TFPT scientific source excerpt Original repository path: experiments/theory-contracts/original-admissible-source-closure-20261025/check_w8_source_twopoint.py Original file SHA-256: b70b38c25833490a15c2847a3612fa31c08e855f147f66418107976860c92851 Inclusive original lines: 216-356 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. 000216 | 000217 | 000218 | def reconstruct_pole_coefficients(level_norms): 000219 | """Solve a_l=sum_(r<=l) binom(l+r-1,2r-1) A_r.""" 000220 | coefficients = [level_norms[0]] 000221 | for level in range(1, MAX_LEVEL + 1): 000222 | remainder = level_norms[level] 000223 | for r in range(1, level): 000224 | remainder -= comb(level + r - 1, 2 * r - 1) * coefficients[r] 000225 | coefficients.append(remainder) 000226 | return coefficients 000227 | 000228 | 000229 | def standard_root_action_check(fold, d_quarters): 000230 | reversal = s.zeros(8) 000231 | for j in range(8): 000232 | reversal[7 - j, j] = 1 000233 | action = s.simplify(-fold * reversal * fold.inv()) 000234 | roots = set(standard_roots()) 000235 | need(s.simplify(action.T * action) == s.eye(8), "quarter transport is not orthogonal") 000236 | need({tuple(action * s.Matrix(root)) for root in roots} == roots, 000237 | "quarter transport does not preserve the actual E8 roots") 000238 | need(s.simplify(action * s.Matrix(d_quarters["1/4"]) - s.Matrix(d_quarters["3/4"])) == s.zeros(8, 1), 000239 | "quarter transport does not map the two actual source charges") 000240 | return action 000241 | 000242 | 000243 | def main(): 000244 | h, variables, state, completion = build_state() 000245 | old = load_old_source() 000246 | lattice, orthogonal, _positive, fold = old.original_transport() 000247 | 000248 | # The W8 polynomial above uses canonical standard E8 coordinates. The 000249 | # old root chart is O*(old physical root), while the source charge is M*delta. 000250 | roots = set(standard_roots()) 000251 | chart_roots = { 000252 | tuple(s.simplify(value) for value in orthogonal * (s.Matrix(root) / s.sqrt(2))) 000253 | for root in lattice["roots"] 000254 | } 000255 | need(chart_roots == roots, "old O chart does not reach the canonical root set") 000256 | 000257 | d_quarters = {} 000258 | preliminary = {} 000259 | for alpha in (s.Rational(1, 4), s.Rational(3, 4)): 000260 | label = str(alpha) 000261 | delta = s.Matrix([(j + alpha) / 8 - s.Rational(1, 2) for j in range(8)]) 000262 | d = tuple(s.Rational(value) for value in fold * delta) 000263 | d_quarters[label] = d 000264 | onepoint, vector, norm, ward = level_one_test(h, state, d) 000265 | expected_onepoint = s.Rational( 000266 | completion["original_quarter_source_cases"][label]["W8_onepoint_times_z8"] 000267 | ) 000268 | need(onepoint == expected_onepoint, "old W8 one-point coefficient changed") 000269 | preliminary[label] = { 000270 | "d_standard": [str(value) for value in d], 000271 | "d_squared": str(s.factor(sum(value * value for value in d))), 000272 | "onepoint": str(onepoint), 000273 | "level_1_vector": [str(value) for value in vector], 000274 | "d_dot_level_1": str(ward), 000275 | "level_1_norm_squared": str(norm), 000276 | "cauchy_lower_bound": str(s.factor(64 * onepoint**2 / sum(value * value for value in d))), 000277 | "level_1_excess_over_cauchy_bound": str(s.factor( 000278 | norm - 64 * onepoint**2 / sum(value * value for value in d) 000279 | )), 000280 | } 000281 | 000282 | action = standard_root_action_check(fold, d_quarters) 000283 | level_one_left = s.Matrix([s.Rational(value) for value in preliminary["1/4"]["level_1_vector"]]) 000284 | level_one_right = s.Matrix([s.Rational(value) for value in preliminary["3/4"]["level_1_vector"]]) 000285 | need(s.simplify(action * level_one_left - level_one_right) == s.zeros(8, 1), 000286 | "level-one vectors do not obey the actual quarter Weyl transport") 000287 | 000288 | cases = {} 000289 | for label, d in d_quarters.items(): 000290 | levels = shifted_levels(state, d) 000291 | level_norms = [fock_norm(level) for level in levels] 000292 | pole_coefficients = reconstruct_pole_coefficients(level_norms) 000293 | need(level_norms[0] == as_fraction(s.Rational(preliminary[label]["onepoint"]))**2, 000294 | "level-zero norm is not the one-point square") 000295 | need(level_norms[1] == as_fraction(s.Rational(preliminary[label]["level_1_norm_squared"])), 000296 | "sparse level-one result differs from the Ward precheck") 000297 | need(pole_coefficients[8] == as_fraction(s.Rational(completion["norm_squared"])), 000298 | "leading collision coefficient is not the W8 vacuum norm") 000299 | # Independent stress-tensor Ward check. For weight8 and c=8, 000300 | # the T coefficient in W8 W8 is (2*8/8)*||W8||^2=2||W8||^2. 000301 | # With =h_d/z^2 this fixes A7=2*h_d*||W8||^2. 000302 | h_d = sum(value*value for value in d)/2 000303 | stress_expected = as_fraction(2*h_d*s.Rational(completion["norm_squared"])) 000304 | need(pole_coefficients[7] == stress_expected, 000305 | "next collision coefficient fails the independent stress Ward identity") 000306 | need(all(value >= 0 for value in level_norms), "a shifted Fock norm is negative") 000307 | cases[label] = { 000308 | **preliminary[label], 000309 | "level_support_sizes": [len(level) for level in levels], 000310 | "level_norms": [rational_text(value) for value in level_norms], 000311 | "pole_basis_coefficients": [rational_text(value) for value in pole_coefficients], 000312 | "independent_stress_Ward_A7": rational_text(stress_expected), 000313 | } 000314 | 000315 | need(cases["1/4"]["level_norms"] == cases["3/4"]["level_norms"], 000316 | "Weyl-related quarters have different level norms") 000317 | need(cases["1/4"]["pole_basis_coefficients"] == cases["3/4"]["pole_basis_coefficients"], 000318 | "Weyl-related quarters have different separated two-point functions") 000319 | 000320 | output = { 000321 | "claim": "MATH.ORIGINAL-SEAM.W8-SOURCE-TWOPOINT.01", 000322 | "status": "PASS_EXACT_IN_EXISTING_SHIFTED_CURRENT_SOURCE", 000323 | "radial_variable": "q_rad=w/z=exp(-tau), |q_rad|<1", 000324 | "radial_generator": "N=Lhat_0-85/256=(Lcal_0^alpha-1/32)/8=K_alpha on the folded neutral sector", 000325 | "original_grading": "H0=Lcal_0^alpha-1/32=8*K_alpha=8*N; h_j(-r) has N-level r and original H0-grade 8r", 000326 | "time_scope": "tau is quotient radial-cylinder separation; this algebraic conversion does not supply an absolute physical-time calibration", 000327 | "pole_variable": "X=q_rad/(1-q_rad)^2", 000328 | "formula": "z^8 w^8 _d=sum_(r=0)^8 A_r X^r", 000329 | "connected_formula": "subtract A_0=_d^2", 000330 | "reconstruction": "a_l=||v_l||^2=sum_(r=1)^l binom(l+r-1,2r-1)A_r for l>0", 000331 | "completeness_reason": "bosonic exchange gives F(q)=F(1/q); after z^8w^8 the only finite pole is q=1 of order at most 16, so F is a degree-at-most-8 polynomial in X", 000332 | "root_chart": "old O maps all 240 old physical roots to the canonical standard E8 roots", 000333 | "quarter_relation": { 000334 | "description": "d_3/4=g d_1/4 with g=-M R M^-1 preserving the actual 240-root set", 000335 | "matrix": [[str(action[i, j]) for j in range(8)] for i in range(8)], 000336 | }, 000337 | "vacuum_control": { 000338 | "d": "0", 000339 | "a_0": "0", 000340 | "a_1": "0", 000341 | "first_possible_level": 8, 000342 | "level_8_norm_squared": completion["norm_squared"], 000343 | }, 000344 | "cases": cases, 000345 | "not_claimed": [ 000346 | "a new state, coupling, direction, or source model", 000347 | "an identification of the two complete CAR modules", 000348 | "a physical measurement or preparation", 000349 | "a full P1 source derivation", 000350 | ], 000351 | } 000352 | print(json.dumps(output, indent=2) + "\n", end="") 000353 | 000354 | 000355 | if __name__ == "__main__": 000356 | main()