TFPT scientific source excerpt Original repository path: experiments/theory-contracts/seam-collar-multifield-20261016/e8_boundary_response.py Original file SHA-256: 8a77bef68666dd18470fce78e04881676862075d216587b291c9e2c3761d5e6b Inclusive original lines: 161-294 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. 000161 | 000162 | 000163 | def polynomial_product(a, b, order): 000164 | return [sum(a[j] * b[k-j] for j in range(k+1)) for k in range(order+1)] 000165 | 000166 | 000167 | def ns_character(order=4): 000168 | """Full 16-Majorana NS coefficients in x=q^(1/2), vacuum factor removed.""" 000169 | p = [1] + [0] * order 000170 | for odd in range(1, order+1, 2): 000171 | factor = [int(sp.binomial(16, k//odd)) 000172 | if k % odd == 0 and k//odd <= 16 else 0 for k in range(order+1)] 000173 | p = polynomial_product(p, factor, order) 000174 | return p 000175 | 000176 | 000177 | def run(): 000178 | rr = roots() 000179 | assert len(set(rr)) == 240 and abs(BASIS.det()) == 1 000180 | assert GRAM.det() == 1 and all(GRAM[i, i] == 2 for i in range(8)) 000181 | for a in rr: 000182 | coordinates(a) 000183 | assert dot(a, a) == 2 000184 | counts = Counter(grade(a) for a in rr) 000185 | assert [counts[i] for i in range(4)] == [52, 64, 60, 64] 000186 | branches = Counter((dot(a[:5], a[:5])/2, dot(a[5:], a[5:])/2, grade(a)) for a in rr) 000187 | assert branches == {(sp.S.One, sp.S.Zero, 0): 40, 000188 | (sp.S.Zero, sp.S.One, 0): 12, 000189 | (HALF, HALF, 2): 60, 000190 | (sp.Rational(5, 8), sp.Rational(3, 8), 1): 64, 000191 | (sp.Rational(5, 8), sp.Rational(3, 8), 3): 64} 000192 | # All root pairs: actual cocycle commutator exactly cancels the integral 000193 | # coordinate exchange sign. No identification with fermion parity is used. 000194 | locality_pairs = 0 000195 | for a, b in product(rr, repeat=2): 000196 | assert epsilon(a, b) == (-1 if int(dot(a, b)) % 2 else 1) * epsilon(b, a) 000197 | assert grade(add(a, b)) == (grade(a)+grade(b)) % 4 000198 | locality_pairs += 1 000199 | conjugate_pairs = 0 000200 | for a in rr: 000201 | assert adjoint_sign(a) * vertex_response((neg(a), a), (2, 1)) == 1 000202 | conjugate_pairs += 1 000203 | a = (HALF,) * 8 000204 | b = (sp.Integer(-1), sp.Integer(-1)) + (sp.S.Zero,) * 6 000205 | c = neg(add(a, b)) 000206 | assert dot(a, b) == -1 and c in rr 000207 | triple = vertex_response((a, b, c), (3, 2, 1)) 000208 | quartic = vertex_response((a, neg(a), b, neg(b)), (4, 3, 2, 1)) 000209 | connected = connected_four((a, neg(a), b, neg(b)), (4, 3, 2, 1)) 000210 | assert triple != 0 and connected != 0 000211 | assert sum(grade(x) for x in (a, b, c)) % 4 == 0 000212 | assert vertex_response((a, b), (2, 1)) == 0 000213 | cartan_root_response = adjoint_sign(a) * current_response( 000214 | (a, neg(a)), (3, 1), ((a, 4),)) 000215 | assert cartan_root_response == sp.Rational(1, 3) 000216 | e0 = (sp.S.One,) + (sp.S.Zero,) * 7 000217 | assert current_response(cartan=((e0, 2), (e0, 1))) == 1 000218 | assert current_response(cartan=((e0, 4), (e0, 3), (e0, 2), (e0, 1))) == sp.Rational(169, 144) 000219 | free_ns = ns_character() 000220 | assert free_ns == [1, 16, 120, 576, 2076] 000221 | dictionary = [] 000222 | for root in rr: 000223 | g = grade(root) 000224 | dictionary.append({ 000225 | "charge": [str(x) for x in root], "quarter_grade": g, 000226 | "quarter_character": ["1", "i", "-1", "-i"][g], 000227 | "weight_D5": str(dot(root[:5], root[:5])/2), 000228 | "weight_A3": str(dot(root[5:], root[5:])/2), 000229 | "weight_total": "1", "boundary_statistics": "bosonic_current", 000230 | "adjoint_sign": adjoint_sign(root), 000231 | }) 000232 | return { 000233 | "scope": "Stipulated E8 level-one vacuum lattice representation; separated-point boundary currents", 000234 | "selection_from_raw_P1P2": False, 000235 | "reverse_bootstrap": { 000236 | "source": "origin_theory.tex:954-981; v6, v15, v350-v352", 000237 | "retained": "Discrete E8 closure fixes g_car=5 and the eight normalization witnesses within its framework", 000238 | "not_inferred": "No automatic raw-kernel field pullback, 4D statistics, field-dependent kinetic operator or source state/time", 000239 | }, 000240 | "quarter": { 000241 | "character_vector": [str(x) for x in V], 000242 | "root_grades": [counts[i] for i in range(4)], 000243 | "current_grades_including_Cartan": [60, 64, 60, 64], 000244 | "internal_character_not_physical_time": True, 000245 | "original_charged_quarter_control": "charged-pair-control.json, kernel-functional-control.json (unchanged)", 000246 | }, 000247 | "dictionary": dictionary, 000248 | "boundary_current_blocks": [ 000249 | {"Spin10_SU4_labels": "(45,1)+(1,15)", "dimension": 60, "grade": 0, 000250 | "meaning": "40 D5 roots +12 D3 roots +8 Cartan currents"}, 000251 | {"Spin10_SU4_labels": "(10,6)", "dimension": 60, "grade": 2, 000252 | "weights": ["1/2", "1/2"]}, 000253 | {"Spin10_SU4_labels": "(16,4) and conjugate, label orientation conventional", 000254 | "dimension_each": 64, "grades": [1, 3], "weights": ["5/8", "3/8"]}, 000255 | ], 000256 | "Cartan_currents": {"count": 8, "weight": "1", "grade": 0, 000257 | "definition": "H^a(z)=i partial X^a(z)", 000258 | "OPE": "H^a(z)H^b(w) ~ delta_ab/(z-w)^2"}, 000259 | "vertex_definition": "V_alpha(z)=e_alpha :exp(i alpha.X(z)):; declared ordered lattice cocycle", 000260 | "response_formula": "delta(sum alpha_i=0) product_(i; current_response computes mixed coefficients only away from diagonals", 000263 | "checks": {"locality_root_pairs": locality_pairs, "positive_adjoint_two_point_pairs": conjugate_pairs, 000264 | "known_vacuum_NS_character_x": free_ns, 000265 | "E8_character_q_first": [1, 248, 4124], 000266 | "E8_character_scope": "Known original v156 control; no new equality of full NS and E8 local fields", 000267 | "positivity_scope": "Root-state norms in the stipulated positive E8 lattice Fock representation, not a raw P1 reflection-positivity proof"}, 000268 | "nonabelian_probe": { 000269 | "charges": [[str(y) for y in x] for x in (a, b, c)], 000270 | "grades": [grade(x) for x in (a, b, c)], 000271 | "three_point_at_3_2_1": str(triple), 000272 | "four_point_a_minus_a_b_minus_b_at_4_3_2_1": str(quartic), 000273 | "connected_four_point": str(connected), 000274 | "adjoint_normalized_H_a_V_a_V_minus_a_at_4_3_1": str(cartan_root_response), 000275 | "meaning": "Current responses are non-Gaussian even in the lattice/free-boson realization; no native Yukawa or independent boson-pair transfer is inferred", 000276 | }, 000277 | "requires_for_physical_P1": [ 000278 | "An original field insertion map into this representation, with local domains, charge, statistics and physical time", 000279 | "The actual map from physical field values to D_rel[Phi], D_ref[Phi] and the original spectral profile", 000280 | "Original absolute amplitude, determinant-line phase and contact/geometry prescription", 000281 | ], 000282 | "new_physics_claim": False, 000283 | } 000284 | 000285 | 000286 | if __name__ == "__main__": 000287 | parser = argparse.ArgumentParser(description=__doc__) 000288 | parser.add_argument("--output", type=Path) 000289 | args = parser.parse_args() 000290 | result = run() 000291 | if args.output: 000292 | args.output.parent.mkdir(parents=True, exist_ok=True) 000293 | args.output.write_text(json.dumps(result, indent=2, ensure_ascii=False)+"\n") 000294 | print(json.dumps({k: result[k] for k in ("scope", "quarter", "checks", "nonabelian_probe")}, ensure_ascii=False))