TFPT scientific source excerpt Original repository path: experiments/theory-contracts/elliptic-moebius-source-bridge-20261026/source_recursion_20261028/check_native_pauli_f20_bridge.py Original file SHA-256: e1200c58403e5854d91a2ad6d20d4844ff077ea2b92d4248ae8b9b1606c32149 Inclusive original lines: 201-316 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000201 | ]) 000202 | omega_b = sp.Matrix([ 000203 | [0, 1, 0, 0], [-1, 0, 1, 0], 000204 | [0, -1, 0, 1], [0, 0, -1, 0], 000205 | ]) 000206 | omega_plus = ii * omega_a + omega_b 000207 | transported = sp.simplify(intertwiner.inv().T * omega_plus * intertwiner.inv()) 000208 | require(is_zero(transported - (1 - 2*ii) * standard_symplectic / 10), 000209 | "intertwiner transports v419 Omega plus to standard form", checks) 000210 | 000211 | # On chi^(-1) Sym2(V), the ten decomposes as 2V4 + chi_1 + chi_3. 000212 | rep10_character: dict[tuple[int, int], sp.Expr] = {} 000213 | for b, a in product(range(4), range(5)): 000214 | matrix = sp.simplify(c_native**a * g_native**b) 000215 | rep10_character[(a, b)] = sp.simplify(ii**(-b) * sym2_character(matrix)) 000216 | one_dimensional_multiplicities: dict[str, str] = {} 000217 | for k in range(4): 000218 | multiplicity = sp.simplify(sum( 000219 | sp.conjugate(ii**(k*b)) * rep10_character[(a, b)] 000220 | for b, a in product(range(4), range(5))) / 20) 000221 | one_dimensional_multiplicities[f"chi_{k}"] = str(multiplicity) 000222 | four_multiplicity = sp.simplify(sum( 000223 | sp.conjugate(sp.trace(c_native**a * g_native**b)) * 000224 | rep10_character[(a, b)] 000225 | for b, a in product(range(4), range(5))) / 20) 000226 | require(one_dimensional_multiplicities == { 000227 | "chi_0": "0", "chi_1": "1", "chi_2": "0", "chi_3": "1" 000228 | } and four_multiplicity == 2, 000229 | "ten decomposes as two fours plus chi one plus chi three", checks) 000230 | commutant_dimension = int(four_multiplicity**2 + 1 + 1) 000231 | require(commutant_dimension == 6, 000232 | "ten dimensional commutant has complex dimension six", checks) 000233 | 000234 | # The earlier determinant-3i rational ten-intertwiner is algebraic, but it 000235 | # is not already an isometry for the field-trace metric. Its canonical 000236 | # metric polar correction is governed by a positive commutant element. 000237 | old_ten_intertwiner = sp.Matrix([ 000238 | [0,0,0,0,0,0,0,0,1,-1], 000239 | [0,0,0,0,ii,0,-ii,0,0,0], 000240 | [0,0,0,0,0,-ii,0,0,0,0], 000241 | [0,0,0,0,0,-1,1,1,-1,0], 000242 | [0,0,0,-1,0,0,0,0,0,0], 000243 | [0,0,-ii,0,0,0,0,ii,0,0], 000244 | [-ii,0,ii,ii,0,0,0,0,-ii,0], 000245 | [-1,1,0,0,0,0,0,0,0,0], 000246 | [0,ii,0,-ii,0,0,-ii,0,0,ii], 000247 | [0,1,-1,0,-1,1,0,0,0,0], 000248 | ]) 000249 | sym_gram = induced_symmetric_gram(field_trace_gram) 000250 | ten_c = symmetric_square(c_v419) 000251 | ten_g = ii * symmetric_square(g_v419) 000252 | require(is_zero(ten_c.conjugate().T * sym_gram * ten_c - sym_gram) and 000253 | is_zero(ten_g.conjugate().T * sym_gram * ten_g - sym_gram), 000254 | "correct induced field trace metric is invariant on twisted ten", checks) 000255 | require(not is_zero(old_ten_intertwiner.conjugate().T * old_ten_intertwiner - sym_gram), 000256 | "old determinant 3i ten intertwiner is not metric preserving", checks) 000257 | polar_square = sp.simplify(sym_gram.inv() * 000258 | old_ten_intertwiner.conjugate().T * old_ten_intertwiner) 000259 | require(is_zero(polar_square * ten_c - ten_c * polar_square) and 000260 | is_zero(polar_square * ten_g - ten_g * polar_square), 000261 | "ten polar square lies in the F20 commutant", checks) 000262 | 000263 | output = { 000264 | "verdict": "EXACT_NATIVE_SOURCE_F20_AND_METRIC_BRIDGE", 000265 | "source_words_rightmost_first": { 000266 | "C_raw": "(I tensor H) CNOT (S tensor I) (H tensor I)", 000267 | "C": "-i C_raw", 000268 | "G_raw": "CNOT (I tensor S) (S tensor I)", 000269 | "G": "(Y tensor Z) G_raw", 000270 | }, 000271 | "exact_G_lifts": exact_lifts, 000272 | "C_native": encode(c_native), 000273 | "G_native": encode(g_native), 000274 | "v419_Gram": encode(field_trace_gram), 000275 | "intertwiner_T": encode(intertwiner), 000276 | "intertwiner_determinant": str(intertwiner.det()), 000277 | "metric_identity": "T^dagger T = 2 (5 I - J)", 000278 | "relative_unitary": "U=T/sqrt(2), U^dagger U=5 I-J", 000279 | "intertwiner_space_dimension": 1, 000280 | "characters": character_rows, 000281 | "conformal_multiplier_G": "i", 000282 | "spin_lift": "zeta8^(-1) G, order 8", 000283 | "transported_alternating_form": "(1-2i)/10 times J_standard", 000284 | "ten_decomposition": "2 V4 + chi_1 + chi_3", 000285 | "ten_multiplicities": { 000286 | **one_dimensional_multiplicities, 000287 | "V4": str(four_multiplicity), 000288 | }, 000289 | "ten_commutant": "M2(C) plus C plus C", 000290 | "ten_commutant_dimension": commutant_dimension, 000291 | "old_ten_intertwiner": "determinant 3i, not metric preserving", 000292 | "ten_polar_repair": ( 000293 | "P^2=H_sym^(-1) A^dagger A; U=A P^(-1), with P the positive " 000294 | "H_sym square root" 000295 | ), 000296 | "unitary_ten_residual_gauge": "U(2) times U(1) times U(1)", 000297 | "checks": checks, 000298 | "scope": ( 000299 | "The F20 is constructed from the frozen v783 native Clifford " 000300 | "generators and source-proved Pauli/mu8 kernel. v783 does not " 000301 | "designate this Singer normalizer as the v419 Galois clock. The " 000302 | "result is a carrier representation bridge, not a physical " 000303 | "four-dimensional field or channel identification." 000304 | ), 000305 | } 000306 | out = Path(__file__).with_name("native-pauli-f20-results.json") 000307 | out.write_text(json.dumps(output, indent=2, sort_keys=True) + "\n") 000308 | print(json.dumps({ 000309 | "verdict": output["verdict"], 000310 | "checks": len(checks), 000311 | "ten_commutant_dimension": commutant_dimension, 000312 | }, sort_keys=True)) 000313 | 000314 | 000315 | if __name__ == "__main__": 000316 | main()