max N=27 <br>│ - Subset sums of 10 Brahim numbers <br>│ - 2^10 - 1 = 1023 non-empty subsets, 369 unique sums <br>│ - Level hierarchy table: <br>│ - Level 1 (singles): 10 scales -> N <= 5 <br>│ - Level 2 (pairs): 42 scales -> N <= 9 <br>│ - Level 3 (triples): 98 scales -> N <= 14 <br>│ - Level 4: 167 scales -> N <= 18 <br>│ - Level 5: 233 scales -> N <= 22 <br>│ - ... <br>│ - All levels: 369 scales -> N <= 27 <br>│ - Conservation at each level: multiples of 214 emerge (214, 428, 642) <br>│ - C(27,2) = 351 <= 369 < 378 = C(28,2) <br>│ - Theorem: The Brahim manifold admits N-body configurations for N up to 27. <br>│ <br>│ Chapter 7: Constraint Geometry vs Force Integration <br>│ <br>│ - Classical approach: solve coupled ODEs with F = -Gm_im_j/r_ij^2 <br>│ - Brahim approach: each body follows independent field equation, geometry from constraints <br>│ - The N-body "solution" is the set of configurations satisfying: <br>│ a. dPsi_i/dt = -(1/Psi_i)(Psi_i - x*) for each body <br>│ b. Pair scales drawn from 369 Brahim subset sums <br>│ c. Mirror conservation laws at each level <br>│ - This is constraint satisfaction, not force integration <br>│ - Analogy: crystal structure (find allowed configurations, not trajectories) <br>│ <br>│ Chapter 8: The Three-Body Example <br>│ <br>│ PROOF 5: Worked example <br>│ - Three bodies at initial states Psi = [phi, 1.0, 1/phi] <br>│ - Each evolves via field equation independently (f = 0) <br>│ - Energy per body = 2pi at every timestep (verified numerically) <br>│ - Total energy = 3 * 2pi = 6pi = 18.849556 (constant) <br>│ - All converge to x* = 0.7104... <br>│ - Pair distance scales {42, 75, 97} define the geometry <br>│ - Conservation: phi^(-42) * phi^(-75) * phi^(-97) = phi^(-214) <br>│ - Full numerical table of evolution <br>│ - Python code for reproduction <br>│ <br>│ Chapter 9: Completeness and Boundaries <br>│ <br>│ - The sequence is complete: 10 numbers, 5 mirror pairs, closed algebra <br>│ - B_2 + B_4 + B_5 = 214 is the single hidden constraint <br>│ - No 11th Brahim number needed <br>│ - N = 27 is the boundary: C(28,2) = 378 > 369 <br>│ - Modular coverage: 93/107 = 86.9% of residues mod 107 <br>│ - 14 missing residues: what they mean <br>│ - Connection to physics constants from the same Brahim numbers <br>│ <br>│ Chapter 10: Conclusions and Open Questions <br>│ <br>│ - Summary: N-body problem solved as constraint geometry on flat potential <br>│ - The 5 proofs and what they establish <br>│ - Open questions: <br>│ - Physical interpretation of the 14 missing modular residues <br>│ - Connection to actual gravitational N-body solutions <br>│ - Extension beyond N=27 (requires new mathematical structure) <br>│ - Relationship between Brahim scales and measured orbital data <br>│ <br>│ Appendix A: Reference Tables <br>│ <br>│ - Constants table (phi, omega, beta, gamma, genesis, x*, 2pi) <br>│ - Brahim numbers table (index, value, mirror, coefficient) <br>│ - Lucas numbers table (dimension 1-12) <br>│ - Subset sum levels table <br>│ - Mirror product table (all 45 pairwise products) <br>│ <br>│ Appendix B: Validation Code <br>│ <br>│ - Full Python script reproducing all 5 proofs <br>│ - Uses only numpy and itertools (standard library + numpy) <br>│ - 65 tests matching the Field Equations validation <br>│ - Additional N-body specific tests </p>