#!/usr/bin/env python3 """ Computation 53 -- C tensor H tensor O multiplication setup (minimal) ===================================================================== Per research/ngen_3.md section 13, Furey 2022's derivation of N_gen = 3 uses the chain algebra of C tensor H tensor O on a 64-dim real space. This script provides the foundational multiplication table and verifies the algebra structure. Dimension counting for the chain algebra (which requires expensive product enumeration on a 64-dim space) is replaced by analytical citation of the structure result: L_{C tensor H tensor O} ~= C tensor H tensor M_8(R) ~= M_8(C tensor H) ~= M_8(M_2(C)) [since C tensor H ~= M_2(C)] ~= M_16(C) Real dimension: 2 * 16^2 = 512 (proper subalgebra of M_64(R), dim 4096). For the actual generation-3-count probe, the relevant question is: how many primitive idempotents does the centralizer of L_{CHO} in M_64(R) admit? That is computed in Computation 54 by explicit construction of the Furey chiral primitive idempotent and its stabilizer in the chain algebra. This script: (1) builds the multiplication table for C ⊗ H ⊗ O (64 basis elements) (2) verifies associativity-breaking comes only from the O factor (3) verifies a small sample of products including imaginary-unit squares """ import numpy as np # Multiplication tables for C, H, O C_T = [[(1, 0), (1, 1)], [(1, 1), (-1, 0)]] H_T = [[(0, 0)] * 4 for _ in range(4)] for _i in range(4): H_T[0][_i] = (1, _i); H_T[_i][0] = (1, _i) for _i in [1, 2, 3]: H_T[_i][_i] = (-1, 0) H_T[1][2] = (1, 3); H_T[2][1] = (-1, 3) H_T[2][3] = (1, 1); H_T[3][2] = (-1, 1) H_T[3][1] = (1, 2); H_T[1][3] = (-1, 2) O_T = [[(0, 0) for _ in range(8)] for _ in range(8)] for _i in range(8): O_T[0][_i] = (1, _i); O_T[_i][0] = (1, _i) for _i in range(1, 8): O_T[_i][_i] = (-1, 0) for _i, _j, _k in [(1, 2, 3), (1, 4, 5), (1, 7, 6), (2, 4, 6), (2, 5, 7), (3, 4, 7), (3, 6, 5)]: O_T[_i][_j] = (1, _k); O_T[_j][_i] = (-1, _k) O_T[_j][_k] = (1, _i); O_T[_k][_j] = (-1, _i) O_T[_k][_i] = (1, _j); O_T[_i][_k] = (-1, _j) def idx(c, h, o): return c * 32 + h * 8 + o def decompose(i): return i // 32, (i % 32) // 8, i % 8 def CHO_mult(b1, b2): """Multiply two basis elements b1, b2 (flat indices in 0..63). Returns (sign, flat_index_of_product).""" c1, h1, o1 = decompose(b1) c2, h2, o2 = decompose(b2) sc, ic_ = C_T[c1][c2] sh, ih = H_T[h1][h2] so, io_ = O_T[o1][o2] return sc * sh * so, idx(ic_, ih, io_) def vec_mult(u, v): """Multiply two arbitrary elements of C ⊗ H ⊗ O given as length-64 vectors.""" out = np.zeros(64) for i in range(64): if abs(u[i]) < 1e-15: continue for j in range(64): if abs(v[j]) < 1e-15: continue s, k = CHO_mult(i, j) out[k] += s * u[i] * v[j] return out def main(): print("=" * 90) print(" Computation 53 -- C tensor H tensor O multiplication setup") print("=" * 90) print() # (1) Sanity checks print(" (1) Multiplication-table sanity checks:") assert CHO_mult(idx(1, 0, 0), idx(1, 0, 0)) == (-1, idx(0, 0, 0)) print(f" i_C * i_C = -1 ✓") assert CHO_mult(idx(0, 1, 0), idx(0, 2, 0)) == (1, idx(0, 3, 0)) print(f" i_H * j_H = k_H ✓") assert CHO_mult(idx(0, 2, 0), idx(0, 1, 0)) == (-1, idx(0, 3, 0)) print(f" j_H * i_H = -k_H ✓") assert CHO_mult(idx(0, 0, 1), idx(0, 0, 2)) == (1, idx(0, 0, 3)) print(f" e_1 * e_2 = e_3 ✓") # i_C commutes with everything for h in range(4): for o in range(8): a, b = idx(1, 0, 0), idx(0, h, o) assert CHO_mult(a, b) == CHO_mult(b, a), f"i_C should commute with ({h},{o})" print(f" i_C commutes with all H, O elements ✓") print() # (2) Verify non-associativity inherited from O print(" (2) Non-associativity check (C tensor H tensor O should inherit from O):") # In O: (e_1 e_2) e_4 != e_1 (e_2 e_4) # In C tensor H tensor O: same when the C and H parts are trivial a = np.zeros(64); a[idx(0, 0, 1)] = 1 # e_1 b = np.zeros(64); b[idx(0, 0, 2)] = 1 # e_2 c = np.zeros(64); c[idx(0, 0, 4)] = 1 # e_4 ab = vec_mult(a, b) ab_c = vec_mult(ab, c) bc = vec_mult(b, c) a_bc = vec_mult(a, bc) print(f" (e_1 e_2) e_4 - e_1 (e_2 e_4) = " f"{int(np.linalg.norm(ab_c - a_bc) > 0)} (non-zero => non-associative)") print(f" => C ⊗ H ⊗ O is non-associative") print() # (3) Verify the algebra has 11 imaginary units squaring to -1 print(" (3) Imaginary-unit structure:") imag_units = [(1, 0, 0)] + [(0, h, 0) for h in [1, 2, 3]] + [(0, 0, o) for o in range(1, 8)] print(f" Total imaginary units: {len(imag_units)}") print(f" Expected: 1 (i_C) + 3 (i,j,k_H) + 7 (e_1..e_7) = 11") bad = 0 for (c, h, o) in imag_units: a = idx(c, h, o) s, k = CHO_mult(a, a) if s != -1 or k != idx(0, 0, 0): bad += 1 print(f" Each squares to -1: {len(imag_units) - bad}/{len(imag_units)}") print() # (4) Cite known structure print("=" * 90) print(" Known algebraic structure (Furey 2014, 2018, 2022)") print("=" * 90) print() print(" As real algebras (with appropriate identifications):") print() print(" C tensor H ~= M_2(C) [biquaternions]") print(" H tensor O (left action) ~= M_8(R) tensor H [as in Furey 2014]") print(" C tensor H tensor O ~= M_8(R) tensor M_2(C)") print(" ~= M_16(C) real dim 512") print() print(" Inside M_64(R) (real algebra of all 64x64 real matrices, dim 4096):") print() print(" L_{C tensor H tensor O} sub-algebra of dim 512") print(" R_{C tensor H tensor O} sub-algebra of dim 512") print(" L cap R = centre of bimodule structure") print() print(" The N_gen = 3 question becomes: does the centralizer of the chiral") print(" primitive idempotent f within L (or R) admit a natural 3-fold") print(" primitive-idempotent decomposition? Computation 54 will answer this") print(" by explicit construction of f and its stabilizer.") print() print(" This script (Comp 53) has built the multiplication table that") print(" Computation 54 will use as the algebraic substrate.") if __name__ == "__main__": main()